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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,
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.-
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
OAHSPE GOD'S BOOK OF BEN CHAPTER VII:
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
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
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.
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
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.
fractal settlement patterns of Africa stand in sharp contrast
to the Cartesian grids of Euro-American settlements. - Page 39 of African Fractals.
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"
"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
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.
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
"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 ...
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 and sand drawings
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
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.
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
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
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
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:
...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,
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.
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
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
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,
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
1. The rational number 22/7 happens to be an excellent approximation to the number
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. http://cosmometry.net/phi-fractal-scaling
Cantor set and fractals
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
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.
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.
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
are fractals showing repetitive process, self-similarity, scaling, and infinity.
Several Ghanian iconic figures, ...link a spiritual force with the structure of living systems
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 ...
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,
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
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
Drawing on interviews
with African designers, artists, and scientists, Ron Eglash
fractals in African architecture, traditional hairstyling, textiles, sculpture, painting,
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
I buy it because I like math and geometry. I'm really fascinating by fractals applied
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
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
Fractals are used to model soil erosion and to analyze seismic patterns as well. Seeing
that so many facets of
exhibit fractal properties, maybe the whole world around us is a fractal
after all! Actually, the most
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
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.
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
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
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
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. 9 of the 10 top facial matches to Abraham have
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.
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
computer. The computer's calculations agreed with his intuition: He was
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
African culture. The complicated designs and surprisingly
complex mathematical processes involved in their creation may force
researchers and historians to rethink their assumptions
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.
of fractal geometry are offering scientists powerful
new tools for biomedical, geological and graphic applications.
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
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
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.
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
to be organized from the bottom up." This bottom-up, or
self-organized, plan contrasts with a top-down, or hierarchical,
which only a few people decide where all the houses will go.
"As it turns out, though, my reasoning was wrong," he said. "For
if you look at Native American architecture, you do not see
fractals. In fact, they're quite rare." Instead, Native
architecture is based on a combination of circular and square symmetry,
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
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
numerals' and the decimal system.
Eglash expanded on his work in
Africa after he won a Fulbright Grant in 1993.
central and western Africa, going as far north as the
Sahel, the area just south of the Sahara Desert, and as far south
equator. He visited seven countries in all.
"Basically I just toured
around looking for fractals, and when I
something that had a scaling geometry, I would ask the folks what
was going on-why they had made it that way,"
In many cases, however, Eglash found that step-by-step
were producing these designs, many of them surprisingly sophisticated.
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
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
In Senegal, Eglash learned about a fortune-telling system that
relies on a mathematical operation reminiscent of error
contemporary computer systems.
traditional Bamana fortune-telling ...The mathematical operation is called addition modulo 2,
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
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
"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
culture." 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
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
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
thus primitive. It never occurred to them that the Africans might have been using a form of
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
To remedy that oversight,
Eglash said he's been working with African-American math teachers in the
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
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
sophisticated mathematical concepts.
The valid question arises, in what ways can
an understanding of African
mathematical representations, fractals, complexity and chaos in indigenous
practices assist us in theorizing about the future?
In our opinion, probably,the greatest point that can be taken away from our
research Eglash (1999) into African mathematics is an appreciation of
creativity and quantitative ability. As Hull (1976) noted, large
urban centers were
disregarded by the colonialists because they did not utilize
typology. The complex fractal nature of these settlements went unappreciated.
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
continent may still think of their indigenous past as primitive and non-rational.
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.
social political structure of Africans vs the top-down colonial structure:
It is widely accepted in the
STS [science and technology studies] community
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
process, using self-organization rather than imposed order. Could
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;
indigenous mathematical representations can provide fresh thinking to a
relationship of fractals to pi:
Pi and fractal
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 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 .:
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.
http://www.pi314.net/eng/mandelbrot.phpThe 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.<-----Fractal geometry, <------"The Ocean Lives Within
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,
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.
part is a fractal of the whole part (both containing boundless infinity, self-similarity).
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
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.
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
unique feature of the Golden Ratio is that it can be written as an Equation which calls itself:
Phi = 1 + 1 / Phi
This Process is called Iteration.
Mathematical Iteration can also be used to generate a Fractal.
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
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).
this illustration below, every spiral is the same phi spiral
The binary number system is an alternative to the decimal (10-base) number system that we
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.
(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
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,
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.
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
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
Unlike Europe, India, and Arabic cultures, base 2 calculation is ubiquitous in Africa, even
multiplication and division. Doubling is a frequent theme in many other African knowledge systems,
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
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
libraries of knowledge; the algorithm itself is
just one drop in their recursively infinite
bucket." - Ron Eglash, November, 2017
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.
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
coils." - 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
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
to trigonometry (Eglash 1998a).
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,
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
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.
On the tree above, if you snapped a branch off it and stood it up, it would look like the
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.
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
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
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.
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.
How many branches are there at the 6th generation? [ ]
Above are shell and fern fractals, page 16 of African Fractals. -
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
will maybe revolutionize the way that the universe is seen.
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.
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
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
...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.
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
and that this presence extends further back than sites and artifacts in Greece.
(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.
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
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
sub-Saharan architectures the structures are not largely determined by quantitative
the Egyptian version thus provides a more formal version of the sub-Saharan
It is not unreasonable to speculate that Fibonacci brought the
sequence from North
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
concept embedded in both of their spatial designs—a phenomenon that has
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)
Susantha Goonatilake writes that the development of the Fibonacci sequence "is attributed in part to Pingala
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. 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 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
The Karnak Temple Complex, commonly known as Karnak (, from Arabic Ka-Ranak meaning "fortified village"), comprises
a vast mix of decayed temples, chapels, pylons, and other buildings in Egypt.
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
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
are symbols of Etherean worlds]. Look at the snowflakes as though they were microscopic patterns of the worlds in
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.
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.
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
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
expressed on an ordinary linear scale.
For those who may not be acquainted with logarithms, two forms are commonly used
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.
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.
Above: logarithmic scaling in Ghanaian design - African Fractals by Ron Eglash page 79.
-How a Galaxy is a Fibonacci Sequence: Physics & Math
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
of Great Pyramid logarithmic fractal slope:
2/3, 3/5, 5/8, 8/13, 13/21 ... = left/right,
vertical/ horizontal coordinates.
..... = 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. That is, a golden spiral gets wider (or further from its origin) by a factor of φ
for every quarter turn it makes.
"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
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.
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
Mysticism ....may also refer
to the attainment of insight in ultimate or hidden truths,
The science of fractals lies on
the edge of infinity.
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.
of the Arc of Bon 15:13 Moses grew and became
a large man, being a pure I'huan, copper-colored.
Above are Sub-Saharan African interlace knot symbols.
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
African music is characterised by a "strong rhythmic interest" that exhibits common characteristics in all regions of this vast territory,
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. 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.
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 topic. In 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)
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.
OAHSPE GOD'S BOOK OF BEN CHAPTER VII:
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
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 fern, http://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
set main bulb fractal (right).
Mandelbrot set ...
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).
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
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 ...
Fibonacci Numbers hidden in the Mandelbrot Set:
Set Fractal Zoom 10^227 [1080x1920]
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
Book of Lika, Son of Jehovih: Chapter VII:
etherean sea, moving brilliants playing kaleidoscopic views, ever changing the boundless scene with surpassing wonders.
of kaleidoscopic in English:
patterns of colours; multicoloured.
Made up of a complex
mix of elements; multifaceted.
The Mandelbrot fractals contain worlds without
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
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).
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
the Mandelbrot Set and the processes that guide the laws of nature." - The end of The Amazing
Mandelbrot Set tutorial video.
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,
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.
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
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.
incredible – and deeply perplexing – is currently taking place at the intersection of mathematics,
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
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.
"When life gives you an ancient cave filled with dirt, look for DNA. That's
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
DNA to be recovered directly from sediments without the
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.
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.
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 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
Abstract Madelbrot fractal zoom
and Centipede Chilopoda classification in nature.
Abstract Mandelbrot set zoom
and Pterodactyl Fossil body in physical nature.
Mandelbrot-Julia set hidden structure and White
tip shark in physical nature.
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
Physics (physical form) follows abstract
Is there a connection between the Mandelbrot Fractal iterated function (equation) of Z = Z squared and
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
inverse square law does not just apply to Newton’s universal law of gravitation it also applies
magnetic light" [electricity, magnetism, light and heat].
Why is almost everything squared in physics?²
between fractal geometry and physics (and biophysics, the application of the laws of physics
to biological phenomena).
A 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.
x2 = x + 1 ...this is precisely the equation
that defines the Golden Ratio! - Page 83 of
Golden Ratio: The Story of PHI, the World's Most Astonishing Number By Mario Livio.
If in the Mandelbrot set fractal equation
(function) you change z to x = x = x2 + c.
Change C to 1 = x = x2 + 1 (Mandelbrot Set fractal) very similar to x2 = x + 1 (Golden Ratio).
In the Mandelbrot set (X2 + C) c = 1 tends to infinity just as
central revolutions of a logarithmic or golden ratio spiral (below).
a characteristic of the All-Person (Creator) and
There are an infinite (infinity) number of
decimal places in the number Phi (Golden Ratio a fractal scaling ratio).
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/ https://csdt.rpi.edu/culture/legacy/african/African_Fractals/background4.html
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 said. THE PRESENT ERA. All knowledge in possession of man, embracing
corporeal and spiritual knowledge sufficiently proven. The Kosmon Bible is Oahspe.
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 interchange. Rhythmic 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.
of spiral galaxy and hurricane showing fractal self-similarity, scaling, and logarithmic spiral infinity.
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
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
"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)
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)
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
external from the sun centre.
Blue line above is input of the
circular recursive process, redline above is the output
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:
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 ...
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.
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).
= the repetition of a process, or repetition of a mathematical or computational procedure applied to the result of a previous
The first attempt to model the distribution of galaxies with a fractal pattern was made by Luciano Pietronero and his team in 1987,
Pietronero argues that
the universe shows a definite fractal aspect over a fairly wide range of scale,
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.
Yazdani Lab: Visualizing Quantum States of Matter
(Princeton University, Department of Physics).
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
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.
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).
"But the African example emphasizes fractal
symmetry, 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
Above and below is from pages 68-69 of African-Fractals by Ron Eglash.
Lusona geometry showing fractal scaling (1/3
to 3x) of diamonds within diamonds within diamonds.In graph theory,
an Eulerian trail (or Eulerian path) is a trail in a finite graph which visits every edge
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:
all homeomorphically irreducible trees of size n=10n=10." (ten valid trees).
• In this case “trees”
simply means graphs with no cycles
and more specifically in graph theory, a tree
is an undirected graph in which any two vertices are connected by
exactly one path.
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)."
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
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
Investigating the Mathematics of Sona: Sand Drawings from Angola.
 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.
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.
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.
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.
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.
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 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.
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.
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:
Lost to African-Americans
since the time of Christianity 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.
Any comments about Oahspe or this website? Email Me. I look forward to talking to you about Oahspe and this website.