This article will start a six-part series on the holographic universe. We have previously discussed this in Cosmic Core, and now we will go into more detail on these concepts.
Revisit articles 101 and 102 on Cosmometry for more information.
In this article we will start with a discussion of fractals, then we will cover the incredibly important geometric concepts of the golden ratio and Fibonacci sequence as they are the basis for fractality. We will then finish with a discussion of holography.
“Like the torus, fractals are a way of visualizing the creative process.” ~ Freddy Silva
Fractals are “a natural phenomenon or mathematical set that exhibits a repeating pattern that displays at every scale”.1
Visually, fractals are a never-ending pattern that can be zoomed in on infinitely.
Fractals have self-similarity on all levels. This is also called ‘expanding symmetry’ or ‘evolving symmetry’.
They appear to be complex and irregular but possess a recognizable statistical self-similarity.
They are a better description of a physical, observable world. You do not need a straight line, you can use a fractal curve.
Fractals are holographic. They stem from the phi ratio, which stems from the pentagon, which stems from the circle. Fractals can be seen in all aspects of nature.
Fractals led scientists to conclude that things that seem to be chaotic are actually organized.
It turns out; highly complex phenomena have a hidden order.
In theory fractals repeat to infinity. In the physical realm they are limited by space and form.
“Blood-vessels do not reduce indefinitely, any more than the whorls within whorls of the fractal cauliflower extend to infinity. Nature uses fractal geometry where it is advantageous.”2
Three Types of Fractals:
- Natural Fractals
- Geometric Fractals
- Algebraic Fractals
Fractal Foundation, found here: http://fractalfoundation.org/resources/lessons/ discusses these three types of fractals in detail.
Natural fractals are observed in nature. They are seen in such structures as mountains, rocks, pebbles, sand, clouds, lightening, rivers, trees, plants, leaves…etc.
They are seen in the branching and rooting of trees and the way river networks branch out.
They are also seen in Lichtenberg “lightning” which is formed by rapidly discharging electrons in lucite.
Credit: Paul’s Lab – https://www.flickr.com/photos/paulslab/
Interestingly, the Grand Canyon looks like a giant Lichtenberg lightning scar.
Here’s a view of the Grand Canyon from the International Space Station:
“The Grand Canyon is approximately 400 kilometers long, 28 kilometers wide, and almost two kilometers deep. Could erosion by the Colorado River be the only factor in its formation?
A few basic facts are necessary to gain a perspective. The Grand Canyon is surrounded by an elevated landscape with the canyon running through it from east to west. The underlying rock strata in the region rises and falls over an area known as the Kaibab Upwarp, while the river descends through an elevation differential of 2100 meters. Water does not flow up over a mountain range nor does it run sideways along sloping terrain, so all theoretical models that insist on water erosion propose that the entire area was slowly uplifted at the same rate as the river eroded the canyon. This process is said to have taken place in a time span of between four million and 400 million years.
The geological models also incorporate natural dams across the river channel that caused reversals in the river flow and were then subsequently breached, allowing the river to resume its previous course. However, a pertinent objection to that theory is that there is no evidence water flowed back into the ends of the giant side channels that join the chasm with the river. Perhaps the most significant challenge to the prevailing theories is the disappearance of almost 1300 cubic kilometers of material that is supposed to have been washed downstream—there is no large delta at the mouth of the Colorado River containing the debris.
Satellite images, as well as pictures taken by astronauts in orbit, seem to indicate that the Grand Canyon is an enormous Lichtenberg figure, in other words, a gigantic lightning scar. As the Electric Universe hypothesis suggests, electric discharge machining (EDM) might account for the Canyon’s appearance: steep walls, thousands of layers, brachiated side canyons at practically every scale, and periodic, hemispherical “nips” cut into each rim.”3
Fractal structures are also seen in the human body in: neurons in the brain, the nervous system, the cardiovascular system, the lymphatic system and branching of the lungs.
In the lungs the trachea branch to the bronchus to the bronchii to the alveoli in a similar fashion as a tree.
Both a tree and the lungs use their surface area to exchange oxygen and CO2.
These same fractal structures are also found in the bodies of other animals.
There are far too many examples of natural fractals to cover here. Just by going outside and taking a look around can endless examples be discovered in nature.
Spirals in Nature
The spiral is another common fractal formed by simple repetition, and combining expansion and rotation to generate its form.
Some examples include:
- Ammonite– a primitive mollusk from 300 million years ago, now extinct. “Their fossil shells usually take the form of planispirals, although there were some helically spiraled and nonspiraled forms.”4
- Plant kingdom – agave cactus, flowers, pineapples, fiddlehead fern…etc.
- Turbulent motion of fluids – soap, oceans, atmosphere…etc.
- Spiral galaxies
Branching patterns in nature are fairly complex systems generated from simple rules.
These are seen throughout nature. For example, lightning strikes resemble river systems resemble animal vascular systems.
All branching patterns involve efficient distribution of energy in one form or another.
They form the simplest way to connect every part of a given area using the shortest overall distance (or least work).
Any branch of a particular size is always outnumbered by those of the next smaller dimension.
Li symmetries are highly common in the natural world. They are related to fractal branching patterns.
They are self-organizing systems in nature caused by the interaction between processes and materials.
“They surround us and pervade the natural world, but it was only in the 1950s that these enigmatic forms of symmetry began to be understood as self-organizing systems through the pioneering work of Alan Turing. The Chinese, however, have been studying them for millennia, and it is from them that they get their name.”5
They are seen in:
- animal markings
- zebra, giraffe, cheetah, snakes, alligators and crocodiles, tropical cone-shell, frogs, lizards, cuttlefish, fish, leopard, ocelot, jaguar, eggshells of common murre and razorbill, seashell patterns, fish scales, section of bones, muscle fibers…
- goliath beetle, wing casing of grasshopper, wing veins, spider webs
- Irish Moss seaweed and other seaweeds, vascular cell structure, tissue-forming parenchymatous plant cells, leaf veins, lichens, gill patterns on mushrooms
- stretch patterns
- tree bark
- landscape patterns
- cloud patterns, Senegalese lagoon dried mud-flat pattern, topography of snowlines, drainage patterns of water and lava, shorelines
- wind over sand – ribbing of sand dunes
- flowing water over sand, silt or clay – ribbing of soil
- crack patterns in clay, ceramic, parched earth, old paints and gels
- ice tracery on windowpanes
- sand and silt patterns settling in water
- Fragmented liquid convection rolls
- Particulate clustering on a liquid medium
- Kerr magneto-optic effect in thin section of barium ferrite
- Magnetic maze-domain patterns in silicon-iron polished crystal
- Magnetic domain patterns
Credit: An inverse transition of magnetic domain patterns in ultrathin films, O. Pormann, A. Vaterlaus & D. Pescia, Nature volume 422, pages 701–704 (17 April 2003), https://www.nature.com/articles/nature01538
- mineral patterns
- agate, malachite, jasper, magnified surface of a diamond, brachiated components in serpentine rock, granite & marble
Geometric fractals are “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.”6
The Sierpinski Triangle is made by repeatedly removing the middle triangle from the prior generation.
The outer triangles repeat by a factor of 3 each step:
1, 3, 9, 27, 81, 243, 729…etc.
The Koch curve is made by repeatedly replacing each segment of a generator shape with a smaller copy of the same generator.
It is similar to a coastline. The length increases the more closely you measure it.
The five Platonic solids are geometric fractals. For instance, each Platonic solid can fit inside itself, and outside itself, perfectly fitting together. This progression can continue inwards and outwards, theoretically, to infinity.
The Platonic solids can also nest together as a set in many different ways. They are made to be able to transition between shapes and to maintain self-similarity on all scales.
Algebraic fractals are made by repeatedly calculating a simple equation over and over.
A famous example is the Mandelbrot Set.
The equation is as follows:
Znew= Zold2 + C
The equal mark is actually a double arrow. It is an Infinite Loop Equation – a system of continual feedback.
The Julia Set is another famous example.
The equation is as follows:
Z = Z2 + C
The exponent can be raised to Z3, Z4, Z5and so on.
The degree of symmetry always corresponds to the degree of the exponent.
Benoit Mandelbrot (1924-2010) was a Polish/French/American mathematician who coined the word ‘fractal’ and discovered the Mandelbrot Set.
He said that things typically considered to be rough, a mess, or chaotic, like clouds or shorelines, actually had a “degree of order.”
Mandelbrot had a 35-year career at IBM and often taught at Harvard. He was hired by AT&T to analyze interference signal patterns. He found a graph for how the electricity was getting stopped.
The pattern he found was the Mandelbrot Set.
Different parts of the Mandelbrot set have different fractal patterns.
The Mandelbrot Set is a visual representation of a simple equation: Znew= Zold2 + C.
The letters stand for numbers. The numbers have both a real and imaginary component.
“You aren’t going to spot the Mandelbrot set out in Nature the way you find the logarithmic spiral in the nautilus shell of the Fibonacci spiral in galaxies, because the Mandelbrot set lies in the complex plane, which isn’t a part of the ordinary world we live in.”7
The Mandelbrot numbers are coordinates – positions on the plane for finding the location of a specific point.
“This series is generated for every initial point Z on some partition of the complex plane. To draw an image on a computer screen the point under consideration is colored depending on the behavior of the series which will act in one of the following ways”8:
- It will decay to zero.
- It will tend to infinity.
- It will oscillate among a number of states.
- It will exhibit no discernible pattern.
Mandelbrot commented on his work, “The form of geometry I increasingly favored is the oldest, most concrete, and most inclusive, specifically empowered by the eye and helped by the hand and, today, also by the computer…bringing an element of unity to the worlds of knowing and feeling…and unwittingly, as a bonus, for the purpose of creating beauty.”
His fractal work contributed to statistical physics, meteorology, hydrology, geomorphology, anatomy, taxonomy, neurology, linguistics, information technology, computer graphics, economics, geology, medicine, physical cosmology, engineering, chaos theory, econo-physics, metallurgy and the social sciences.9
How the Mandelbrot Set Works
Using the equation Znew= Zold2 + C:
Plug a value for C into the equation and receive an answer = Znew.
Plug the answer in as Zold.
Repeat this process.
Most values of C get bigger due to the squaring.
Some, however, do not – they get smaller or alternate between a set of fixed values.
These are the points inside that are colored black.
The colors are proportional to the speed at which they expand.
Explanation of the Mandelbrot Set
“When you zoom in on a piece of the Mandelbrot set, you realize that that piece contains, and consists of, another Mandelbrot set. Zoom in again, and you see that that piece also contains and consists of another Mandelbrot set. Zoom in again. Same thing. In fact, you can zoom in forever and you will always see more Mandelbrot sets!
The Mandelbrot set doesn’t iterate over these simple numbers. Instead it iterates over complex numbers.
Complex numbers come in two parts: a real part and an imaginary part.
The real part is easy to grasp. They are regular numbers that you know and love: 1, 0, -5, 4.534343, 232423432.4787865, -0.0000000000002, etc.
The imaginary part of a complex number is a real number (like above) multiplied by a unique little number called i. [i = Square root of -1]
For the Mandelbrot set, we start Z at zero…but our choice for C will be a complex number. Our exact choice for C is what determines the Mandelbrot set. The Mandelbrot set consists of all the choices for C we can find (where Z starts at zero and C is a complex number) so that the iterations never grow beyond the number 2.
That is the mathematical definition of the Mandelbrot set.”10
The Mandelbulb is a 3D manifestation of the Mandelbrot Set discovered by Daniel White and Paul Nylander in 2009.
They used a spherical coordinate system and ingenious math to discover the Mandelbulb. This is discussed and illustrated on their website http://mandelbulb.com/.
The 3D Mandelbulb starts as a sphere. It iterates outward from there to form the Mandelbulb in its various complexities.
Interestingly, the Platonic solids can be found in the 3D Mandelbrot set. You can play connect the dots with the bumps on the shape and get the Platonic solids.
The above image shows the ancient Scottish solids: spherical versions of the five Platonic solids that can be found in the Mandelbulb.
Nested Platonic solids fractalize out to create the spheres within spheres (nested spheres) that create the 3D Mandelbulb.
This shows how the Mandelbrot Set actually describes a 3D spherical structure. The Platonic solids as close-packed spheres is the big secret of the Mandelbrot/Mandelbulb fractal.
Above are close-packed spherical versions of the Platonic solids: the tetrahedron, octahedron, cube, icosahedron and dodecahedron. Each of these are found in the Mandelbulb in a nested formation.
The Mandelbrot/Mandelbulb fractal is an illustration of the fractalized cosmos.
From the side the Mandelbulb looks like a series of close-packed spheres. Going inside the structure you find the fractalized geometry – the tendrils of the universe.
Zooming in on the Mandelbrot set or Mandelbulb is like flying through the Universe to see all the fractal branching of moons, planets, solar systems, stars, galaxies, galaxy clusters and voids.
Take a look at the painting ‘God as Architect/Builder/Craftsman’ from the frontispiece of Bible Moralisee from France circa 1220-1230.
“God has created the universe after geometric and harmonic principles, to seek these principles was therefore to seek and worship God.”
Notice how similar the construction of the Universe resembles a Mandelbrot Set!
Wikipedia tells us, “the Buddhabrot is a fractal rendering technique related to the Mandelbrot set. Its name reflects its pareiodolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehead mark (tikka) and traditional topknot (ushnisha).” It was discovered by Melinda Green.
We have already seen that Platonic solids can be made out of close-packed spheres. This is illustrated at D’Source: http://www.dsource.in/course/geometry-design/concepts-3-dimensional and in Keith Critchlow’s brilliant book Order in Space.
Platonic Solids as Spheres
Each Platonic Solid can be illustrated as close-packed spheres:
- tetrahedron – 4 spheres
- octahedron – 6 spheres
- cube – 14 spheres
- dodecahedron – 32 spheres
- icosahedron – 32 spheres
- cuboctahedron (a.k.a. vector equilibrium – an Archimedean solid) – 13 spheres (12-around-1)
The Platonic Solids faces can fractalize out in infinite iterations until they create perfect spheres.
Above is an example of the tetrahedron fractilizing out into the star tetrahedron, then the isotropic vector matrix, and then yet another more complicated version of the IVM. If we were to keep fractilizing this shape outwards it would eventually form a perfect sphere.
The Platonic Solids are fractal structures and the Platonic solids and the Mandelbrot set are inextricably connected.
Phi – The Golden Proportion – Φ
The Phi Golden Ratio is an extremely important concept to understand in mathematics and geometry. It is discussed in great detail in other articles in the Number/Geometry section.
It is a three-term proportion constructed from two terms.
Phi is a proportion, not a number, “a proportion,” as Robert Lawlor writes, “upon which the experience of knowledge (logos) is founded.”
The golden ratio is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion.
This results in a continuous geometric proportion.
Whole = Large Part = Φ
Large part small part
It is also described as the proportion a : b :: b : (a+b) or approximately 1.618.
It demonstrates that Number is above all a relationship, not a quantity.
Symbolically the golden ratio reveals how we can be a ‘part of the whole’ and ‘the whole’ simultaneously; how we can be finite and Infinite at the same time.
Φ = 1.618033988749894848204586834365638117720309180…
φ = 0.61803398875… (pronounced ‘fee’) φ = 1/2(√5-1)
Φ = 1.61803398875… (pronounced ‘fi’) Φ = 1/2(√5+1)
The Fibonacci Sequence
The Fibonacci series is a self-generating number series that is:
Additive: each number is the sum of the previous two
Multiplicative: each number approximates the previous number multiplied by the golden section
It begins with zero and unity, 0 & 1, nothing and everything, the Unknowable and the manifest Monad, time/space and space/time.
It begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…
“The series grows by accruing terms that come from within itself, from its immediate past, taking nothing from outside the sequence for its growth.”11
Each number is the sum of the two preceding:
- 1 + 1 = 2
- 1 + 2 = 3
- 2 + 3 = 5
- 3 + 5 = 8
- 5 + 8 = 13
- 8 + 13 = 21…etc.
Fibonacci & Phi
When you divide any pair of adjacent numbers in the sequence the result achieves an increasing accuracy to phi the farther out in the sequence you go.
In a sense, the Fibonacci numbers oscillate around the golden ratio, never quite reaching it, but getting closer and closer the farther out you go.
8/5 = 1.6
55/34 = 1.617647…
144/89 = 1.617977…
You can see in the graph above how the ratio bounces above and below the golden ratio, eventually getting closer and closer, but never quite reaching it…
Fractal Scaling Ratios
Octave Doubling is found in the primary structure of the Aether.
It is found in the isotropic vector matrix (tessellating tetrahedra & octahedra) and the cosmic octave hierarchy.
It is found in the structure of music and the electromagnetic spectrum.
It is found in the 64-based (8-octave) nature of binary digital systems.
It is also found in the octave-based growth of the chemical periodic table of elements.
Phi expansion and contraction is found extensively in nature, in both structural and flow forms.
Phi Boundary Condition
The Phi Boundary Condition is the phenomenon of a circle being created when a phi spiral is “spun” around a center point.
The unexpected synergetic quality of the whole emerges as a “boundary” defining an inside and an outside of the resulting field pattern.
Dan Winter describes fractality as the ultimate or perfected state of coherence.
Coherence is the difference between a flashlight (low coherence) and a laser (high coherence).
Coherence is when all the soldiers marching across the bridge are in step.
Coherence is when the different harmonics within a complex wave (like a heartbeat) are musically locked into phase.
Coherence is the only way waves of an infinite number of different wavelengths (harmonic diversity or inclusiveness) can be all locked together without destroying one another.
The Golden Ratio & Fractality
Dan Winter goes on to teach that the golden ratio is the ultimate state of fractality or self-similarity.
It defines the state of the “inside that looks like the outside.” (Metaphysically this manifests as compassion.)
“Although you are divided, you are still connected.”
Or as Henry Miller wrote, “We are of one flesh, but separated like stars.”
A golden mean spiral is the only equi-angular curve in which a wave can re-enter itself without hurting itself.
The wave can ‘eat its own tail’. This is called ‘self-reentry’. Symbolically this is seen from ancient times as the ouroborus – the serpent eating its own tail.
This is a picture of recursion. Recursiveness means something happens inside itself.
“Light when folded back on itself, comes to know itself.”
Holographic means the whole is present everywhere.
The entire cosmos is contained at every point in the cosmos.
All points in space and time are connected.
Dennis Gabor was a Nobel Prize winner in the 1940s for his discovery of holography.
He was the first engineer to win the Nobel Prize in physics. He had been working on the mathematics of light rays and wavelengths when he made his discovery.
He discovered: “If you split a light beam, photograph objects with it, and store this information as wave interference patterns you could get a better image of the whole than you could with the flat 2D you get by recording point-to-point intensity, the method used in ordinary photography”.12
Holograms: Fractals in Action
A hologram is made by shining two beams of coherent laser light onto a glass plate coated with a light-sensitive emulsion. One beam is the unchanged original laser light (the reference beam). The second beam bounces off an object, then onto the plate (the object beam).
It is the interference pattern of these two beams that make a holographic image.
“It’s likely not possible to have such a thing as a hologram exist within a non-holographic universe. The simple fact that we have defined and proven the science of holographic interference patterns in light means that the universe must be holographic, otherwise we couldn’t make something so fundamental happen that is outside the actual nature of the cosmos itself.”13
This is well worth contemplation.
Components of a Hologram:
- A single laser split into two separate beams
- A beam splitter
- A mirror. This creates the interference patterns.
- A film plate to record the image. The interference pattern is recorded on the holographic film plate.
Interference patterns are crisscrossing patterns that occur when two or more waves ripple through each other. They can be constructive interference or destructive interference, the two waves superposing to forming a wave of greater, lower, or the same amplitude.
Phase – the point the wave is at on its oscillating journey
Amplitude – a measure of the waves change over a single period – basically the height of the wave
Frequency – the number of occurrences of a repeating event per unit time
Constructive Interference: “If a crest of a wave meets a crest of another wave of the same frequency at the same point, then the magnitude of the displacement is the sum of the individual magnitudes.”
Destructive Interference: “If a crest of one wave meets a trough of another wave then the magnitude of the displacements is equal to the difference in the individual magnitudes.”
Holography is shorthand for wave interference (the language of phase, amplitude and frequency).
Interference patterns account for a constant accumulation of information.
Waves have a virtually infinite capacity for storage – far more than the 280 quintillion bits of information supposedly constituting the average human memory through an average lifespan.
Light from every point on the object is captured at every point on the holographic plate.
The holographic image will be 3-D.
You get self-similarity at all levels. If you cut the plate in small pieces and hit it with a laser beam you will still get the full image, though the image will get dimmer. All of the information in the hologram stays consistent.
Holograms possess a huge capacity for information storage.
A 1 square inch of film can store the amount of info contained in 50 Bibles.
Holography uses Fourier transforms.
Jean B.J. Fourier
Jean B.J. Fourier was an 18th century French mathematician who developed a mathematical way to convert any pattern, no matter how complex, into a language of simple waves.
He called these Fourier transforms. They are equations used to convert images into wave forms and back again.
The image above shows a simple Fourier transform of a rectangle at two different positions.
Russell and Karen DeValoisare Berkeley neurophysiologists who published their findings about Fourier Transforms in their 1979 book Spatial Vision.
They found that brain cells did not respond to original patterns, but to the Fourier translations.
“The brain was using Fourier mathematics – the same mathematics holography uses – to convert visual images into the Fourier language of wave forms.”14
Fourier was able to break down and precisely describe patterns of any complexity into a mathematical language describing the relationships between quantum waves.
Any optical image could be converted into the mathematical equivalent of interference patterns, the information that results when waves superimpose on each other.
In this technique, you also transfer something that exists in time and space into ‘the spectral domain’. The ‘spectral domain’ equates to metaphysical time/space. It is a kind of timeless, spaceless shorthand for the relationship between waves, measured as energy.
A Sawtooth Fourier Analysis. The upper graph is translated into a series of waveforms seen in the bottom graph.
You can also do the equations in reverse.
In this article we have taken a slightly deeper look at the concept of fractals. Understanding fractals and holography is extremely important in understanding how a universe that appears to be concrete and physical can be a fractal-holographic universe with self-similarity on all scales.
Understanding the golden ratio is another important aspect in understanding how fractality works.
It is interesting here to note how the golden ratio is embedded within the proportions of an equilateral triangle in a circle, and a tetrahedron in a sphere.
If photons are spiraling tetrahedra, then they will automatically create the golden proportion in every movement they make. Spiraling light is of the golden proportion. The inward/outward opposing yet harmonious pressures of the Aether – gravity and radiation – are embedded within the golden proportion.
The golden proportion is embedded within light itself, and as light is the fundamental constituent of matter, the golden proportion is inherent in all matter and life.
At its core, the golden ratio is a fractal-holographic process – one of the fundamental processes of a fractal-holographic universe and the way infinite information storage takes place in the universe.
We will continue our discussion of the holographic universe over the course of the next five articles.
- Wade, David, Symmetry: The Ordering Principle, Walker Publishing Company Inc, 2006
- Armstrong, Michael, The Grand Canyon: Part One, 29 September 2008, https://www.thunderbolts.info/tpod/2008/arch08/080929grandcanyon.htm
- Lundy, Miranda, Sacred Geometry, Wooden Books, 2001
- Bourke, Paul, An Introduction to Fractals, May 1991, http://paulbourke.net/fractals/fracintro/
- Francis, David (Bowdoin College’s web programmer), The Fractal is a Beautiful Gift to Mathematicians and Stoners Alike, Popular Science, 20 April 2016, http://www.popsci.com/fractal-is-beautiful-gift-to-mathematicians-and-stoners-alike
- Schneider, Michael, A Beginner’s Guide to Constructing the Universe, HarperCollins Publishers, 1994
- McTaggart, Lynn, The Field, HarperCollins Publishers, 2001
- Lefferts, Marshall, Fractal Holographic Synergetic Universe, http://www.cosmometry.net/fractal-holographic-synergetic-universe
- Talbot, Michael, The Holographic Universe, Harper Perennial, 1991