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Also see bottom page of this website titled "Misc. 3, Sacred mathematics, geometry, music, harmonics, cosmology, cycles, fractals, chaos".
most schools teach classical geometry -- the study of simple shapes like circles or squares -- not fractal geometry, Eglash said.

"fractal geometry can take us into the far reaches of high tech science, its patterns are surprisingly common in traditional African designs, and some of its basic concepts are fundamental to African knowledge systems."
...although fractal designs do occur outside of Africa (Celtic knots, Ukrainian eggs, and Maori
raftors have some excellent examples), they are not everywhere. Their strong prevalence in Africa
(and in African-influenced southern India) is quite specific. - Page 7 of African Fractals Modern
Computing and Indigenous Design (1999) by Ron Eglash.
African impact on India is found in artefacts, ancient texts, genetic foot-prints, physical
resemblance amongst populations, cultural and linguistic similarities, gastronomic affinities,
and a common world view
. It is generally agreed today by scientists and historians that an early
migration of African population first settled the coastal areas of south India, then spread
gradually inland ...
3. I am the living mathematics;
Rensselaer Professor Ron Eglash. Eglash, a professor in our Department of Science and Technology Studies, has made fractals a keystone in his efforts to show minority students the cultural relevance of the STEM (science, technology, engineering and mathematics) fields.
"African fractals are not just the result of spontaneous intuition; in some cases they are
created under rule-bound techiniques equivalent to western mathematics." - pages 68-69 of
African Fractals by Ron Eglash.
Mandelbrot coined the term "fractals" for this new geometry, and it is now used in every
scientific discipline from astrophysics to zoology
. - Page 15 of African Fractals by Ron
Fractals are characterized by the repetition of similar patterns at ever-diminishing scales. Fractal geometry has emerged as one of the most exciting frontiers on the border between mathematics and information technology and can be seen in many of the swirling patterns produced by computer graphics. It has become a new tool for modeling in biology, geology, and other natural sciences.
Anthropologists have observed that the patterns produced in different cultures can be characterized by specific design themes. In Europe and America, we often see cities laid out in a grid pattern of straight streets and right-angle corners. In contrast, traditional African settlements tend to use fractal structures-circles of circles of circular dwellings, rectangular walls enclosing ever-smaller rectangles, and streets in which broad avenues branch down to tiny footpaths with striking geometric repetition. These indigenous fractals are not limited to architecture; their recursive patterns echo throughout many disparate African designs and knowledge systems.
Drawing on interviews with African designers, artists, and scientists, Ron Eglash investigates fractals in African architecture, traditional hairstyling, textiles, sculpture, painting, carving, metalwork, religion, games, practical craft, quantitative techniques, and symbolic systems. He also examines the political and social implications of the existence of African fractal geometry. His book makes a unique contribution to the study of mathematics, African culture, anthropology, and computer simulations. - From the book African Fractals by Ron Eglash.
The fractal settlement patterns of Africa stand in sharp contrast to the Cartesian grids of Euro-American settlements. - Page 39 of African Fractals.
Comparing the Mayan snake pattern with an African weaving based on the cobra skin pattern (fig. 3.3b), we see
how geometric modeling of similar natural phenomena in these two cultures results in very different representations.
The Native American example emphasizes the Euclidean symmetry within one size frame
('size frame" because the term "scale" is confusing in the context of snake skin). This Mayan pattern is composed
of four shapes of the same size, a four fold symmetry
. But the African example emphasizes fractal
, which is not about right/left or up/down, but rather similarity between size frames. The
African snake pattern shows diamonds within diamonds within diamonds. - Page 43 of African Fractals.
...but with the impressive exception of the Pacific Northwest carvings, fractals are almost entirely absent from Native American designs. - Page 45 of African Fractals.
Below: Kwakiutl Native Americans and design. 
"Finally, there are three Native American designsthat are both indigenous and fractal. The best case is the abstract figurative art of the Haida, Kwakiutl, Tlingit, and others in the Pacific Northwest (Holm 1965). These figures, primarily carvings, have the kind of global, nonlinear self-similarity necessary to qualify as fractals and clearly exhibit recursive scaling of up to three or four iterations." - Page 43 of African Fractals.
Above: Tlingit Native Americans and design.
Ascher (1991) has analyzed some of the algorithmic properties of Warlpiri (Pacific Islander) sand drawings. Similar structures are also found in Africa where they are called Lusona. But while the Lusona tend to use similar patterns at different scales, the Warlpiri drawings tend to use different patterns at different scales. Ascher concludes that the Warlpiri method of combining different graph movements is analogous to algebraic combinations, but the African Lusona are best described as fractals. - Page 47 of African Fractals.
Moses was involved in the freedom of his people.
In Kosmon all people need freedom:
"Free Your Mind and Your Ass Will Follow" - 1970 by George Clinton.
What does "free your mind and your ass will follow" mean?
"It means change your state of thought and your actions will change as well" - Mar 22, 2017 by
GrailGuardian Space Age Hustle.

Fractal Spirituality-The Infinite Within Our Souls.
How can a Mystery as large as the Universe find expression within the smallness of our souls? How can we tiny beings experience the Infinite? I found a new way to think about this question when I learned about fractal geometry. Fractals are never ending patterns, with self-similarity at all sizes.
Love this. I did my undergraduate work in mathematics – which I fell in love with because of the fractals. The beautiful mystery of patterns repeating themselves at scale and in unrelated contexts was a hint at making meaning of the complexity of the universe.

"We will see not only in Architecture, but in traditional hairstyling, textiles, sculpture, painting, carving, metalwork, in religion, games and practical craft, in quantitative techniques and symbolic systems, Africans have used the patterns and abstract concepts of Fractal geometry." - page 7 of African fractals.
"the first time I submitted a journal article on African fractals, one reviewer replied that Africans could not have "true" fractal geometry because they lacked the Advanced mathematical concept of infinity ...we have already seen another example of an infinity icon in the Nankani architecture discussed in chapter 2. There the coils of a serpent of infinite length, sculpted into the house walls, make use of the same association between prosperity without end, and a geometry length without end. ..And unlike the naturally occurring shells, the packing of this infinite length into a finite space (the Nankani describe it as "coiling back on itself indefinitely") cannot be mistaken for mere mimicry of nature; it is rather the artifice of fractals." - from Ron Eglash pages 147-149 of African Fractals. 
Among the Nankani people in Northern Ghana ...
"...an African working with a system of mythological symbols is performing the same cognitive operations as a European working with a system of computer code symbols." - Claude Levi-Strauss, page 188 of African Fractals by Ron Eglash.
Claude Lévi-Strauss was a French anthropologist and ethnologist whose work was key in the development of the theory of structuralism and structural anthropology.
Often known as “the “father of modern anthropology”, he revolutionized the world of social anthropology by implementing the methods of structuralist analysis developed by Saussuro in the field of cultural relations.
cognitive operations, such as memory, reasoning and planning.
1/3 to 3x scaling used in African fractal architecture are harmonics of 3.
Pages 4 and 8 of THE UNIFIED CYCLE THEORY (How Cycles Dominate the Structure of the Universe and Influence Life on Earth) by Stephen J. Puetz, 2009 says "Cycles exhibit harmonics of three. Take the period of cycles, multiply by three, and the next larger cycle results [33 x 3 = 99-100]. Conversely, take cycle...divide it's frequency by three, and the next smaller cycle appears [33 / 3 = 11].
Oahspe Book of Cosmology and Prophecy: Chapter IX:
2. These fall under the divisor, or multiple, 3 ...corresponding to the times, 11, 33, 99, and so on.

He was a brilliant scholar, a man of profound wisdom, an adept at occultism, and a bound
devotee of the false Osiris...he must build a temple to Osiris, ...But first Thothma was required
to drive the Faithists out of the land, and to make slaves of those who remained. Their numbers
now amounted to three millions in the land of Egypt. Thothma therefore levied a powerful army,
driving out the Listians or Shepherd Kings with great slaughter, and enslaving the children of
Abraham, the Israelites...
Above: faces of original Hebrew Israelites tribe.
Above: Ancient Hebrew Israelites in Egyptian Captivity (slavery).
they were oppressed by cruel laws and penalties, and were forced to reveal the mathematical
science which had been preserved with them from their distant ancestors the I'hins, to whom
it was committed by the angels in the first ages of mankind. Humanity was now ripe for this
development, but the surrender of their secrets was a further blow to the Israelites.
Both the knowledge and the slave labour of the Israelites were now thus impressed into the
 construction of the temple of Osiris, the Great Pyramid.
And the etherean dawn in which
Moses would bring them deliverance was still five hundred years away.
[2053 - 500 = 1553 B.C.E.]. - Pages 189-190 of Darkness, Dawn And Destiny
(Drawn from Oahspe) 1965 by Augustine Cahill.
Oahspe Book of Wars Against Jehovih Chapter XLIX:
2. ...Thothma, made the following laws, to wit:
4. ...And of thy arts, of measuring and working numbers, thou shalt not keep them secret longer, or thy blood be upon thee.
"measuring and working numbers" = mathematics.
Definition of mathematics for Students. : the science that studies and explains numbers, quantities, measurements ...
Algebra, arithmetic, calculus, geometry, and trigonometry are branches of mathematics.
Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles.

Thothma did not have the mathematical knowledge of the Israelites until he forced the Israelites to REVEAL it to him.
This mathematical knowledge was lost to the Egyptian slave masters after the death of Thothma.
The true Israelites kept this sacred mathematical knowledge up to modern times.
Above: Original-Israelites-Abraham-86% and 90% facial match.Above: Original-Israelites-Moses-93% and 92% facial match.
Concerning the Great Pyramid of Khufu, the theory proposed in my essay turns
out to make 22/28 a very logical choice as the inverse-slope for the slant-angle
of the faces
. Now I will discuss the pyramid exercises from the Rhind Papyrus.
This papyrus as well as the other extant mathematical papyri were written
hundreds of years after the 4-th dynasty. One can ask how accurately they
represent the mathematical knowledge of the architect who built the Great
Pyramid. It is obvious that the architects undertaking the building of a
pyramid would need a good mathematical knowledge of the geometry associated
with such a structure, and one can indeed find this in these later papyri.
Two of the five pyramid exercises from the Rhind papyrus can be found here.
As these exercises show, the seked is represented as a certain number of palms
and fingers. This is really the horizontal change in the distance for each change
of one cubit in the vertical distance. The exercises make it clear that one cubit
is equal to 7 palms and that one palm is equal to 4 fingers. Therefore, one cubit
is equal to 28 fingers. (Not so different from the English measurement system in
which one yard is 3 feet and one foot is 12 inches.) Thus, for example, a seked
of five palms, two fingers would correspond to an inverse-slope of
(22 fingers)/(28 fingers), or 22/28. As I mentioned above, the Great Pyramid
indeed has this seked, and with a high degree of accuracy.
...the famous relationship between &pi and the Great Pyramid of Khufu ...has
its roots in two facts - one purely mathematical and the other historical, but
both involving the number 7:

1.  The rational number 22/7 happens to be an excellent approximation to the number pi.

2.  The Egyptian measurement system involves dividing one unit of measurement(the cubit) into 7 equal units (palms)

Concerning the first fact, it is actually somewhat remarkable that an irrational
number such as &pi can be approximated so well by a rational number with a small
denominator. (The denominator is 7 in this case.).

There is also a frequently mentioned relationship between the Great Pyramid and
the number φ, ..The legend that the architect who designed the Great Pyramid of
Khufu intentionally incorporated the Golden Mean (which is this number φ) into
the proportions of that structure ...


...there is a specific proportional ratio that is found throughout nature. This ratio is called Phi ....
This ratio is nature's most ubiquitous fractal scaling ratio and is easy to see present in plants,
animals, seashells, vortices of water and air, and many other phenomena of Earth. It is also
present at both the atomic and galactic scales.


Cantor set and fractals

I have read that the Cantor set is considered a fractal. I am referring to the Cantor set
in which the middle third of a real line is removed recursively...think of a fractal as
some geometrical object with scale invariance such that if you "zoom in" on the object it
will look the same/similar...a classic fractal such as the Koch Snowflake or the Sierpinski
triangle. I do see that Cantor's set would look the same if you kept "zooming in".
 If you restrict your sight to [0,13] then the picture of Cantor's set is exatly the same as it
is in the whole [0,1]. Again, restrict to [0,1/9] and you get the same picture. I think this is
the main property of a fractal: a picture which repeats itself.

Remove the (open) middle third of it, i.e. get (1/3, 2/3). Now remove the middle thirds of each of
the remaining intervals,i.e. get (1/9, 2/9) and (7/9, 8/9). Continue this process ad infinitum.
The points left over form a fractal called the standard Cantor Set.
 Figure 1  


Cantor Set -- Math Fun Facts - HMC Math - Harvey Mudd

 Check out C'vorkum light-years numbers divided into 1/3 and 1/9 (Cantor Set fractal numbers) symmetrical number parts.


"The logarithmic spiral and the Golden Ratio go hand in hand." - page 118 of THE GOLDEN RATIO The Story of Phi by Mario Livio, P.h.D astrophysics.

Logarithmic spiral - Wikipedia

Spirals in Nature - Fractal Foundation Online Course - Chapter 1 ...


Logarithmic spirals are fractals showing repetitive process, self-similarity, scaling, and infinity

Logarithmic Spirals
Several Ghanian iconic figures, ...link a spiritual force with the structure of living systems
through logarithmic spirals
. - page 78 of African Fractals by Ron Eglash.
Above images show fractal design in Oahspe and fractal spirals in computer graphics.Above: Spider web geometric fractal designs. Below: fractal architecture in Oahspe:

OAHSPE: Book of Fragapatti CHAPTER IX:
3. Go build me an avalanza capable of carrying three thousand million angels, with as many rooms,
capable of descent and ascent, and east and west and north and south motion, and prepare it with
a magnet, that it may face to the north, whilst traveling.
4. The builders saluted, and then withdrew, and went and built the vessel. And it was two hundred
thousand paces east and west, and the same north and south; its height was one thousand lengths,
and the vesture around it was a thousand paces thick; ...The floor was woven in copy of a
spider's net, extending from the centre outward, and with circular bars at crosses
; ...
Relative to science, fractals are essentially geometric shapes or forms that are represented in natural
objects, from a fern leaf or tree, to a spider web ...

African Fractals: Modern Computing and Indigenous Design.
Fractals are characterized by the repetition of similar patterns at ever-diminishing scales.
Fractal geometry has emerged as one of the most exciting frontiers on the border between
mathematics and information technology and can be seen in many of the swirling patterns produced
by computer graphics. It has become a new tool for modeling in biology, geology, and other
natural sciences
Anthropologists have observed that the patterns produced in different cultures can be
characterized by specific design themes. In Europe and America, we often see cities laid out in
a grid pattern of straight streets and right-angle corners. In contrast, traditional African
settlements tend to use fractal structures
-circles of circles of circular dwellings, rectangular
walls enclosing ever-smaller rectangles, and streets in which broad avenues branch down to tiny
footpaths with striking geometric repetition. These indigenous fractals are not limited to
architecture; their recursive patterns echo throughout many disparate African designs and
knowledge systems

Drawing on interviews with African designers, artists, and scientists, Ron Eglash investigates
fractals in African architecture, traditional hairstyling, textiles, sculpture, painting,
carving, metalwork, religion, games, practical craft, quantitative techniques, and symbolic
. He also examines the political and social implications of the existence of African
fractal geometry. His book makes a unique contribution to the study of mathematics, African
culture, anthropology, and computer simulations
I buy it because I like math and geometry. I'm really fascinating by fractals applied at design
and architecture. - Daniele De Rosa
I have used this book several semesters for teaching philosophy of science, social science
methods, and Southern African political economy. It quickly demonstrates that the colonizers
understood little or nothing about 'messy, irregular' African villages;
it was Euclidean
geometry which kept them from seeing
. African engineering using fractals, such as the
fractal-measured fence weave to match the wind, is amazing. We still have very much to learn
from African peoples and this book gets Americans started on a journey long past due
. Read this
book if you want a different way of viewing the world, from hairstyles to sculpture to urban
planning. - Carol Thompson
The book makes no assumptions in knowledge and will cleanly bring in the topic of fractals in
african culture.
The concept is quite intriguing and shatters many of the held perceptions of "the hierarchy of
Ron Eglash is a great man and I know he loves talking with people that share
similar interests in mathematics or cybernetics. - Lorne E. Nix


STS = Science, technology and society, or science and technology studies, a field of social science and the humanities.
Fractals are used to model soil erosion and to analyze seismic patterns as well. Seeing that so many facets of
mother nature exhibit fractal properties, maybe the whole world around us is a fractal after all! Actually, the most
useful use of fractals in computer science is the fractal image compression.
Why is fractal geometry important?
Fractals help us study and understand important scientific concepts, such as the way bacteria grow, patterns in
freezing water (snowflakes) and brain waves, for example. Their formulas have made possible many scientific
breakthroughs. ... Anything with a rhythm or pattern has a chance of being very fractal-like.Oct 13, 2011

...As computers get smaller and faster, they generally produce more heat, which needs to be dissipated or else the computers will overheat and break. The smaller they are, the more this becomes a problem. Engineers at Oregon State University have developed fractal pattern that can be etched into a silicon chip to allow a cooling fluid (such as liquid nitrogen) to uniformly flow across the surface of the chip and keep it cool.

The fractal pattern above derived from our blood vessels provides a simple low-pressure network to accomplish this task easily. http://fractalfoundation.org/OFC/OFC-12-2.html

... the most useful use of fractals in computer science is the fractal image compression. This kind of compression uses the fact that the real world is well described by fractal geometry. By this way, images are compressed much more than by usual ways (eg: JPEG or GIF file formats). An other advantage of fractal compression is that when the picture is enlarged, there is no pixelisation. The picture seems very often better when its size is increased. http://kluge.in-chemnitz.de/documents/fractal/node2.html

Modern medicine often involves examining systems in the body to determine if something is malfunctioning. Since the body is full of fractals, we can use fractal math to quantify, describe, diagnose and perhaps soon to help cure diseases.

Teaching computers to use mathematical processes to tell the difference between healthy lungs and lungs suffering from emphysema promises to help make faster, more reliable diagnoses. The fractal dimension of the lung appears to vary between healthy and sick lungs, potentially aiding in the automated detection of the disease.

Cancer is another disease where fractal analysis may not only help diagnose but also perhaps help treat the condition. It is well known that cancerous tumors - abnormal, rapid growth of cells - often have a characteristic growth of new blood vessels that form a tangled mess instead of the neat, orderly fractal network of healthy blood vessells. http://fractalfoundation.org/OFC/OFC-12-4.html

This work introduces a new design approach for the use of fractal structures for low-resistance large area transistors structures. Aspects of layout with adapted current density and high-area utilization are considered. Furthermore the work presents a realization of fractal structures in AlGaN/GaN technology. Both static and dynamic behaviors are characterized. The fabricated devices achieve a breakdown voltage of V BR >; 700V and on-state currents of I D = 40A at V GS = 1V.

Published in: 2012 24th International Symposium on Power Semiconductor Devices and ICs


In addition to novel fractal transistors and sensors, we are developing fractals circuits for human implants and solar cells. In each case, we use the principle of biomimicry to exploit the functionality of nature’s fractals to provide technological advances. The fractal circuits are ideal for bioeletronics because they mimic the neurons they are designed to stimulate and measure. They also replicate the light-harvesting properties of fractal trees for the solar cells. These two projects represent the most important targets for future physics research – safeguarding human health and the Earth’s environment. For example, fractal electronics could address neurological disorders such as Parkinson’s disease and depression, and also improve nerve connections to prosthetic limbs.


Cyberneticist Ron Eglash has suggested that fractal geometry and mathematics are prevalent in African art, games, divination, trade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.[32][82] 


Fractal patterns can also be found in commercially available antennas, produced for applications such as
cellphones and wifi systems by companies such as Fractenna in the US and Fractus in Europe. The self-similar
structure of fractal antennas gives them the ability to receive and transmit over a range of frequencies, allowing
powerful antennas to be made more compact.
Above: Fulani wedding blanket (textile) showing fractal design.
Above the Mandiack weavers of Guinea-Bissau have also created an abstract design...but choose
to emphasize the fractal characteristics [textile mathematical art]. Page 44 of African Fractals.
Above is a Ashkenazi Jew with genetic (L2a1) roots in Guinea-Bissau sub-Sahara Africa.
Your mtDNA HVR1 exact matches may be recent, but they may also be hundreds or thousands of years in the past.
Above biometric (living mathematics) 78% facial match of pure I'huan Israelite Abraham and
Sandra Araujo Miss world Guinea-Bissau 2016. 8 of the 10 top facial matches to Abraham have
been African-American or sub-Saharan African.
IN 1988, RON EGLASH was studying aerial photographs of a traditional
Tanzanian village when a strangely familiar pattern caught his eye.
    The thatched-roof huts were organized in a geometric pattern of
circular clusters within circular clusters, an arrangement Eglash
recognized from his former days as a Silicon Valley computer engineer
Stunned, Eglash digitized the images and fed the information into a
computer. The computer's calculations agreed with his intuition: He was
seeing fractals
    Since then, Eglash has documented the use of fractal geometry-the
geometry of similar shapes repeated on ever-shrinking scales-in
everything from hairstyles and architecture to artwork and religious
practices in African culture. The complicated designs and surprisingly
complex mathematical processes involved in their creation may force
researchers and historians to rethink their assumptions about
traditional African mathematics
. The discovery may also provide a new
tool for teaching African-Americans about their mathematical heritage
In contrast to the relatively ordered world of Euclidean geometry
taught in most classrooms, fractal geometry yields less obvious
patterns. These patterns appear everywhere in nature, yet mathematicians
began deciphering them only about 30 years ago
The principles of fractal geometry are offering scientists powerful
new tools for biomedical, geological and graphic applications
. A few
years ago, Hastings and a team of medical researchers found that the
clustering of pancreatic cells in the human body follows the same
fractal rules that meteorologists have used to describe cloud formation
and the shapes of snowflakes.
   But Eglash envisioned a different potential for the beautiful
fractal patterns he saw in the photos from Tanzania: a window into the
world of native cultures.
 Eglash had been leafing through an edited collection of research
articles on women and Third World development when he came across an
article about a group of Tanzanian women and their loss of autonomy in
village organization. The author blamed the women's plight on a shift
from traditional architectural designs to a more rigid modernization
program. In the past, the women had decided where their houses would go.
But the modernization plan ordered the village structures like a
grid-based Roman army camp, similar to tract housing
    Eglash was just beginning a doctoral program in the history of
consciousness at the University of California at Santa Cruz. Searching
for a topic that would connect cultural issues like race, class and
gender with technology, Eglash was intrigued by what he read and asked
the researcher to send him pictures of the village.
    After detecting the surprising fractal patterns, Eglash began going
to museums and libraries to study aerial photographs from other cultures
around the world
    "My assumption was that all indigenous architecture would be more
fractal," he said. "My reasoning was that all indigenous architecture
tends to be organized from the bottom up." This bottom-up, or
self-organized, plan contrasts with a top-down, or hierarchical, plan in
which only a few people decide where all the houses will go.
    "As it turns out, though, my reasoning was wrong," he said. "For
example, if you look at Native American architecture, you do not see
fractals. In fact, they're quite rare." Instead, Native American
architecture is based on a combination of circular and square symmetry,
he said
    Pueblo Bonito, an ancient ruin in northwestern New Mexico built by
the Anasazi people, consists of a big circular shape made of connected
squares. This architectural design theme is repeated in Native American
pottery, weaving and even folklore, said Eglash.
    When Eglash looked elsewhere in the world, he saw different
geometric design themes being used by native cultures. But he found
widespread use of fractal geometry only in Africa and southern India
leading him to conclude that fractals weren't a universal design theme.
    Focusing on Africa, he sought to answer what property of fractals
made them so widespread in the culture.
"use of fractal geometry only in Africa and southern India". Southern India
is where the concept of zero (used in numerical computing) came from along
with "arabic numerals' and the decimal system
Eglash expanded on his work in
Africa after he won a Fulbright Grant in 1993.
    He toured central and western Africa, going as far north as the
Sahel, the area just south of the Sahara Desert, and as far south as the
equator. He visited seven countries in al
    "Basically I just toured around looking for fractals, and when I
found something that had a scaling geometry, I would ask the folks what
was going on-why they had made it that way," he said.
In many cases, however, Eglash found that step-by-step mathematical procedures
were producing these designs, many of them surprisingly sophisticated
 Eglash realized that many of the
fractal designs he was seeing were consciously created. "I began to
understand that this is a knowledge system
, perhaps not as formal as
western fractal geometry but just as much a conscious use of those same
geometric concepts," he said. "As we say in California, it blew my
mind." In Senegal, Eglash learned about a fortune-telling system that
relies on a mathematical operation reminiscent of error checks on
contemporary computer systems
In traditional Bamana fortune-telling ...The mathematical operation is called addition modulo 2,
which simply gives the remainder after division by two. But in this case, the two
"words" produced by the priest, each consisting of four odd or even
strokes, become the input for a new round of addition modulo 2. In other
words, it's a pseudo random-number generator, the same thing computers
do when they produce random numbers. It's also a numerical feedback
loop, just as fractals are generated by a geometric feedback loop
"Here is this absolutely astonishing numerical feedback loop, which is indigenous," said Eglash.
"So you can see the concepts of fractal geometry resonate throughout many facets of African
Lawrence Shirley, chairman of the mathematics department at Towson (Md.) University,
lived in Nigeria for 15 years and taught at Ahmadu Bello University in Zaria, Nigeria. He said
he's impressed with Eglash's observations of fractal geometry in Africa.
"It's amazing how he was able to pull things out of the culture and fit them into mathematics
developed in the West," Shirley said. "He really did see a lot of interesting new mathematics
that others had missed." Eglash said the fractal design themes reveal that traditional African
mathematics may be much more complicated than previously thought. Now an assistant professor of
science and technology studies at Rensselaer Polytechnic Institute in Troy, Eglash has written
about the revelation in a new book, "African Fractals: Modern Computing and Indigenous Design."
Recent mathematical developments like fractal geometry represented the top of the ladder in most
western thinking
, he said. "But it's much more useful to think about the development of
mathematics as a kind of branching structure and that what blossomed very late on European
branches might have bloomed much earlier on the limbs of others.
"When Europeans first came to Africa, they considered the architecture very disorganized and
thus primitive. It never occurred to them that the Africans might have been using a form of
mathematics that they hadn't even discovered yet."
Eglash said educators also need to rethink
the way in which disciplines like African studies have tended to skip over mathematics and
related areas.
To remedy that oversight, Eglash said he's been working with African-American math teachers in the
United States on ways to get minorities more interested in the subject.
Dr. Ron Eglash:
Assistant Professor .
Department of Science and Technology Studies
Rensselaer Polytechnic Institute (RPI)

Troy, NY 12180-3590
4-3-2-1-0 (Walter Russell and Michael James 9-8-7-6-5) wave pattern number system found in African mathematics:
numeric systems in Africa:
Players in Ghana use  the  term "marching  group" for a self-replicating  pattern, 
such  as  the  example  below. Here the number of counters in a series of cups each
decrease by one (e.g. 4-3-2-1).
As simple as it seems, this concept of a self replicating pattern is at the heart of some
sophisticated  mathematical  concepts.
The valid question arises, in what ways can an understanding of African  
mathematical representations, fractals, complexity and chaos in indigenous cultural
practices assist us in theorizing about the future
In our opinion, probably,the greatest point that can be taken away from our 
recent  research  Eglash  (1999) into African mathematics is an appreciation of 
African  indigenous  creativity  and  quantitative ability. As Hull (1976) noted, large 
urban  centers  were  disregarded  by  the  colonialists  because  they  did  not  utilize 
Cartesian   typology.  The  complex fractal nature of these settlements went unappreciated. 
This  point  is  crucial  in  any  discussion  of  possible  applications of current research

The sheer redemptive power of knowledge is at play here. Even  today, people living on the 
African  continent may still think  of their indigenous past as primitive and non-rational. 
An  understanding  of  the  fractal  characteristics of indigenous culture enables an
appreciation  of  the  complexity  of  the  ‘mundane’ indigenous artifacts. This singular
understanding can act as a powerful motivator for rethinking modernity.
"Bottom-up" social political structure of Africans vs the top-down colonial structure:
It is widely accepted in the STS [science and technology studies] community that indigenous
communities often posses  tacit  knowledge  that  may  be  invaluable  in  problem  solving
(Wynne  1996, Epstein 1996) but is usually deprivileged within the dominant discourse.
We have  shown, for example, that many traditional African villages were structured in a
“bottom-up” process, using  self-organization rather than imposed order. Could the top-down
hierarchal approaches that linger on in so many post-colonial African countries – often due to
the legacies of  colonialism – also give way to more bottom-up self-organizing social processes

There is an obvious need for a change in the methodologies of modernity on the African continent;
perhaps hybridizing indigenous mathematical representations can provide fresh thinking to a
persistent problem.

Fractals a Part of African Culture - Fractal Enlightenment

Out of Africa: Using Fractals To Teach At-Risk Students - APS Physics

A discussion of the relationship between pi and the Great Pyramid of Khufu. ... we know about ancient
Egyptian mathematics (based primarily on the Rhind Papyrus), .... the mathematical knowledge of the
architect who built the Great Pyramid.
It’s possible that pi , phi or both, as we understand them today, could have been the factors in the design of the pyramid.
A detail of the geometries and calculations is below:
relationship of fractals to pi:
Pi and fractal sets
The Mandelbrot set - Dave Boll - Gerald Edgar
In 1991 David Bolle tried to verify if the narrowing we can see at (-0.75,0) was actually infinitely thin. That is to say
that that however wide a non-zero width vertical line would be passing through that point it would meet the fractal
set before the x-axis.
And D Bolle then had the idea of using the point c=(-0.75,X) for the quadratic iteration and to make X tend to 0.
And there, what was his surprise when he counted the number of iterations before which the series diverged
and by discovering the following table .:
1.0  3
0.1  33
0.01  315
0.001  3143
0.0001  31417
0.00001 314160
0.000001 3141593
0.0000001 31415928
Yes, it was Pi that was appearing magnificientely ! As he could not manage to prove this he posted it in
1992 on the sci.maths newsgroup. Gerald Edgar from a university of Ohio answered it on 27 march 1992
by bringing an intuitive explanation of this result. This has been put lower down in the "Trial" section.

The Mandelbrot Set--

The relationship of the All-person to individual man or woman is a fractal.
Jehovih saith "I AM within man and man is within me" = a fractal relationship
"I AM within all things centering them, and I AM without all things controlling them" - Walter Russell.

Fractals-zoom.gif<-----Fractal geometry, <------"The Ocean Lives Within The Drop"--
Fractal geometry (mathematics) = sacred mathematics (geometry) of the I'hins and the Israelites:
Not only does fractal geometry and fractal mathematics describe much of the natural universe, but
fractals also describe the relationship of the Creator and spiritual man, such as:
Self-similarity (exactly or approximately similar, sameness)
Scaling (tiny section looks similar to whole and Vice versa)
Infinity (unlimited extent, boundless)
Recursion (repetitive process)
Fractional Dimension (such as
1.26 dimensions, infinite length in finite boundary)
Above are the five essential components of fractal geometry - pages 17 -18 of African Fractals by Ron Eglash.
Oahspe Book of Inspiration Chapter I:
1. ...I am Light; I am Central, but Boundless, saith Jehovih.
The central part is a fractal of the whole part (both containing boundless infinity, self-similarity).
Oahspe Book of Cosmogony and Prophecy ch 2:
26. one light, with a central focus. [The Father-Creator is One Boundless Light with a central focus, his central focus is man. Man is the focal point of the Father God]
Man is the fractal of the Creator-God.
Oahspe Book of Inspiration Chapter I:
12. As out of corpor I made thy corporeal body, so, out of My Light, which is My Very Self, I built thee up in spirit, with consciousness that thou art.
"out of My Light, which is My Very Self" = self-similarity = fractal.
Definition of very in US English:
adjective 1 Actual; precise (used to emphasize the exact identity of a particular person or thing). 'those were his very words'
"exact identity of a particular person" = SELF-SIMILARITY (fractal).

◄ John 14:11 ► of the Bible:
"I am in the Father and the Father is in me" = fractal relationship.
The fractal (mathematical) universe:
Oahspe Book of Sue, Son of Jehovih: Chapter V
16. God: I see nothing in all the universe but Thee! All selfs are but fractions of Thyself, O E-o-ih!
1/1000 = fraction = 1/1000 of Infinity = Infinity = a self-similarity fractal.
Google frac·tal: MATHEMATICS 1. a curve or geometric figure, each part of which has the same statistical character as the whole.
Word Origin & History: fractal, 1975, from Fr., from L. fractus "broken," pp. of frangere "to break" (see fraction). Coined by Fr. mathematician Benoit Mandelbrot in "Les Objets Fractals."

Relationship of fractals to phi (golden ratio):
2: The Golden Ratio as a Continued Fraction
A unique feature of the Golden Ratio is that it can be written as an Equation which calls itself:
Equation 1:   Phi = 1 + 1 / Phi
This Process is called Iteration.
Mathematical Iteration can also be used to generate a Fractal.
The above equation ( equation 1 ) can be used to generate Phi as a continued fraction.
From The Definition of a Fractal:
"A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays
at every scale. It is also known as an expanding symmetry or evolving symmetry.
C'vorkum light-years numbers (when rounded to nearest whole number) show a fractal pattern of
expanding symmetry.
Example 2727 (one whole C'vorkum) is a symmetrical expansion of 909 (1/3 C'vorkum) 9 x 3 (expanded) = 27.
Both 2727 and 909 are symmetrical numbers (same numbers to the left and right of center, 9-9 or 27-27).
In this illustration below, every spiral is the same phi spiral repeated:
The binary number system is an alternative to the decimal (10-base) number system that we use
every day. Binary numbers are important because using them instead of the decimal system
simplifies the design of computers and related technologies
Below: Binary code.
Above: Binary punched card.
"In Africa, on the other hand base-2 calculation was ubiquitous, even multiplication and division.
...The implications of this trajectory--from sub-Saharan Africa to North Africa to Europe are
quite significant for the history of mathematics. - Page 99 of African Fractals.
Zaslavsky (1973) shows archaeological evidence suggesting that ancient Egypt's use of base-2 calculations derived from the use of base-2 in Sub-Saharan Africa. - Page 89 of African Fractals.
Eglash explained that diviners use base-two arithmetic, just like the ones and zeros in digital circuits, and bring each output of the arithmetic procedure back in as the next input.
...the binary code appears to have a distinct African origin (Eglash 1997a)
The modern binary code, essential to every digital circuit from alarm clocks to super-computers,
was first introduced by Leibnitz around 1670. Leibniz had been inspired by the binary-based
"logic machine" of Raymond Lull, which was in turn inspired by the alchemists’ divination
practice of geomancy (Skinner 1980). But geomancy is clearly not of European origin.
It was first introduced there by Hugo of Santalla in twelfth century Spain, and Islamic scholars had
been using it in North Africa since at least the 9th century, where it was first documented in
written records by the Jewish writer Aran ben Joseph. The nearly identical system of divination
in West Africa
associated with Fa and Ifa was first noted by Trautmann (1939), but he assumed
that geomancy originated in Arabic society, where it is known as ilm al-raml ("the science of sand").
The mathematical basis of geomancy is, however, strikingly out of place in non-African systems.
Unlike Europe, India, and Arabic cultures, base 2 calculation is ubiquitous in Africa
, even for
multiplication and division. Doubling is a frequent theme in many other African knowledge systems,
particularly divination. The African origin of geomancy -- and thus, via Lull and Leibnitz, the
binary code -- is well supported

Oahspe The Lord's Fifth Book: Chapter VI:
29....and the seer sat therein, with a table before him, on which table sand was sprinkled. And the Lord
wrote in the sand, with his finger, the laws of heaven and earth.

Below: first three steps in Bamana sand divination.
Below: 4th and 5th steps in Bamana sand divination.
"I did receive permission from my teacher to make the Bamana algorithm public. The TED format
only gives you 17 minutes, and in that time I had to explain what fractal geometry is, how to
simulate fractals in African material design, and what the connection is between the fractal
design and the indigenous knowledge. So I had to cut a lot of details. I should also mention
that the divination priests told me that I was just reading a sentence or two from whole
libraries of knowledge; the algorithm itself is just one drop in their recursively infinite
." - Ron Eglash, November, 2017
...the Bamana use recursion to generate a binary code in their divination,...
- Page 124 of African Fractals by Ron Eglash. 
On Study of Oahspe
Misc. 3, Sacred mathematics, geometry, music, harmonics, cosmology, cycles page: about 1/5 down:
Cymatics harmonic sound made visible show very similar geometric fractal pattern to snow flakes and etherean worlds.


Cymatics can be traced back at least 1000 years to African tribes who used the taut skin of drums sprinkled
with small grains to divine future events. 3
3. Encyclopedia of Religion Volume 4. Eliade, Mircea (Editor)



Oahspe Book of Apollo, Son of Jehovih: Chapter XI:
2. Cim'iad was a small woman, dark, and of deep love, most jovial of Goddesses; and had long looked forward with joy to her pleasure of bringing so large a ship to deliver two thousand million of Jehovih's Brides and Bridegrooms into etherean worlds.
Practical fractals: recursion in construction techniques...Williams goes on to note that much
African metal work, unlike European investment casting, uses a "spiral
technique" to build up structures...resulting in ..."helical coils formed from smaller helical
- Page 112 of African Fractals by Ron Eglash.
Figure 8.2-h A single iteration of a three-dimensional version of the recursive triangle
construction, created by Akan artists in Ghana. [Ghana Akan worshipped the Great
Spirit like the Israelites
Above: Kitwe community clinic fractal design in Zambia, Africa by David Huges and Alex Nyangula.
Above: Jola fractal settlement of Mlomp, Senegal - Page 163 of African Fractals by Ron Eglash.
Eglash described an ivory hatpin from the Democratic Republic of the Congo that is decorated with carvings of faces. The faces alternate direction and are arranged in rows that shrink progressively toward the end of the pin. Eglash determined that the design matches a fractal-like sequence of squares where the length of the line that bisects one square determines the length of the side of the following square.

Above: Geometric analysis of Mangbetu iterative squares structure of ivory sulpture
Pages 66-68 of African Fractals figure 5.5 ...the construction algorithm can be continued
...applied to a wide variety of math teaching applications from simple procedural construction
to trigonometry
(Eglash 1998a).
noun: iteration.
  1. the repetition of a process or utterance.
    • repetition of a mathematical or computational procedure applied to the result of a previous application, typically as a means of obtaining successively closer approximations to the solution of a problem.
    • a new version of a piece of computer hardware or software.
Fractal geometry is a field of maths born in the 1970’s and mainly developed by Benoit Mandelbrot.

The process by which shapes are made in fractal geometry is amazingly simple yet completely
different to classical geometry. While classical geometry uses formulas to define a shape,
fractal geometry uses iteration. It therefore breaks away from giants such as Pythagoras, Plato
and Euclid and heads in another direction. Classical geometry has enjoyed over 2000 years of
scrutinisation, Fractal geometry has enjoyed only 40.

How to make a fractal shape

The rules are as follows:

1. Split every straight line into 3 equal segments.

2. Replace the middle segment with an equilateral triangle, and remove the side of the triangle
corresponding to the initial straight line.

After this has iterated an infinite amount of times the fractal shape is defined. This may sound
bewildering but it is still possible to analyse it mathematically and visually you can see what
the shape starts to look like. The gif below (from Wikipedia) is a good illustration of what the
curve looks like by zooming in on it:
The von Koch curve [above looks exactly like a snowflake] is a great example of a fractal: the rule
you apply is simple, yet it results in such a complex shape. This kind of shape is impossible to
define using conventional maths, yet so easy to define using fractal geometry.
Fractal trees:
On the tree above, if you snapped a branch off it and stood it up, it would look like the
original tree. If you took a twig from the branch and stood it up, it would still look like the
original tree [self-similar]
. Again, this is a property that occurs in nature, but until fractal
geometry there was not a good way to put it into maths.
Not only do these shapes look like natural objects, but the process of iteration sounds
intuitive when thinking about nature. When a tree is growing, its trunk will create branches,
these branches create further branches, these branches create twigs. It’s as if the function is
a genetic code telling the branch how to grow and repeat itself
, eventually creating shapes that
are ‘natural’.
Fractal fibonacci numbers (sequence) and iterations:
A realistic map of the branching of a tree (or a variety of other plants too) is shown in the
figure below. In this case, the tree grows from the bottom up, and the rule here is that a
branch grows one unit long in each iteration
. When a branch is two units long, it is strong
enough to support a node, which is where a new branch splits off. The branches alternate on the
left and the right, and very quicky a recognizable plant pattern emerges.
The rules that generate this fractal are really identical to the rules for the rabbit family
tree above, since it takes two iterations for a branch in the rabbit family tree to bifurcate,
and it also takes a tree branch two iterations before it is strong enough to bifurcate. The same
kind of self-similar pattern emerges, because at any step you can look at a new rabbit pair, or
branch, as being the beginning of an entirely new sequence. A little branch on a tree can be cut
off and planted and will form a whole new tree
. A grandchild rabbit can turn into the grandparent

of many rabbits. Any unit, at any iteration, is just a scaled version of any other unit in the system.Questions:
How many branches are there at the 6th generation? [ ]
Above are shell and fern fractals, page 16 of African Fractals. 
In theory, one can argue that everything existent on this world is a fractal:
the branching of tracheal tubes,
the leaves in trees,
the veins in a hand,
water swirling and twisting out of a tap,
a puffy cumulus cloud,
tiny oxygene molecule, or the DNA molecule,
the stock market
Fractals will maybe revolutionize the way that the universe is seen.
A dissident group of scientists claims that the structure of the universe is fractal at all scales.
If this new theory is proved to be correct, even the big bang models should be adapted.
The real world is well described by fractal geometry.

Oahspe Book of Sue, Son of Jehovih: Chapter V
16. God: I see nothing in all the universe but Thee! All selfs are but fractions of Thyself, O E-o-ih!
fractals can occur over time as well as space—one example is how hearts beat across time. Robust hearts have fractal heartbeats, according to Ary Goldberger, a professor at Harvard Medical School. Using graphs of heart rate time series (like the 30-minute time series show below), he quantified the "fractal-ness" of heartbeats using a method called detrended fluctuation analysis, which identifies similarities in curves across different scales.
It may seem like sudden spikes and falls in the stock market are anomalous flukes, but they happen all too often to just be random, according to Benoit Mandelbrot, a mathematician who is often called "the father of fractals." Based on his belief that market fluctuations follow fractal geometry, he has created fractal-based financial models that better account for extreme events than traditional portfolio theory, which is based on a normal bell curve. Furthermore, these models can be applied to any timescale, from years to hours.
Climate cycles and the rules of prophecy in Oahspe are fractal in nature. 3.7 year cycle is a 1/3 fractal of the 11.1 year cycle. The 11.1 cycle is a 1/3 fractal of the 33.3 year cycle. The 33.3 year cycle is a 1/3 fractal of the 99.9 year cycle. The 1000 year cycle is a 1/3 fractal of the 3000 year cycle. The 11.1 year cycle divided into three 3.7 years is self-similar to the 33.3 year cycle divided into three 11.1 years, etc...
See "Cycles, Predictions, Earth events, A'ji" page of this website.
Using the golden ratio gets you the same proportion no matter what scale or how big or small you
go…it is infinite in keeping the same proportion throughout whatever scale. Think in terms of
the geometry and the Golden Ratio Spiral. Each revolution retains the same proportion and is
therefore self similar, i.e. fractal.
Fibonacci's Fractals (they are actually African in artistic & mathematic origin).
Indeed, these are not "Fibonacci's" fractals any more than Georg Cantor's "Cantor set" was "Cantor's" i.e. European in origin. In both cases the earliest documented human creative reproduction of nature's fractals are found in Black Africa.
Badawy (1965) noted what appears to be use of the Fibonacci series in the layout of the temples of ancient Egypt. Using a slightly different approach, I [professor Ron Eglash] found a visually distinct example of this [Fibonacci] series in the successive chambers of the temple of Karnak, as shown in the figure 7.22. Figure 7.2b shows how these numbers can be generated using a recursive loop. This formal scaling plan may have been derived from the nonnumeric versions of scaling architecture we see throughout Africa. - Page 87-89 of African Fractals.
There is no evidence that ancient greek mathematicians knew of the Fibonacci Series. There was use of the Fibonacci series in Minoan design, but preziose (1968) cities evidence indicating that it could have been brought from Egypt by Minoan architectural workers employed at Kahun." - Page 89 of African Fractals.

Does the Parthenon really follow the golden ratio?


...Known as recursive. The Fibonacci sequence was the first such recursive sequence known in Europe. - Page 97 of THE
GOLDEN RATIO The Story of Phi by Mario Livio (Ph.D astrophysics).
In the 1202 AD, Leonardo Fibonacci wrote in his book “Liber Abaci” of a simple numerical sequence that is the foundation
for an incredible mathematical relationship behind phi.
"but when it comes to the use of the Fibonacci series in the contemporary history of mathematics (cf. B rooke 1964), 
there is actually no evidence of a direct contribution from Fibonacci himself. By all accounts, German astronomer 
Johanes Kepler rediscovered the series independently in 1611, and it was only in the mid-1800s, with the formal 
publication of Liber Abaci, that french mathematician Edouard lucas found the Pisan historical predessor and named 
it accordingly. This fact has received little attention, and most texts present Fibonacci's discovery as if it were
in a direct intellectual line of descent rather than an honary title given to a well deserving but disconnected
antecedent." - Page 206 of African Fractals.
The golden ratio, a mathematical relation that often arises in fractals and other 
scaling geometries, ...structures derived from the golden ratio that organize typographic 
compositions and even structure living spaces also has presence in traditional African 
architecture, and that this presence extends further back than sites and artifacts in Greece.
...Eglash (1999). The first is the chief’s palace in Logone-Birni, Cameroon. This historical 
architectural site has a golden ratio scaling pattern embedded in its spatial design. The second 
is a similar scaling pattern in the Temple of Karnak from ancient Egypt.
As Eglash notes, there is no evidence that ancient Greek mathematicians knew of the Fibonacci 
series (89). However Badaway (1965) found a use of the Fibonacci series (1, 1, 2, 3, 5, 8, 13…) 
in the layout of temples in Ancient Egypt.
Greek mathematician Euclid did describe dividing a line such that it created a golden ratio, but never discussed constructing a two-dimensional figure with that ratio. The most famous representative of Greek architecture, Vitruvius, worked exclusively with proportions that can be expressed in whole numbers, rather than irrational proportions that the golden ratio would require. This was in keeping with the ancient Greek distaste for irrational numbers and the infinite series, which clashed with Platonic notions of mathematics as static perfection existing in an eternal, unchanging ‘realm of the Forms.’ African designs, on the other hand, deliberately make use of such iterative sequences, since they fit well with the African cultural emphasis on fecundity, fertility and cyclic aspects of life.
One can hear this dynamicism in African polyrhythmic music, and see it in iterative architectural
designs such as Karnak and Logone-Birni.
The Temple of Karnak from ancient Egypt shows successive chambers with lengths determined by 
iterations of the Fibonacci Series. The altar in the temple of Karnak depicted in Figure 3 
creates the initial value for the generation of its form, just as we see for altars in other 
cases of self-generating architectural forms in Sub-Saharan Africa. Since archaeological 
evidence shows that Egyptian civilization was founded when groups traveled down the 
Nilotic valley,[if the hebrews were originally nilotic (which they probably were), haplo L2a (or possibly L2a1)
would likely be their founding maternal haplogroup.
it is no surprise that these traditions of recursive form were continued in Egypt. In the 
original sub-Saharan architectures the structures are not largely determined by quantitative 
formula; the Egyptian version thus provides a more formal version of the sub-Saharan recursive 
tradition. It is not unreasonable to speculate that Fibonacci brought the sequence from North 
Africa where it was used in the weights of a scale balance as well as architecturally.
As shown in Figure 4 [above], we can postulate then that the golden ratio originated in Sub-Saharan 
Africa, migrated north possibly to Egypt, and then traveled to Italy and onward around the rest 
of the world.
The two historical African sites discussed in this paper are significant because of the 
well-known mathematical concept embedded in both of their spatial designs—a phenomenon that has 
relevance to the history of communication design and African-American design identity.
Length of blocks of Logone-Birni Cameroon palace spiral path:
44 / 3.384= 13.0 for largest
28 / 3.384= 8.2 for middle
18 / 3.384= 5.3 for smallest
Fibonacci sequence = ...5, 8, 13, ...
Logone-Birni palace in Cameroon and 3 golden rectangles of different sizes (3x and 1/3 scaling) in design. 
The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[8][14
Susantha Goonatilake writes that the development of the Fibonacci sequence "is attributed in part to Pingala
(200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150)".[7]
Parmanand Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and cites scholars who interpret
it in context as saying that the cases for m beats (Fm+1) is obtained by adding a [S] to Fm cases and [L] to the
Fm−1 cases. He dates Pingala before 450 BC.[15] However, the clearest exposition of the sequence arises in the
work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):

Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters
of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be
followed in all mātrā-vṛttas [prosodic combinations].[16]

Outside India, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Fibonacci.[6 
The Karnak Temple Complex, commonly known as Karnak (/ˈkɑːr.næk/[1], from Arabic Ka-Ranak meaning "fortified village"),
comprises a vast mix of decayed temples, chapels, pylons, and other buildings in Egypt.
Construction at the complex began during the reign of Senusret I [from 1971 BC to 1926 BC] in the Middle Kingdom and
continued into the Ptolemaic period, although most of the extant buildings date from the New Kingdom [between the 16th
century BC and the 11th century BC].
"Reevaluating all of the existing material in this light, I have to conclude that it is very unlikely that the
Babylonians discovered the Golden Ratio." - Page 47 of THE GOLDEN RATIO The Story of Phi, The World's Most 
Astonishing Number by Mario Livio. 
The Fibonacci sequence, logarithmic spiral and Golden Ratio scaling (a fractal scaling ratio) go hand in hand. 
Above: Golden Ratio Fibonacci spiral and Oahspe cyclic coil of the Great Serpent of the solar system.
Both showing decreasing in size twists.
Above: Fractals of neurons in brain (see dark matter network below) and passages in lungs.
Above: Etherean worlds and roadways have fractal architecture. Oahspe God's Book of Ben Plate 44 -SNOW FLAKES [Snow flakes are symbols of Etherean worlds]. Look at the snowflakes as though they were microscopic patterns of the worlds in high heaven.
Above: many traditional sub-Saharan African villages have fractal architecture.
A traditional branching fractal settlement in Senegal, page 35 of African Fractals. 
...a wide variety of African settlement architectures can be characterized as fractals. Their physical construction makes use of scaling and iteration, and their self-similarity is clearly evident from comparison to fractal graphic simulations...fractals in African architecture...is linked to conscious knowledge systems that suggest some of the basic concepts of fractal geometry,...we will find more explicit expressions of this indigenous mathematics in astonishing variety and form. - page 38 of African Fractals.

1.1 Definition of Fractal The formal mathematical definition of fractal is defined by Benoit Mandelbrot. ... Generally, we can define a fractal as a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole.
Below logarithmic spirals and fractals:
Smaller pyramid within smaller spiral in center is a fractal of larger pyramid within larger spiral.
Below: Logarithmic scaling and Great Pyramid of Egypt:
Figure Three shows a graph of the mean orientation among structures on a log scale. This
produces a linear decline according to pyramid number. The measured values hold amazingly well
to a mathematical equation expressing the decay, with small variance from the regression line.

The mathematical expression for this line is:
Y = A (intercept at 0 pyramid number) - B (slope) (pyramid number).
Y = A - B X P where P denotes the pyramid number.
Note that we are now dealing with logarithmic numbers on the vertical scale, not linear numbers.
The orientations are expressed in the natural logarithm of the original number, while the pyramid
numbers are expressed on an ordinary linear scale.
For those who may not be acquainted with logarithms, two forms are commonly used in mathematics
and the physical sciences
. The first is called "common logs" based on multiples of 10. 100 = 1,
101 = 10, 102 = 100, and so on. The second is called "natural logs" based on an important
physical constant we symbolize by the letter e = 2.71828. e0 = 1, e1= 2.71828, e2 = 7.389, and
so on. e most often expresses the rate of natural decay processes.
In following discussion I shall use the symbolism of log to denote common logs, and ln to denote
natural logs, following common practice in modern scientific and technical fields.
We might argue that the data from the pyramids are not natural; they do not come from natural
processes. They were devised by human intelligence and control. However, the data plots show
that the designer prearranged his structures to express a curve similar to natural decay. He
probably did so because he knew that anyone competent enough to detect the form of the curves
would be familiar with natural processes
. He also had to be familiar with natural decay
processes, and how they are expressed mathematically, otherwise he could not imitate them.
After I calculated the intercept and the slope of the orientations from the data on the basis
of a natural decay curve I bumped into another amazing value. The intercept was 10 Pi and the
slope was very nearly 1/Pi or possibly Pi/10
. I show both the regression line calculated from
the data (solid) and the two theoretical lines (dashed) on Figure Three.
We saw in the Great Pyramid Pi chamber that the designer used (Pi X ln 10) for one of his
dimensions to show his knowledge of higher mathematics. Here he displays it with the intercept
values and the slope of the logarithmic decay.

https://postimg.cc/image/rqphotbn9/ Above: P2,P3,P4...P10,P11,P12 on the logarithmic spiral corresponds
to P2,P3,P4...P10,P11,P12 on the Great Pyramid scalene triangle.
Above: logarithmic scaling in Ghanaian design - African Fractals by Ron Eglash page 79.
How a Galaxy is a Fibonacci Sequence: Physics & Math
Fibonacci Logarithmic Pyramid slope:
Great Pyramid Logarithmic slope graph:
https://postimg.cc/image/9idvre3qz/ (How the Great Pyramid is a Fibonacci Sequence).
Above I have mapped the slope angle of the Great Pyramid of Egypt using a basic coordinate system similar to the way longitude (vertical) and latitude (horizontal) coordinates are demarcated on a map. Left side vertical coordinates / right side horizontal coordinates
Coordinates of Great Pyramid logarithmic fractal slope:
2/3, 3/5, 5/8, 8/13, 13/21 ... = left/right, vertical/ horizontal coordinates.
0,1,1,2,3,5,8,13,21,34, ..... = Fibonacci Golden ratio (Phi) sequence.
Notice the vertical coordinates / horizontal coordinates of the Great pyramid
Logarithmic slope are Fibonacci (Phi) numbers. The coordinate numbers approximate the Golden Ratio (Phi) very close, just as a spiral galaxy
is a close approximation of a Fibonacci golden ratio logarithmic spiral.
In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio.[1] That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.
The logarithmic spiral is also known as the growth spiral, equiangular spiral, and spira mirabilis. This spiral is related to Fibonacci numbers, the golden ratio, and the golden rectangle, and is sometimes called the golden spiral.
"Generating the Fibonacci series requires a feedback loop or, as mathematicians call it, iteration.
In iteration, there is only one transformation process, but each time the process creates an output,
it uses the result as the input for the next iteration, as we've seen in generating fractals." - Page 110
of African Fractals.
Fractal mathematics is a part of the religion and SPIRITUALALITY of sub-Saharan Africans (original Israelites) considered sacred by the priests they did not want to voluntarily teach it to professor Ron Eglash until he showed them the Cantor set.
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.

The Cantor set is the prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated. More precisely, there are two functions, the left and right self-similarity transformations, {\displaystyle T_{L}(x)=x/3} and {\displaystyle T_{R}(x)=(2+x)/3}, which leave the Cantor set invariant up to homeomorphism: {\displaystyle T_{L}({\mathcal {C}})\cong T_{R}({\mathcal {C}})\cong {\mathcal {C}}.}    Repeated iteration of T_{L} and T_{R} can be visualized as an infinite binary treeCantor set - Wikipedia

upside-down Cantor set shows that they are not simply applying mod 2 again and again in a mindless fashion. The self-similar physical structure of the shortcut method vividly illustrates a recursive process. African divination can be elsewhere linked to re-cursion, as in Devisch's (1991) description of the Yaka diviners' ...
Ron Eglash suggests that the marks made by Bamana sand diviners resemble the Cantor set. And Yoruba IFA divination, while it has mathematical implications, is based on 256 verses in an oral literature, which must be memorized by the diviner and recalled in conjunction ...
Reflecting on his fieldwork realized among Bamana (or Bambara) diviners, the author compares their use of recursion, where the iterative function is addition modulo 2, with Cantor's recursion (Cantor set), and hypotheses that an African concept of self-generated fecundity is the shared origin of both the Bamana divination and transfinite set theory.
Georg Cantor's Theory of Transfinite Numbers. ... The HarperCollins Dictionary of Mathematics describes "transfinite number" as follows: "A cardinal or ordinal number used in the comparison of infinite sets, the smallest of which are respectively the cardinal (Aleph -null) and the ordinal (omega).
Egyptian Column capital with pattern like Cantor set:
...you can see the Cantor set in the capitals of their columns.
Philae, Arabic Jazīrat Fīlah (“Philae Island”) or Jazīrat al-Birba (“Temple Island”), island in the Nile River between the old Aswan Dam and the Aswan High Dam, in Aswān muḥāfaẓah (governorate), southern Egypt.
the earliest structures known are those of Taharqa (reigned 690–664 bce), the Cushite 25th-dynasty pharaoh. The Saites (664–525 bce) built the earliest-known temple, found dismantled and reused in the Ptolemaic structures. Nectanebo II (Nekhtharehbe [reigned 360–343 bce]), last pharaoh of the 30th dynasty and last independent native ruler of Egypt prior to 1952, added the present colonnade. The complex of structures of the Temple of Isis was completed by Ptolemy II Philadelphus (reigned 285–246 bce) and his successor, Ptolemy III Euergetes (fl. 246–221 bce). Its decorations, dating from the period of the later Ptolemies and of the Roman emperors Augustus and Tiberius (30 bce–37 ce), were, however, never completed.
Since Philae was said to be one of the burying-places of Osiris, it was held in high reverence both by the Egyptians to the north and the Nubians (often referred to as "Ethiopians" in Greek) to the south. It was deemed profane for any but priests to dwell there and was accordingly sequestered and denominated "the Unapproachable" (Ancient Greek: ἄβατος).[10][11]
African Fractals
Culture 11: Ethiopian cross
"...the processional crosses of Ethiopia also indicate a threefold iteration. Perczel (1981) reports that related designs can be found on ornaments excavated from the city of Axum in Northern Ethiopia in the second half of the first millennium B.C.E., so we should not assume that the threefold iteration was originally related to the Christian trinity ..." - page 135 of African Fractals. 
The false religions of the false Gods are those that worship a savior God in the image of a man.
Sacred fractal mathematics teaches us that the small part contains all the statistical data of
the whole so no savior or worship of another God is needed.
◄ John 14:11 ► of the Bible:
"I am in the Father and the Father is in me"
(Cosmic Consciousness) = fractal relationship (The Infinite ALL within him).
As Jesus (Yeshu -Joshu) said "I say ye are Gods" John 10:34. The Infinie All within each one of you also.
2) Brain regions involved with mystic experiences.
...are the amygdala and hippocampus. A mystic whose experiences appear from an unusually responsive right hippocampus is expected to report experiences dominated by right hippocampal (RH) phenomena. The RH role in spatial reasoning and memory implicates it in experiences of 'infinity', the "infinite void", spaciousness, and the experience that the space occupied by the sense of self is limitless ("one with the universe"). The RH role ...Its cognitive functions implicate it in the experience of 'knowingness', and 'insight', in which understandings appear spontaneously.
Factors that determine who are the Israelites:
DNA (genetics), facial recognition, and mathematics:


1. Enhanced hippocampal volume (rs7294919 C;C, C;T):
3.5% of CEU(C) Northern and Western European ancestry share the Enhanced hippocampal volume (rs7294919 C;C) gene. Mexican ancestry MEX(M) had about the same percentage (3.5%) as the Northern and Western European CEU(C) and the Chinese CHD(D) for Enhanced hippocampal volume (rs7294919 C;C).
15.8% to 22.7% of African-Americans ASW(A) and sub-Saharan Africans LWK(L), MKK(K), YRI(Y), share the Enhanced hippocampal volume (rs7294919 C;C) gene (higher than other populations sampled around the world).
2. Higher intracranial volume (rs10784502 C;C,C;T) :
3. mtDNA Haplogroup L2a and L2a1:
...the I'hins became extinct as a separate people or merged with such races as the Hebrews [Israelites] and the Vedic tribes of India ...page 81 of DARKNESS, DAWN AND DESTINY by Augustine Cahill.
(Oahspe First Book of God 8:4-6)
4. These, then, are the generations of the line whence came Abram,
"In Sadr the line was lost, but through his daughter Bar-bar regained through the I'hins"
"here the line ran by female heirs, beginning in Rac-ca's daughter"
"from whom was descended seven generations in su-is; and it was lost in We-ta-koo, but regained again through I'hin seed"
"the line ran by female heirs" = maternal lineage = mtDNA = L2a and L2a1.

 the Mbuti Pygmies, L2a (fig. 2).
L2 is the most common haplogroup in Africa,...The highest frequency occurs among the Mbuti Pygmies (64%).[8]
L2a2 is characteristic of the Mbuti Pygmies.[8]
Shortest tribe
The smallest pygmies are the Mbutsi from Zaire, with an average height of 1.37 m (4 ft 6 in) for men
and 1.35 m (4 ft 5 in) for women, with some groups averaging only 1.32 m (4 ft 4 in) for men and 1.24 m
(4 ft 1 in) for women. According to medical research, pygmy children are not significantly shorter than
the children of other tribes. But they do not grow in adolescence because they produce only a limited
amount of the hormone called insulin-like growth factor (IGF).
Interesting Facts About the People of Africa
The shortest people in the world, the pygmies, live in Africa.
Peaceful Societies: Alternatives to Violence and War. Mbuti Pygmies,
Haplogroup L2a1 was found in two specimens from the Southern Levant Pre-Pottery Neolithic B site at Tell Halula, Syria,
dating from the period between ca. 9600 and ca. 8000 BP or 7500-6000 BCE.[17]
L2a is widespread in Africa and the most common and widely distributed sub-Saharan African Haplogroup and is also
somewhat frequent at 19% in the Americas among descendants of Africans (Salas et al., 2002).
4. Facial recognition:

5. Traditional mathematics, fractal geometry:
they were oppressed by cruel laws and penalties, and were forced to reveal the mathematical
science which had been preserved with them from their distant ancestors the I'hins, to whom
it was committed by the angels in the first ages of mankind. - Pages 189-190 of Darkness, Dawn And Destiny
(Drawn from Oahspe) 1965 by Augustine Cahill.
The complicated designs and surprisingly
complex mathematical processes involved in their creation may force
researchers and historians to rethink their assumptions about
traditional African mathematics
. The discovery may also provide a new
tool for teaching African-Americans about their mathematical heritage
When Eglash looked elsewhere in the world, he saw different
geometric design themes being used by native cultures. But he found
widespread use of fractal geometry only in Africa and southern India
leading him to conclude that fractals weren't a universal design theme.
    Focusing on Africa, he sought to answer what property of fractals
made them so widespread in the culture.
"the I'hins ...merged with such races as the Hebrews [Israelites] and the Vedic tribes of India" = "he found widespread use of fractal geometry only in Africa and southern India".
Above: synchronously the highest male and female facial recognition matches to the Oahspe drawing of Moses are ancestrally from sub-Sahara Africa, and India the only 2 places where professor Ron Eglash found widespread use of fractal geometry.
Pankaj Advani (far left photo above) is a 19-time World Champion English billiards and snooker player from India.
Michael James (2nd photo from left above) is a African American author of this website. Audrey Pulvar (3rd photo from left above)
is a Black French journalist. Last (4th) photo above is a Cobra Gypsy woman from India. The two people above from India match higher in facial recognition to south Indian person than to north Indian person (see facial recognition proof below).
Recent mathematical developments like fractal geometry represented the top of the ladder in most
western thinking
, he said. "But it's much more useful to think about the development of
mathematics as a kind of branching structure and that what blossomed very late on European
branches might have bloomed much earlier on the limbs of others.
"When Europeans first came to Africa, they considered the architecture very disorganized and
thus primitive. It never occurred to them that the Africans might have been using a form of
mathematics that they hadn't even discovered yet."
Assistant Professor .
Department of Science and Technology Studies
Rensselaer Polytechnic Institute (RPI)

Troy, NY 12180-3590
6. Slavery and Persecution:
OAHSPE: Book of the Arc of Bon CHAPTER XIII:
9. Yokebed feared, for in those days male children of Israelitish parentage were outlawed, nor could any man be punished for slaying them.
Jim Crow was the name of the racial caste system which operated primarily, but not exclusively in southern
and border states, between 1877 and the mid-1960s. Jim Crow was more than a series of rigid anti-Black laws.
It was a way of life. Under Jim Crow, African Americans were relegated to the status of second class citizens.
Jim Crow represented the legitimization of anti-Black racism. Many Christian ministers and theologians taught
that Whites were the Chosen people
, Blacks were cursed to be servants, and God supported racial segregation.
Lynching was the practice of killing, usually by a hanging resulting from extrajudicial mob action. Lynchings in the United States occurred after the American Civil War in the late 1800s, the emancipation of slaves, and chiefly from the late 1800s through the 1960s. Lynchings took place most frequently against African American men and women in the South, with lynchings also appearing in the North during the Great Migration of blacks into Northern areas. The political message—the promotion of white supremacy and black powerlessness—was an important element of the ritual, with lynchings photographed and published as postcards which were popular souvenirs in the U.S.[1][2] As well as being hanged, victims were sometimes burned alive and tortured, with body parts removed and kept as souvenirs.[3]Lynchings occurred most frequently from 1890 to the 1920s, a time of political suppression of blacks by whites, with a peak in 1892.
From 1882-1968, 4,743 lynchings occurred in the United States.  Of these people that were lynched 3,446 were black.  The blacks lynched accounted for 72.7% of the people lynched.  These numbers seem large, but it is known that not all of the lynchings were ever recorded.  Out of the 4,743 people lynched only 1,297 white people were lynched.  That is only 27.3%.  Many of the whites lynched were lynched for helping the black ...

Emmett Louis Till (July 25, 1941 – August 28, 1955) was a 14-year-old African-American who was lynched in Mississippi in 1955. The brutality of his murder and the fact that his killers were acquitted drew attention to the long history of violent persecution of African Americans in the United States.
In September 1955, Bryant and Milam were acquitted by an all-white jury of Till's kidnapping and murder. Protected against double jeopardy, the two men publicly admitted in a 1956 interview with Look magazine that they had killed Till. https://en.wikipedia.org/wiki/Emmett_Till

For more than eight decades — between 1882 and 1964, to be exact — the extrajudicial killing of African-Americans known as lynching took the lives of at least 3,445 men, women and kids.
At its worst, in 1892, an astounding 161 African-Americans were hanged, shot, beaten or burned to death by whites across the United States. In a horrific 10-year period from 1891-1901 — the worst decade for lynching in American history — an average of more than 100 African-Americans per year were brutally lynched. Not one person was convicted of first-degree murder in over 1,000 lynchings during this period.
Just one white man was convicted of murdering African-Americans during the 82-year period when lynching was commonplace, according to Douglas Blackmon’s Pulitzer Prize-winning work, “Slavery by Another Name.”
Mother fears for her son's life . - "Yokebed feared" - OAHSPE: Arc of Bon Book CHAPTER 13v9.
Oahspe Book of the Arc of Bon.: Chapter I:
16. And in those days, whoso was of the seed of the worshippers of the Great spirit, Ormazd, was outlawed
in receiving instruction. So that the chosen, the Faithists, were held in ignorance, lest a man of learning might
rise up amongst them and deliver them.
The Israelites themselves, being slaves in Egypt, were of course
an illiterate people, and remained so for centuries thereafter. When
they came to collect their own history, they were obliged to derive it to
some extent from their enemies. - Page 205 of DARKNESS, DAWN, AND DESTINY
(DRAWN FROM OAHSPE) 1965 by Augustine Cahill.
il·lit·er·ate 1.unable to read or write.
1. a person who is unable to read or write.


Fearing that black literacy would prove a threat to the slave system whites in the Deep South passed laws forbidding slaves to learn to read or write and making it a crime for others to teach them.
Blacks were denied the right to read and were punished for reading:
Oahspe - Book of Lika, Son of Jehovih: Chapter IV:
4. For thou shalt find My chosen a scattered people, persecuted and enslaved, the most despised of all the races of men. But I will show My power with them; I will raise them up; the things I do through them, and the words I speak through them, even in their ignorance and darkness, shall become mighty.
8. But My chosen, who are their slaves, and are as nothing in the world, shall speak, and their words shall not be forgotten; shall write, and their books will be a new foundation in the world. Because My hand will be upon them, My wisdom shall come forth out of their mouths.

7. Artwork:

Thutmosis III was the sixth Pharaoh of the Eighteenth Dynasty. (c. 1543–1292 BC) during the Arc of Bon cycle and the time of Moses.
8. Oahspe description:
Oahspe First Book of God 8:7: Abram was of pure blood, an I'huan;
Oahspe Book of the Arc of Bon 15:13 Moses grew and became a large man, being a pure I'huan, copper-colored.
Although northern India tends toward Euclidean architecture, explicit recursive design is seen
in several temples in southern India--the Kandarya Mahadeo in Khajuraho is one of the clearest
examples--and is related to recursive concepts in religious cosmology. These same areas in
southern India also have a version of lusona drawings, and many other examples of fractal design.
Interestingly, these examples from southern India are the products of Dravidian culture, which is
suspected to have significant historical roots in Africa.
Most traditional European fractal designs, like those of Japan and China, are due to imitation 
of nature ...There are at least two stellar exceptions, however, that are worth noting. One is
the scaling iterations of triangles in the floor tiles of the Church of Santa Maria in Cosmedin
Rome (see plate 5.7 in Washburn and Crowe 1988). I have not been able to determine anything about
their cultural origins, but they are clearly artistic invention rather than imitation of some
natural form. The other can be found in certain varieties of Celtic interlace designs. Nordenfalk
(1997) suggests that those are historically related to the spiral designs of pre-Christian Celtic
, where they trace the flow of a vital life force. They are geometrically classified as
an Eulerian path, which is closely associated with mathematical knot theory (cf. Jones 1990, 99).
Above is a Sottish-American and a Celtic interlace knot fractal of round squares and crosses.
Above are Sub-Saharan African interlace knot symbols. 
Fractal structure is by no means universal in the material patterns of indigenous societies. In
Native American designs, only the Pacific Northwest patterns show a strong fractal characteristic;
Euclidean shapes otherwise dominate the art and architecture. Except for the Maori spiral designs,
fractal geometry does not appear to be an important aspect of indigenous South Pacific patterns
either. That is not to say fractal designs appear nowhere but Africa--southern India is full of
fractals and Chinese fluid swirl designs and Celtic knot patterns are almost certainly of
Independent origin.  - pages 47-48 of African Fractals by Ron  Eglash.
Above Maori spiral design tattoo and Maori warriors from New Zealand.
"Anything with a rhythm or pattern has a chance of being very fractal-like."
Rhythm in Sub-Saharan African culture
Sub-Saharan African music is characterised by a "strong rhythmic interest"[1] that exhibits common characteristics in all regions of this vast territory,
Many sub-Saharan languages do not have a word for rhythm, or even music. Rhythms represent the very fabric of life and embody the people's interdependence in human relationships.
The African focus on fractals emphasizes their own cultural priorities: it can even be heard in their polyrhythmic music (similar simultaneous rhythms at different scales).
Polyrhythm is traditional West African music considered by musicologists to be the most rhythmically complex music in the world. Rhythms and counter rhythms in the common African tradition of call and response complement and communicate with one another with different drum lines, other musical instruments, bodies and voices contributing rhythmic elements. This element of instrumental call and response is also evident in the polyrhythmic quality of jazz. By contrast, most traditional European music has a flat linearity.
In popular music
Nigerian percussion master Babatunde Olatunji arrived on the American music scene in 1959 with his album Drums of Passion, which was a collection of traditional Nigerian music for percussion and chanting. The album stayed on the charts for two years and had a profound impact on jazz and American popular music.[citation needed] Trained in the Yoruba sakara style of drumming, Olatunji would have a major impact on Western popular music.[citation needed] He went on to teach, collaborate and record with numerous jazz and rock artists, including Airto Moreira, Carlos Santana and Mickey Hart of the Grateful Dead. Olatunji reached his greatest popularity during the height of the Black Arts Movement of the 1960s and 1970s.
Among the most sophisticated polyrhythmic music in the world is south Indian classical Carnatic music. A kind of rhythmic solfege called konnakol is used as a tool to construct highly complex polyrhythms and to divide each beat of a pulse into various subdivisions, with the emphasised beat shifting from beat cycle to beat cycle.
"Keep in mind that, hands down, the most rhythmically complex music known is commonly considered by musicologists to be West African. (The next, IMO, is likely the ragas of India.) It seems that some appropriate mention of the cultural context of the phenomenon is in order here -- perhaps a mention of how European classical music is, and much of Western music (before African/African-American influence) was, heavily linear and flat/unsyncopated." - deeceevoice 12:21, 5 December 2005 (UTC)
"Since no one else saw fit to add anything about the true origins of polyrhythmic musical expression in Western music, I did so. It's outrageous to mention Zappa and white musicians and no mention of the source, Africa. Someone might also want to mention the ragas of India, where the sounds of the tabla correspond to spoken sounds." - deeceevoice 02:04, 27 July 2006 (UTC)
"In West African music, polyrhythm is part of the basic music vocabulary, and has been so for a very long time. The basic musical forms taught to beginning musicians are usually accompaniment parts of polyrythmic ensemble pieces. Polyrhythm, quite simply, is an African Music 101 topicIn Western classical music, polyrhythm really is a fringe curiosity, and when it shows up, it is treated as an advanced topic.
There's also the fact that citing 20th century classical composers as an example is problematic on other grounds—20th century European art has in many cases looked to African arts for inspiration ..."
- (talk) 19:59, 17 July 2008 (UTC), Spot-on. Co-signing. My point(s) precisely! ;) deeceevoice (talk) 12:20, 3 August 2009 (UTC)
"A good example of this nonlinear scaling can be seen in the logarithmic spiral." - page 76 of African Fractals. 
Mathematics deals with abstract thought.
Abstract mathematical reasoning:

Fractal mathematics of E-O-IH (creator's name):
E-O-IH = 9-8-5 = 9/5, 8/5, 5/5 = 1.8, 1.6, 1.0.
E+O+IH = 1.8 + 1.6 + 1.0 = 4.4.
Generate a fractal by recursive procedure.
Recursion = output at one stage becomes the input for the next.
output synonym = amount/quantity produced.
sum = the total amount resulting from the addition of two or more numbers.
4.4 = fractal, having 4 of the essential components of fractals below:
4.4 = recursion, output (3.4) becomes input for the next (3.4 + 1.0).
4.4 = scaling, .4 is 4 on a smaller 1/10th scale (1 order of magnitude less)
4.4 = self similarity, 4 is exactly similar to .4
4.4 = fractional dimension, .4 is 2/5 fraction.
Fractals in nature don't go on for infinity like in computer graphics.
corpor = nature
OAHSPE: God's Book of Ben Chapter III:
24. Why sayest thou nature? Now I say unto thee, the soul of all things is Jehovih; that which thou callest nature is but the corporeal part.
OAHSPE'S BOOK OF COSMOGONY and Prophecy Chapter 8:
1. LET ethe stand as one; ...and corpor as four.
4.4 is the fractal number of nature (corpor) which is a quality (or part) of E-O-IH.
The number 4 is the number of the wave. the secret of creation.
Above image of traditional Camaroon braids hairstyle of branching fractals, page 114 of African Fractals. 
"...African fractals have a surprisingly strong utilization of recursion. Indeed, in Mandelbrot's seminal text,
The Fractal Geometry of Nature (1977), the index lists "recursion" only twice, and the terms iteration,
self-reference, self-organization, and feedback are entirely absent. As we will see this absence is not accident; it reflects a European historical trend ...Europeans traditionally placed such little importance on recursion." - page 176 of African Fractals.
"...recursion is prominent in African fractals ...In the historical appendix to The Fractal Geometry of Nature,
Mandelbrot Provides an erudite history of mathematical developments that led to his work: recursion is never mentioned." - Page 214 of African Fractals by Ron Eglash. 
Above magnification zooms of Mandelbrot set showing bifurcation (branching) fractals.
The logarithmic spiral depicts growth and expansion in the universe, and the edges of the Mandelbrot set fractal depict growth and expansion in infinite space.
Above (branching and logarithmic spiral) shows as the spiral expands the number of branches in the tree increases.

3. I am the living mathematics;
How do cells duplicate themselves, and why? In all complex multicellular organisms (eukaryotes), cell duplication occurs by a process called "mitosis" or cell division.
Growth. We all started out as a single cell; the fusion of a sperm from dad and an egg from mom. That original cell divided repeatedly until you grew and differentiated into an organism composed of billions of cells. But this raises an interesting problem. When one cell divides into two, both must have a copy of the genetic information. Therefore, before cell division occurs, the genes must also make duplicates of themselves so that all of the important genetic information ends up in each of the new cells. The first cell divides into two, and each of those two divide again, and this process continues geometrically along the following progression: 1, 2, 4, 8, 16, 32, 64, 128, and so on into the billions. That's growth. 
Computer code bits is in harmonic multiples and divisions of 8. The two (2) base system (bi-nary, base 2 calculation) is a harmonic division of 8. Multiply 2 x 2 x 2 x 2 = 4, 8, 16, 32 which are harmonic parts (divisions) or multiplications of 8 which are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, etc...start off with 1 whole and multiply x 2, x 2, x 2, x 2, x 2, x 2, x 2, etc... you get all divisions or multiples of 8, like harmonic musical notes or computer bits.
Computer binary code 8 bit-system (each number below represents a iteration in the recursive branching fractal process):
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768.

"There's a new branch of mathematics available to all scientists and that application will stretch on through the centuries now as the primary tool for descriptive physical science". -
 Michael Barnsley, The Colours of Infinity: The Beauty and Power of Fractals.

The Colours of Infinity: The Beauty and Power of Fractals.

Michael Fielding Barnsley, born in 1946, is a British mathematician, researcher and an entrepreneur who has worked on fractal compression; he holds several patents on the technology. Wikipedia
"But there will be new devices, extraordinary new devices based on the principles of Fractal Geometry, that will emerge over the
next centuries
." - Michael Bransley, The Colours of Infinity: The Beauty and Power of Fractals.
" If you want to reveal the handiwork of God and see his design and order in all things, fractals will reveal that design to you."
WHY DO FRACTALS OCCUR BOTH IN MATH, WHICH IS ABSTRACT(not made up of atoms) AND THE PHYSICAL WORLD, WHICH IS PHYSICAL(is made up of atoms)? The same designer (Jehovih)
Mandelbrot image (left) and branching lightning (right). 
Mandelbrot image (left) and snowflake (right).
Barnsley fern (left) and nature fern (right). One sort of fractal is known as the Iterated Function System, or IFS. This fractal system was first explored by Michael Barnsley at the Georgia Institute of Technology in the 1980s. You start with shapes plotted on a graph, and iterate the shapes through a calculation process that transforms them into other shapes on the graph. Starting with four shapes, one of which is squashed into a line segment (this becomes the fern's rachis or stalk), you apply the shapes to the calculation to generate more shapes, feed them back into the calculation process, etc. Eventually a pattern emerges that bears a startling resemblance to a fernhttp://www.home.aone.net.au/~byzantium/ferns/fractal.html 
Math generated fractal (left) and window frost (right).
Mathematical graph shape (left) and Romanesco broccoli (right). 
Mandelbrot set logarithmic spiral (left) and spiral galaxy (right).
Above: African Ba-ila settlement fractal of Zambia early 1940s (left), 
fractal circles similar to Mandelbrot set (center),
Mandelbrot set main bulb fractal (right). 
The Mandelbrot set ...
This fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups.[2] On 1 March 1980, at IBM's Thomas J. Watson Research Center in Yorktown Heights, New YorkBenoit Mandelbrot first saw a visualization of the set.[3]
Above: fractal pattern of Songhai village in Mali (left and center). Others suggest the village is an conscious expression of a fractal structure that became a cultural construct of many regions in Africa. Notice the similarity of the African Mali village layout to the Julia set fractals (right).
Arthur C Clarke - Fractals - The Colors of Infinity.
From 7:50 to 8:21 of video:
the seeds to this discovery were in fact sown decades before the M set was first seen, in Paris in 1917 a mathematician
called Gaston Julia
published papers connected with so-called complex numbers, the results of his endeavors eventually
became known as Julia sets although Julia himself never saw a Julia set he could only guess at them
[his guesses were way off] and it wouldn't be until the advent of modern computers that Julia sets
could be seen for the first time
The Mandelbrot set (named after Benoit Mandelbrot) is the most famous fractal of all, and the first one to be called a fractal.
Ron Eglash was exposed to the fact that the knowledge and application of fractals had been alive for millennia in Africa.
This concept of infinity had for long, before Cantor, been part of the African divination system. In Africa, Eglash encountered some of the most complex fractal systems that exist in religious activities, such as the sequence of symbols used in sand divination, a method of fortune telling found in Senegal. The concept of infinity had a metaphysical link with infinity.
The relevant point is that fractals existed in nature and before Mandelbrot there was Koch and Cantor. Before Koch and Cantor there were many people in Africa who understood fractal geometry ...

Causality in Fractals - shapestacking explained - YouTube
From 1:40 to 2:25 into video we see base 2 calculation in the Mandelbrot set, 1,2,4,8,16,32,64,128,...infinity.

Fibonacci Numbers hidden in the Mandelbrot Set:
Mandelbrot Set Fractal Zoom 10^227 [1080x1920]
"the Mandelbrot main structure is repeated infinitely, but each structure has more pattern than the previous one".
"This is beyond amazing".
"I have a hard time believing this is real".
Eye of the Universe - Mandelbrot Fractal Zoom (e1091) (4k 60fps)
Oahspe Book of Lika, Son of Jehovih: Chapter VII:
1...in the etherean sea, moving brilliants playing kaleidoscopic views, ever changing the boundless scene with surpassing wonders.
Definition of kaleidoscopic in English:
Having complex patterns of colours; multicoloured.
Made up of a complex mix of elements; multifaceted.
The Mandelbrot fractals contain worlds without end.
Book of Ouranothen CHAPTER 1:
8. First, then, His Living Presence I declare to you; that He is now, always was, and ever shall be present in all places, worlds without end.
Oahspe Book of Discipline: Chapter XIV:
21. ...by Jehovih, Creator, Ruler and Dispenser, worlds without end. Amen!
The Mandelbrot fractal is like Infinite reflections in a mirror:A computer can magnify (zoom-in) infinitely (or to 227 orders of magnitude) so you can see more.

Scale of the Universe: 62 Orders of Magnitude (1062) | Montessori ...

Mandelbrot Zooms Now Surpass the Scale of the Observable Universe

Infinite self-similarity, Infinite complexity, Infinite scaling, Infinite repetitions, Infinite universe contained within a fractal.
The Mandelbrot set fractal is created by recursion and iterations of a mathematical formula Z = Z2 + C (x2 + c).
HD Mandelbrot Fractal Tour Guide - YouTube...Tour of the famous Mandelbrot Fractal, visiting 16 popular areas in the fractal. 
The Amazing Mandelbrot Set tutorial:
"Modern computers have given us the ability to peer deep into the complex plane opening up a whole new world of
mathematical wonder." "But computers can't give us the power to understand what we may discover a link between
the Mandelbrot Set and the processes that guide the laws of nature." - The end of The Amazing Mandelbrot Set tutorial video.

Could our universe be fractal? - YouTube

3:39 into video: "The easiest way to simulate a world as realistic as possible is to use fractal formulas".

3:50 to 4:29 into video: "The first completely computer generated movie seen in a feature film was the fractal 

animation of a planet in Star Trek 2 The Wrath of Khan. The Lucas film group responsible for this was later

acquired by Steve Jobs, from this he created PIXAR thus revolutionizing Hollywood, they're animated movies

look so realistic because the generated landscapes are based on the fractal principle of self-similarity,

progress has continued and today these elaborate and costly movie seens have evolved into real-time walkable 

game worlds like the landscapes of Minecraft". 

Holographic = realistic 3D simulation.

Fractal formulas = mathematical recursion, iteration, feedback. 

A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.
Study reveals substantial evidence of holographic universe.
The idea is similar to that of ordinary holograms where a three-dimensional image is encoded in
a two-dimensional surface, such as in the hologram on a credit card. However, this time, the
entire universe is encoded!”
Although not an example with holographic properties, it could be thought of as rather like watching
a 3D film in a cinema. We see the pictures as having height, width and crucially, depth – when in fact
it all originates from a flat 2D screen. The difference, in our 3D universe, is that we can touch objects
and the ‘projection’ is ‘real’ from our perspective.


Living in a MATRIX hologram – scientists say EVERYTHING we see could be an illusion
Jan 30, 2017 - EVERYTHING in the universe could be a "complex hologram" meaning all that we see in 3D is actually an illusion.

Oahspe God's Book of Ben: Chapter II:
24. Nor is there aught in thy corporeal knowledge that thou canst prove otherwise, save it be thy presence;
and even that that thou seest is not thy presence, but the symbol and image of it, for thou thyself art but as
a seed, a spark of the All Light, that thou canst not prove to exist.
Chapter V
17. Uz said: All thou seest and hearest, O man, are but transient and delusive
Something incredible – and deeply perplexing – is currently taking place at the intersection of mathematics,
religion, design technology and computer science; something that could fundamentally alter the way we perceive
Over the past few years, several prominent mathematicians have claimed to have discovered a connection
between a mysterious mathematical sequence and the very structure of our universe, speculating that it is possible
to express nature's immutable laws in a complex geometric image
the Mandelbrot set, may be a geometric depiction of an "eternally existing self-reproducing chaotic and
inflationary universe
" and can only now be rendered in full detail by using state of the art computer technology.
Mandelbrot used fractal geometry and funky color schemes to demonstrate mathematically that infinity is real and
exists even in a world that appears finite to the naked eye
This is where shit gets deep - evoking the concept of God and the idea of eternity.
Roger Penrose. He is a mathematical Platonist, and believes that both the fractals worlds (such as the Mandelbrot set) and the mathematical truths
(such as Fermat’s last theorem) are discovered. In his view, the mathematical truths have an eternal, unchanging, objective existence in some kind of
Platonic ideal world, independent of human observers.

(Mathematics) a concept of quantity ....
Definition of concept. 1 : something conceived in the mind : thought.
fractal simulations for natural objects are created with a "seed shape that undergoes recursive replacement". - page 12
of African Fractals. A'su seed shape undergoes recursive replacement by adding angelic genes (I'hin) into the mix. 
Like fractals the Kosmon race in Oahspe is created by additive recursive feedback loop iterations:
0. A'su = no mixbreeding
1. start with seed shape A'su, breed A'su with Ethereans (who died in fetal or infancy)  = output I'hins.
2. take the output I'hins and breed back to A'su = output Druk
3. take the output Druk and breed back to I'hin = output I'huan
4. take the output I'huan and breed back to I'hin = output Ghan
5. take the output Ghan and breed back to I'huan = output Kosmon.
Number of iterations on the left (above). Takes 5 iterations of breeding to make the Kosmon race, the goal of Jehovih. 
5 = balance (center of 0-9, 1-10). 5 is a Fibonacci number.
The most balanced complete "perfect" man (race) was created using fractal principles of recursion and feedback loop according to Oahspe and the fossil record (see The Mysterious Origins of Hybrid Man: Crossbreeding by Susan Martinez Ph.D. Anthropology), this is the same way Jehovih created the fractal universe of nature which is different from the Biblical story in Genesis
Like a fractal the races of man in Oahspe are generated by a recursive feedback loop of 5 iterations from I'hin to Kosmon.
Recursive feedback loop formula for generating the present and future races of mankind:
A'su = A, I'hin = C, Druk = D, etc...
A + B = C. (C is the output).
C + A = D (C is the first input that was an output).
D + C = E
E + C = F
F + E = G
Generating the present and future races of mankind involves recursion, self-similarity, scaling, and infinity = Fractal.
I'hin, I'huan, Ghan, and Kosmon are self-similar to each other with everlasting (infinity) life. The I'hin was half the size (scaling) of the I'huan, Ghan, and Kosmon (1/2 and 2x). The races of man in Oahspe are generated by recursion (the output at one stage becomes the input at the next stage). The output is a previous race, the input is the creation of a new (or different) race. In the Bible, Man (Adam) is created whole complete and "perfect" at the beginning with no build up or use of recursion or mix-breeding, the same with Noah. In Oahspe Adam (A'su) was not whole or complete and neither was the I'hin. The way Oahspe says the races of man were created agrees with the fractal processes of nature we observe everyday, the way the Bible says the races of man was created does not agree with the fractal processes of nature we observe everyday. 
What scientists call human evolution is just iterations of a recursive process of mix-breeding.
How the races of man in Oahspe is like a Fibonacci sequence (recursion):
Each race is the sum of its two predecessors:
A + B = C
A + B = two predecessors or input, C = sum or output.
The output becomes the input for the next number.
C + A = D and so on.
According to Oahspe even though the first manlike creation (A'su) was not the result of
mix-breeding he was the result of recursion (sum of its predecessors).
19. Nevertheless, I had given to man, and thus made him, out of the dissolved elements
of every living thing that had preceded him.
Ouput becomes the input (recursive process of generating a fractal):
"out of the dissolved elements of every living thing that had preceded him". 
Oahspe Book of Inspiration Chapter I:
8. Thou wert nothing; though all things that constitute thee, were before.
9. These I drove together, and quickened.
Here (above) Oahspe explains how Jehovih created man (recursion), not from nothing, but
from that which was before, these parts Jehovih brought together (a creation). 

"When life gives you an ancient cave filled with dirt, look for DNA. That's what

paleontologists and those involved in the study of ancient humans will likely be

doing in the future following the revelation of a breakthrough technique that

enables hominin DNA to be recovered directly from sediments without the

need for fossils."

Ancient human DNA found in dirt, not fossils - New Atlas The study has been

published in Science.

Soil is composed of a matrix of minerals, organic matter, air, and water.


1 Corinthians 15:47 The first man was of the dust of the earth, the second man from heaven. 

OAHSPE: First Book of the First Lords CHAPTER I. 4. And the name of the first race was 

Asu (Adam), because they were of the earth only; and the name of the second race was 

I'hin (Abel), because they were capable of being taught spiritual things. 

The Mandelbrot set fractal is created by recursion and iterations of a mathematical formula Z = Z2 + C.
Physical structure follows mathematical fractal laws
Physical form on the right follows mathematical recursive iterated function (Z = Z2 + C) on the left. Images on the left are self-similar to the images on the right.

Abstract Mandelbrot fractal image and spiral galaxy M-100 in physical space.
Abstract Mandelbrot zoom and  Hubble Space Telescope image of quasar and host galaxies.
Abstract Mandelbrot fractal zoom and Duplex 

Cotton Candy Nebula

 in physical space.
Abstract Mandelbrot Fractal Zoom and M102 lenticular galaxy and planet Saturn in physical space.
Abstract Mandelbrot fractal zoom and 8 petal flower in physical nature.
Abstract Mandelbrot fractal zoom and 4 petal white flower in physical nature.

Mandelbrot fractal zoom and Planetary Nebula Ethos 1 (circle and cross) and stars in space.

Mathematical Mandelbrot fractal zoom and Elephants with coiled trunk and walking in line in physical nature.
Abstract Mandelbrot fractal zoom and Autumn forest trees with no leaves in nature.
Abstract Madelbrot fractal zoom and Centipede Chilopoda classification in nature.
Abstract Mandelbrot set zoom and Pterodactyl Fossil body in physical nature.
Abstract Mandelbrot-Julia set hidden structure and White tip shark in physical nature.
Abstract Mandelbrot-Julia set fractal zoom and Sunflower Asteraceae plant in physical nature.
Mandelbrot fractal zoom and Luidia australiae seastar.Mandelbrot fractal zoom and DNA double helix scanning electron microscope image.
https://www.youtube.com/watch?v=PD2XgQOyCCk (5 min 37 seconds into Mandelbrot Zoom video).
Physics (physical form) follows abstract mathematical function:
Is there a connection between the Mandelbrot Fractal iterated function (equation) of Z  = Z squared and
the many equations in physics that are squared? "Could the reason why so many equations in physics are squared
represent aspects of a single truth based on just one geometrical process?" what is significant is that
the inverse square law does not just apply to Newton’s universal law of gravitation it also applies to electric
magnetic light
" [electricity, magnetism, light and heat].

Why is almost everything squared in physics?²

A connection between fractal geometry and physics (and biophysics, the application of the laws of physics to biological phenomena).
function is an equation that has only one answer for y for every x. A function assigns exactly
one output to each input of a specified type.

Mathematical infinity a characteristic of the All-Person (Creator) and fractals
There are an infinite (infinity) number of decimal places in the number Phi (Golden Ratio a fractal scaling ratio).
1.61803398874989484820458683436563811772030917980576286213544862270526046281890 ... 
In a Logarithmic spiral there are an infinite number of revolutions, with the distance between them decreasing
as the curve approaches the center. 

Mandelbrot fractal set, for example, seemingly infinite complexity is achieved with a very simple looking equation:

Z = Z2 + C.

Pay special attention to the double arrow equal sign. This is very important because it signifies the recursive nature of fractals, and the fact that there’s a built-in feedback loop. This simple equation, given enough iterations, can produce patterns that look as complex and as beautiful as the images... https://blog.kareldonk.com/the-holographic-and-fractal-universe/ 


Fractal geometry mathematics contains the divine knowledge of Nature, the universe. and the Creator. In ancient times this divine knowledge was passed on to the I'hins, then the Israelites and their close relatives in Africa and India, then since the late 1970s modern man was inspired to discover fractals, for this is the Kosmon cycle of universal knowledge (corporeal and spiritual) and universal fellowship in all nations.

 Kosmon, or, Kosmon said. THE PRESENT ERA. All knowledge in possession of man, embracing corporeal and spiritual knowledge sufficiently proven. The Kosmon Bible is Oahspe.



Fractal principles explain spiritual law and God and the Creator.

Physical form follows mathematical function (computational code or program). A fractal is created by recursion A + B = C (output), you take the output C and bring it back in to repeat the process, where the output C becomes the input C + A = D, then you repeat the process again, over and over. The output becomes the input which is KARMA, you get back what you put out, a spiritual law which Walter Russell called rhythmic balanced interchangeRhythmic means regular repetition or cyclic, balanced means equal, interchange means giving and receiving or output and input. We Know we exist, so there is something in the universe that is self-similar to us but on a bigger or larger scale. The largest scale would be the Infinite One, the All, the whole. The whole is in the part and the part is in the whole (a fractal). 

A vortex with logarithmic spirals (like a hurricane or spiral galaxy) is very fractal like. Logarithmic spirals contain 4 essential components of fractal geometry. Logarithmic spirals have recursion, scaling, self-similarity, and infinite turns in the center.

The universe is full of recursive patterns & self-similarity on all scales. 

Above: images of spiral galaxy and hurricane showing fractal self-similarity, scaling, and logarithmic spiral infinity.
“The same flattening of the rotational curve is observed in the magnetohydrodynamics of stars and even hurricanes on earth, both of which are vastly different in both scale and density. Finding self-similar characteristics in rotating bodies across such enormous differences in scale points to a common underlying mechanism.”

Hurricane diameter (500 miles) is 10^5 orders of magnitude (1 magnitude = 1-10 meters, 3-30 feet)

Milky Way Galaxy diameter (100,000 light-years) is 10^20 orders of magnitude.

Milky Way and Whirlpool Galaxy (86,000 light years) are 15 orders of magnitude larger than a hurricane.


Whirlpool Galaxy - Wikipedia


OAHSPE BOOK OF COSMOGONY IX:1 says "The same force, vortexya, pervadeth the entire universe but different, according to volume, velocity and CONFIGURATION. 

Page 17 of African Fractals by Ron Eglash: 

RECURSION...fractals are generated by a circular process ...


Oahspe Plate 47 - THE CYCLIC COIL:

Jehovih...He is the circle without beginning or end... 

"Recursion is the motor of fractal geometry; it is here that the basic transformations - whether numeric or spatial - are spun into whole cloth, and the patterns that emerge often tell the story of their whirling birth." - Page 109 of African Fractals. 

Definition of whirl: to move in a circle ...

Fractal recursive mathematics applies to physics, astronomy and other sciences.

The Oahspe vortex unified field theory of the universe:

Like a fractal a vortex is recursive, a circular process. A vortex is sub-atomic particles (infinitesimal needles)

in rotary (circular) motion. Magnetism is a manifestation of a recursive vortex force. Positive vortex'ya (vortex force) 

is the input of the vortex current. Gravity is the input of the vortex force. Negative vortex'ya (m'vortex'ya) is 

the output of the vortex current. When the vortex force is charged or stored in iron or steel it is called magnetism.

The current of the vortex force is called electricity. When the vortex current causes the sub-atomic particles to

line up in one direction it is called light. When the stored up vortex force is liberated it is called fire or heat.

A nuclear explosion is liberated stored up atomic vortex force. Atomic mass is the manifestation of stored up

vortex force. Chemical elements are the manifestation of the velocity, pressure,  configuration, and volume of

the vortex force.

Above: vortex with currents and lines of force, output at bottom, input at top. 
Above left: magnet showing magnetic field lines (output N becomes input S)
Above center: vortex, output at bottom becomes input at top. 
Above right: solar system vortex, output at bottom past Neptune becomes input at top center or sun.
Mathematical recursive vortex equation:
A + B = C, next step or iteration of the recursive process = C + A or B = D. 
A + B = input, C = output, A + B = positive vortex'ya, C = negative or m'vortex'ya.
The center of the vortex between the input and the output is neutral, corpor, atom, planet or sun. 
Above: blue line is positive vortex'ya, red line is negative or m'vortex'ya
Oahspe Book of Cosmology and Prophecy: Chapter I:
35. ..."the master's infinitesimal needles remain poised from the sun centre outward,
even to the earth, and may be compared to telegraph wires"... = redline above.
15. The positive force of the vortex is, therefore, from the external toward the internal;
and the negative force of the vortex is toward the poles, and in the ascendant toward the
pole external from the sun centre
Blue line above is input of the circular recursive process, redline above is the output of
the circular recursive process.
Magnetic vortex - experimental proof.
The video linked here shows experimental proof of the existence of a magnetic vortex. The direction of rotation changes when magnetic polarity is reversed.
Usually, we see magnetic field lines shown as bending straight back from one end of the magnet to the other. Correctly, what should be shown is magnetic lines of force in a vortex configuration, with flow spiraling into the magnet (or out of it) in a right-hand or left-hand turning motion, depending on the magnetic polarity.
Magnetic vortex spin proof:
Above is from page 142 of African Fractals by Ron Eglash, the Vodun god Dan and periodic snake-like cycles 
notice how very similar it is to the Oahspe concept of Dan, the Great Serpent and Cyclic coil. 
In West Africa they call Dan the "cyclic Dan" and also "Dangbe", this is very similar to Oahspe calling
Dan a "cyclic dawn" and "synonymous with dang". See below for proof:
Me calling a "warm up-spike year a time of Dan" is equivalent to the West African concept of "Regular
phenomena in nature--the periodic aspects of weather ...are attributed to the action of Dan." 
Above we see from Oahspe the Roadway of the Solar Phalanx showing the snake-like great serpent of the solar system going through cyclic dawns of Dan. The Oahspe great serpent and Dan (Dang) is very similar to the West African symbolic concept of Dan (Dangbe)
Vodun cosmology centers around the vodun spirits and other elements of divine essence that govern the Earth, a hierarchy that range in power from major deities governing the forces of nature and human society ...
West African Vodun is practiced by the Fon people of Benin, and southern and central Togo; as well in Ghana,andNigeria.
In West Africa Dan is pictured as "a serpent biting its tail". In Oahspe Plate 48.--THE CYCLIC COIL you can see the spiral currents of the vortex spiral inward to the sun-center and then spiral and go out from the center toward the tail. Oahspe says "a very long serpent in spiral form, constantly turning its head in at one pole, and its tail at the other".
Above is an image of a recursive feedback loop, the output at the bottom stage becomes the input at the top stage of cycle.
If one could imagine a very long serpent in spiral form, constantly turning its head in at one pole, 
and its tail at the other, and continuously crawling upon its own spirality, such a view would somewhat illustrate the currents of a vortex. - Book of Cosmogony and Prophecy Ch III: 25.
Above West African Dan (Dangbe) snake symbol swallowing his tail = mathematical iteration = cycle.
Iteration, a dog chasing his tail or a snake swallowing it's tail = the output at one end becomes the input at the other = the end of one cycle (arc of Bon) = the beginning of the next cycle (Kosmon).
Iteration = the repetition of a process, or repetition of a mathematical or computational procedure applied to the result of a previous application.

The Colours of Infinity: The Beauty and Power of Fractals


The first attempt to model the distribution of galaxies with a fractal pattern was made by Luciano Pietronero and his team in 1987,[1] 

Pietronero argues that the universe shows a definite fractal aspect over a fairly wide range of scale,



…fractal patterns exist at many scales in nature. Physicists believe that fractals also exist in the quantum world, and now a group of researchers in the US has shown that this is indeed the case. Quantum repetition, Fractal patterns enter the quantum world.

This image shows the fractal pattern that results when the waves associated with electrons start to interfere with each other. A fractal is a geometric entity whose basic patterns are repeated at ever decreasing sizes.

Fractal patterns spotted in the quantum realm – Physics World

Yazdani Lab: Visualizing Quantum States of Matter (Princeton University, Department of Physics).

Yazdani Lab - Princeton University

Above is quantum world image enlarged in square (original image just to the right upper). Notice

the enlarged image (212%) is very similar to larger scale orange, green, and blue images in the 

original (100%). This shows self-similarity and scaling (on different size scales) of the 

quantum world. 

Mpc meaning in astronomy = A distance of one million parsecs is commonly denoted by the 
megaparsec (Mpc). Astronomers typically express the distances between neighbouring galaxies 
and galaxy clusters in megaparsecs. One parsec is equal to about 3.26 light-years
(30 trillion km or 19 trillion miles) in length.

Parsec - Wikipedia

Large scale distribution of galaxy clusters above shows a fractal pattern very similar to neuron networks

in the brain (like a Cosmic Consciousness).

"Mathematics is the language with which God wrote the Universe." – Galileo.

Anything that can be measured is mathematics. Thoughts and Words are mathematics. 

Fractal Geometry is the mathematics in which the Creator wrote the Universe. - Michael James.

In the beginning was the Word (mathematical concept) and the mathematical concept was 

with the Creator (Programmer) and the mathematical code was the Programmer 

(underlying recursive code that programed the universe of physical form). 

Above and below is from pages 68-69 of African-Fractals by Ron Eglash.

In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph which visits every edge exactly once.


Will Hunting solves a graph theory problem on a board in the hall posed by Professor Gerald Lambeau.

In the movie "Good Will Hunting" there is a particular mathematics problem in the movie:

"Draw all homeomorphically irreducible trees of size n=10n=10." (ten valid trees).

The rules

• In this case “trees” simply means graphs with no cycles


In mathematics, and more specifically in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path.

Tree (graph theory) - Wikipedia

In graph theory (mathematics) a tree is a network of dots and lines (Numberphile Youtube). 

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges,

In mathematics, graphs are useful in geometry and certain parts of topology such as knot theory. Algebraic graph theory has close links with group theory.


Above are Celtic knots containing Eurlarian paths.

Page 48 of African Fractals "...certain varieties of Celtic interlace designs. Nordenfalk
(1997) suggests that those are historically related to the spiral designs of pre-Christian Celtic
, where they trace the flow of a vital life force. They are geometrically classified as an Eulerian path
which is closely associated with mathematical knot theory (cf. Jones 1990, 99)." 


Migration map of interlace knot designs and Eurlarian path knowledge from Africa to Celtic Europe.

Oahspe God's Book of Eskra 50:30-33: "Now, after the fall of the great empire, Egupt, her people

migrated westward, hundreds of thousands of them, and they settled in western Uropa, where these

people married with the aborigines. Their offspring were called Druids, Picts, Gales (Gaelic),

Wales (Welsh), Galls (Gauls), and Yohans (Johns), all of which are Eguptian names, preserved to

this day. " Interlace knots and Eurlarian path knowledge predate the mathematician Leonhard Euler.

Relating to the Celts or their languages, which constitute a branch of the Indo-European family and include Irish, Scottish Gaelic, Welsh, Breton, Manx, Cornish, and several extinct pre-Roman languages such as Gaulishhttps://en.oxforddictionaries.com/definition/celtic 

The plaited mat design in Figure 24 is well known from the central African cultural area. Among the Kuba it is called ‘namba’, the bowels (Torday, p. 101).  

Investigating the Mathematics of Sona: Sand Drawings from Angola.


[1] Paulus Gerdes, Drawings from Angola: Living Mathematics, Morrisville, NC: Lulu.com, 2007.


...the Tchokwe [sona] drawing tradition, threatened with extinction during the colonial occupation...

The sona tradition has been vanishing: "What we find today is probably only the remnant, becoming more and more obsolete, of a once amazingly rich and varied repertoire of symbols" [Kubik, 1987, p. 59].

l succeeded in reconstructing important mathematical elements of the sona tradition. 

...probably original sona have been reconstructed (cf. my analysis of the Tamil drawing tradition in South India that is technically related to the Tchokwe tradition. Gerdes [1989a]).

3. Educational and mathematical potential

Initially I was mostly interested in the reconstruction of the mathematical knowledge that had been present at the invention of the sona.

Many reported and reconstructed (sub)classes of Tchokwe ideograms satisfy a common construction principle.

By filming them starting the curve at one point, one sees a geometrical algorithm at work.

The study of the mathematical properties of these curves constitutes a new and attractive research field. 

Possibly they may be applied in the codification of information, in the development of laser memory circuits for optical computers, in the study of the topology of large scale integration chips, etc.

Probably the "akwa kuta sona" who invented this and other construction rules [Gerdes, 1989c] knew why they were valid, i.e. they could prove in one way or another the truth of the theorems that these rules express.

On the other hand, an analysis of Tchokwe sona stimulates the development of new mathematical research areas.  


"not only in Angola, but also in other African countries, such as Ghana and the Congo, many adults and children know immediately how many lines are needed – you only have to show them the array. When they see a rectangular array of points of dimensions 4 by 6, they quickly say that 2 lines are needed to construct the lioness. When you ask them how many lines are needed to encompass in the same way all the points of an array of dimensions 5 by 7, they immediately respond that one line will be enough.


Oahspe The Lord's Fifth Book: Chapter VI:
29....and the seer sat therein, with a table before him, on which table sand was sprinkled. And the Lord
wrote in the sand, with his finger, the laws of heaven and earth.

The Mathematics of Sona:

Mathematical ideas that readily appear include:

Graph Theory: what is a graph, Euler’s Formula.

A particularly challenging question, which seems to have been investigated extensively, and experimentally, by the Chokwe is "What layouts of dots, or dots and walls, can give rise to one-line drawings, when following the "bouncing ball" rule above?" Mathematicians investigating these drawings have discovered a variety of theorems addressing parts of this question. The two most fundamental theorems discovered seem to have been known, in one form or another, to the Chokwe artists.


Network theory origins in graph theory.

Network theory was first introduce as “graph theory” by Euler. Rather than “links” and “nodes” he called them “vertices” and “edges.”

Network theory origins in Lusona? Recall that lusona, the African design practice of drawing graphs in the sand, had exactly the same constraints: a connected path that never re-traces. Could we also credit the Tchokwe with the origins of network theory?


Lusona is a Eulerian Path mandala [a geometric figure] that shows up in the traditions of African and Indian cultures.


The branch of mathematics that studies knots is known as knot theory, and has many simple relations to graph theory.

Knot (mathematics) - Wikipedia

Perhaps surprisingly, there will always be some knotted loop in a sufficiently complicated network. Such “sufficiently complicated” networks are called intrinsically knotted graphs. Very complicated graphs are always intrinsically knotted, and very simple graphs are always not, but graphs in between may be harder to identify

When is a Graph Knotted?

When graph theory meets knot theory Joel S. Foisy and Lewis D. Ludwig Abstract. Since the early 1980s, graph theory has been a favorite topic for undergraduate research due to its accessibility and breadth of applications. By the early 1990s, knot theory was recognized as another such area of mathematics, in large part due to C. Adams’ text, The Knot Book. In this paper, we discuss the intersection of these two fields and provide a survey of current work in this area, much of which involved undergraduates.

When graph theory meets knot theory - Denison University

Application of Graph Theory in real world


Chemical graph theory is a branch of mathematics which combines graph theory and chemistry. Graph theory is used to mathematically model molecules in order to gain insight into the physical properties of these chemical compounds. Some physical properties, such as the boiling point, are related to the geometric structure of the compound. This is especially true in the case of chemical compounds known as alkanes. Alkanes are organic compounds exclusively composed of carbon and hydrogen atoms.

The resulting representation of ethane is the carbon tree shown in Figure 2 . This carbon tree can be represented as a graph by replacing the carbon atoms with vertices. Chemical bonds are then represented as an edge in the graph.


Above rotating model of the diamond cubic crystal structure.  More images for diamond molecular model.

Above diamond (crystal carbon) molecular model is a network of dots and lines

made up of vertices, nodes, or points which are connected by edges


There are many useful applications to Euler circuits and paths. In mathematics, networks can be used to solve many difficult problems, like the Konigsberg Bridge problem. They can also be used to by mail carriers who want to have a route where they don't retrace any of their previous steps. Euler circuits and paths are also useful to painters, garbage collectors, airplane pilots and all world navigators, like you!


Applications of Graph Theory in Everyday Life and Technology

About Moses below:
doctrine of one Great Spirit, Whom they worshipped secretly under the name Jehovih.
Lost to African-Americans since the time of Christianity, slavery, and Islam in Sub-Saharan Africa is
the traditional belief in the Great Spirit and the mathematical knowledge of fractal geometry,
I would like for both to be rebuilt and shared with the rest of the world.

We need to break the shackles of Christianity and the Constantine Bible.


Slavery Timeline 1400-1500.


1441: Start of European slave trading in Africa. The Portuguese captains Antão Gonçalves and Nuno Tristão capture 12 Africans in Cabo Branco (modern Mauritania) and take them to Portugal as slaves.


8 August 1444: de Freitas lands 235 kidnapped and enslaved Africans in Lagos, the first large group of African slaves brought to Europe.


18 June 1452: Pope Nicholas V issues Dum Diversas, a bull authorising the Portuguese to reduce any non-Christians to the status of slaves.



1483: Diogo Cão discovers the Congo river. The region is later a major source of slaves.

8 January 1454: Pope Nicholas V issues Romanus Pontifex, a bull granting the Portuguese a perpetual monopoly in trade with Africa. Nevertheless, Spanish traders begin to bring slaves from Africa to Spain. 


22 April 1500: Pedro Cabral of Portugal discovers Brazil, landing at Porto Seguro, southern Bahia.


Slavery Timeline 1501-1600.


1502: Juan de Córdoba of Seville becomes the first merchant we can identify to send an African slave to the New World. Córdoba, like other merchants, is permitted by the Spanish authorities to send only one slave. Others send two or three.


1509: Columbus's son, Diego Cólon, becomes governor of the new Spanish empire in the Carribean. He soon complains that Native American slaves do not work hard enough.


22 January 1510: the start of the systematic transportation of African slaves to the New World: King Ferdinand of Spain authorises a shipment of 50 African slaves to be sent to Santo Domingo.


18 August 1518: in a significant escalation of the slave trade, Charles V grants his Flemish courtier Lorenzo de Gorrevod permission to import 4000 African slaves into New Spain. From this point onwards thousands of slaves are sent to the New World each year.


Maximum extension of the Viceroyalty of New Spain, with the incorporation of the Louisiana (1764 - 1803). In light green the territory not controlled effectively, but claimed as part of the Viceroyalty.
Maximum extension of the Viceroyalty of New Spain, with the incorporation of the Louisiana (1764 - 1803). In light green the territory not controlled effectively, but claimed as part of the Viceroyalty.
StatusViceroyalty of Spain
CapitalMexico City
Common languagesSpanish (official),NahuatlMayan,Indigenous languages,French (Spanish Louisiana), Philippine languages
ReligionRoman Catholicism

1500 to 2000 is 500 years.

Both the knowledge and the slave labour of the Israelites were now thus impressed into the construction of the temple of Osiris, the Great Pyramid. And the etherean dawn in which Moses would bring them deliverance was still five hundred years away.[2053 - 500 = 1553 B.C.E.]. - Pages 189-190 of Darkness, Dawn And Destiny (Drawn from Oahspe) 1965 by Augustine Cahill.


The original religion of the African people, however, was animism (the worship of spirits), and

there is no doubt that most of the slaves were animists.


an·i·mism animism

    2. the belief in a supernatural power that organizes and animates the material universe.


"a supernatural power that organizes and animates the material universe" = The Great Spirit (Jehovih).

Shamanism is the practice that involves a shaman reaching an altered state of consciousness in order

to meet up with and interact with the spirit world and channel mystical and supernatural energies

into this world, or this reality.

Shamanism shares the animistic worldview that non-human entities such as animals, plants, and

inanimate objects or phenomena, all possess a spiritual essence or soul.

Shamanism also shares the animistic belief in a supernatural power that organizes and animates

the material universe, namely – The Great Spirit or God.


Some Common Lakota Words and Terms

Taku Wakan: Wakan Relatives. (“Relatives” relates to all things in nature, not just to blood relatives or other humans.)


Oahspe God's First Book: Chapter XXVII:

42. From far over the water will I bring their kindred (Jews)[Israelites], who worship none born of woman;

a wise speaking people. A people that war not; who kill not off weaker nations. 

Testimony from the Bible and Thomas Paine and Oahspe:
The Bible: 1 Chronicles 4:40:
English Standard Version
"where they found rich, good pasture, and the land was very broad, quiet, and peaceful,
for the former inhabitants there belonged to Ham."
"The Managers of that Trade themselves, and others, testify, that many of these African nations 
inhabit fertile countries, are industrious farmers, enjoy plenty, and lived quietly, averse to 
war, before the Europeans debauched them
with liquors, and bribing them against one another; and 
that these inoffensive people are brought into slavery, by stealing them, tempting Kings to sell 
subjects, which they can have no right to do, and hiring one tribe to war against another, in 
order to catch prisoners." - African Slavery in America by Thomas Paine (1775).

Oahspe The Lord's Fifth Book: Chapter III:
13. The Lord said: Having designed Ham for teaching the barbarian world of me and my dominion,

Ham. Cosmological name of Egypt. The followers of Abram bestowed that country's name on him, after they settled there. One who is black with sunburn
The Bible refers to Egypt as "the land of Ham" in Psalms.78:51; 105:23,27; 106:22; 1 Chronicles 4:40.

Ham (son of Noah) - Wikipedia 

American Indians and Africans before slavery were shamanists and animists and believed in the Great Spirit.

Oahspe Book of Jehovih Chapter I:

2. He said, I Am! And He comprehended all things, the seen and the unseen. Nor is there aught in all the universe but what is part of Him.

3. He said, I am the soul of all; and the all that is seen is of My person and My body

The Message of The Divine Iliad, By WALTER RUSSEL:

 “For behold, said the Universal One, I am within all things centering them". 

The Babylonian geometric patterns in stone above show two-fold and four-fold symmetry
(up/down, left/right), they are NOT fractal.
Above is Clay panel with pattern of squares and triangles - Ishtar Gate from Babylon -
Pergamon museum in Berlin, Germany.

Fractals, Complexity, and Connectivity in Africa.

Page 102...a year of fieldwork in West  and Central Africa involving interviews with the artisans provide convincing evidence that they are better classified as intentional, conscious designs: that something analogous to fractal construction techniques – recursive scaling practices – exists as an African indigenous knowledge system.

page 106...an understanding of African mathematical representations, fractals, complexity and chaos in indigenous cultural practices ...


Using Sierpinski fractal to analyze and investigate pore distribution and heat transfer performance of metal foam.

Effect of Pore Distribution on Melting Behavior of Paraffin in Fractal Metal Foam*

An unsteady model of melting heat transfer in metal foam filled with paraffin is developed and numerically analyzed. In the model, the pore distribution of metal foam is described by Sierpinski fractal. By this fractal description, six types of metal foams with the identical porosity, different pore distributions are reconstructed. The effect of pores distribution on the heat transfer performance of the paraffin/copper foam composite is emphatically investigated. The solid-liquid distribution, the evolution of the melting front, the dynamic temperature response and the total melting time in these six fractal structures are all examined and analyzed. The results indicate that, the pores distribution has a significant impact on the melting heat transfer of the paraffin in the metal foam. When the first-level fractal pore is located near the heat source boundary, the thermal hysteresis effect of the paraffin on the solid matrix heat transfer is more significant. If the cross-sectional length of the pores along the normal direction of heat transfer is larger, the heat transfer barrier of the solid matrix is greater. It is favorable for the paraffin heat transfer when the pores specific surface area is larger.




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