How do Fractals Contribute to Beauty?

by JagmOHaN Singh Sekhon June 29, 2021

What do mountains, broccoli and the stock market have in common? The answer to that question may best be explained by fractals, the branch of geometry that explains irregular shapes and processes, ranging from the zigs and zags of coastline to Wall Street market risk.
Fractal is a new branch of mathematics and art. Perhaps this is the reason why most people recognize fractals only as pretty pictures useful as backgrounds on the computer screen or original postcard patterns. But what are they really?
With computers, we can generate beautiful art from complex numbers. These designs are called fractals. Fractals are produced using an iteration process. This is where we start with a number and then feed it into a formula. We get a result and feed this result back into the formula, getting another result. And so on and so on Fractals start with a complex number. Each complex number produced gives a value for each pixel on the screen. The higher the number of iterations, the better the quality of the image.
For the most part, when the word fractal is mentioned, you immediately think of the stunning pictures you have seen that were called fractals. But just what exactly is a fractal? Basically, it is a rough geometric figure that has two properties: First, most magnified images of fractals are essentially indistinguishable from the unmagnified version. This property of invariance under a change of scale if called self-similarity. Second, fractals have fractal dimensions.
Fractal geometry is an extension of classical geometry. It can be used to make precise models of physical structures from ferns to galaxies. Nature is rough, and until very recently this roughness was impossible to measure. The discovery of fractal geometry has made it possible to mathematically explore the kinds of rough irregularities that exist in nature.
Fractal geometry has permeated many area of science, such as astrophysics, biological sciences, and has become one of the most important techniques in computer graphics. For example, fractal patterns have appeared in almost all of the physiological processes within our bodies.
While Mandelbrot will always be known for his discovery of fractal geometry, he should also be recognized for bridging the gap between art and mathematics, and showing that these two worlds are not mutually exclusive.
Because fractal geometry is relatively new — the term was coined in 1975 by the late Benoit Mandelbrot, — it is a concept not well understood by a portion of the population.
Casey Donoven, one of Montana State University’s newest recipients of the prestigious Goldwater Scholarship for excellence in science and math, uses fractals in his research to understand variations in heartbeats.
Donoven, who hails from a family farm outside Kremlin-Gilford, first learned about fractals while a student at Havre High School. At MSU, he has been studying under math professor Lukas Geyer.

What is a fractal?

A fractal is a geometric pattern that repeats at every level of magnification. Another way to explain it might be to use Mandelbrot’s own definition that “a fractal is a geometric shape that can be separated into parts, each of which is a reduced-scale version of the whole.” Think of Russian nesting dolls.
Fractals are common in nature and are found nearly everywhere. An example is broccoli. Every branch of broccoli looks just like its parent stalk. The surface of the lining of your lungs has a fractal pattern that allows for more oxygen to be absorbed. Such complex real-world processes can be expressed in equations through fractal geometry. Even to the everyday person, fractals are generally neat to look at even if you don’t understand what a fractal is. But to a mathematician, it is a neat, neat subject area.

Why are fractals important?

Fractals help us study and understand important scientific concepts, such as the way bacteria grow, patterns in freezing water (snowflakes) and brain waves, for example. Their formulas have made possible many scientific breakthroughs. Wireless cell phone antennas use a fractal pattern to pick up the signals better, and pick up a wider range of signals, rather than a simple antenna. Anything with a rhythm or pattern has a chance of being very fractal-like.

Why do we hear so much about fractals now?

Actually, some fractals were understood long before Mandelbrot coined the term. He popularized the concept with computer graphics and pictures of fractal patterns in nature. While, the Mandelbrot set and Julia sets (two well-known fractals) were investigated in the early 20th century, they never left the mathematical/physical “ghetto” until fast computers and good computer graphics came along, which in turn led to a wave of new research and better understanding.
Fractal geometry is characterized by self similarity and endless iteration. Using dance, the human experiences the fractal as a simple form that varies by repeating an infinite range of combinations. Using this experimental process, dance interrelates all the human systems and has the outcome of simplifying the entire experience in a melodic form of metaphor.
Dance is a friendly entry point for using the fractal. Actually dance is already built on the fractal with its full experience in every direction, every mood, every dynamic, every rhythm.
Dance is defined as the principles that were selected through evolution and hard wired along with music into the earliest humans. Dance movement is similar to music which uses only twelve notes and is infinitely able to express in melodic statements the unity of a person, a generation, a people.
The artistic process like the scientific process is built on simple elements that are combined into an infinite number of experiments. Dance deserves our focus because it uniquely interconnects every human system physical, personal, motivational. Dance goes beyond mechanics to the quality of movement and to the interrelationship of emotion and the building of the self.
Fractal concepts are applied in dance by the use of pattern, interconnecting signals throughout the entire person through perception, rhythm, timber, and melody.
Pattern are added to pattern down to the smallest size and to the slowest speed. Basic shapes are triangles and spirals.
Signals are differentiated through the nervous and limbic systems. Signals initiated from specific areas of the spine define movement qualities and dynamics of mood and emotion.
Quality is defined as variations of initiation and extension giving concepts such as sustained or staccato, flicking and floating.
Dynamics is a use of increase and decrease, emphasis to initiate force or ease.
Anticipation precedes initiation with either arc or continuing line. The variations interrelate the processes of perception, integration of new forms as they emerge, and the process of metaphor to simplify and express the building of the self and the community.
Rhythms are constructed of sound and silence. Rhythms are visualized with shape using a binary mode made of short and long durations.
Timber is identified with imagery of sensation, color and taste.
Melody is the formation of a metaphor to represent a person, a people.
Perception differentiates incoming signals and builds awareness and interconnection of each human system. Perception may begin with the eyes, ears, nose, taste, skin, then progress to the sense of chemical and electrical modulations within the body, then may become interrelated to the processes of anticipating and initiating transformations and finally to the building of the self through metaphor. Perception may be within a range of both familiar or unfamiliar experience. All experiences are integrated through fractal processes of identifying simple forms, varying these forms in infinite repetition of combinations.

Fractality Relationships

Ancient texts and traditions refer to a canon of numbers, geometry, and measurements that were enshrined in ancient works of stone and represented the cosmos. Many allude to the ubiquitous presence of the golden section throughout nature as evidence for divine geometry. As far back as Greek Mystery schools 2500 years ago, we learn that the ancients were taught that there are five perfect 3-dimensional forms -The tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron.
Collectively these are known as The Platonic Solids — and are the foundation of everything in the physical world. Modern scholars ridiculed this idea until the 1980’s, when Professor Robert Moon at the University of Chicago demonstrated that the entire Periodic Table of Elements — literally everything in the physical world — is based on these same five forms! In fact, throughout modern Physics, Chemistry, and Biology, the sacred geometric patterns of creation are being rediscovered, but a unifying matrix of these as related to the cosmos as a whole has yet to be derived.
Attempts have been made with the study of ancient works, earth grid schemes, harmonics, and astronomical cycles and have produced models of the planets, plants, and other living forms as well as nuclear and electron physics shell models, but all fall short in finding metrics that will apply to the cosmos as a whole. This is a first attempt to define a cosmic matrix that has the potential of unifying our knowledge from the physical to the spiritual levels of reality.

Introduction:

We begin our examination of nature using the golden section, also known as PHI, the golden mean, and the golden proportion which we find ever present in the fractalality of nature. The self-similar repetition of a proportion using the golden mean or the Fibonacci sequence of numbers is found in physical constants, plants, animals, man, and ancient architecture. As we investigate the ubiquitous presence of the golden section in nature, we are impressed with the idea that it is an essential and integral part of the Cosmic Matrix.
The Golden Section is a ratio based on a phi
The Golden Section is also known as the Golden Mean, Golden Ratio and Divine Proportion. It is a ratio or proportion defined by the number Phi ( = 1.618033988749895… )
It can be derived with a number of geometric constructions, each of which divides a line segment at the unique point where:
the ratio of the whole line (A) to the large segment (B)
is the same as
In other words, A is to B as B is to C.
This occurs only where A is 1.618… times B and B is 1.618… times C.1
Leonardo Fibonacci discovered the series which converges on phi
In the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical relationship behind phi.
Starting with 0 and 1, each new number in the series is simply the sum of the two before it.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
The ratio of each successive pair of numbers in the series approximates phi (1.618. . .) , as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60.
The table below shows how the ratios of the successive numbers in the Fibonacci series quickly converge on Phi. After the 40th number in the series, the ratio is accurate to 15 decimal places.
1.618033988749895 . . .
Compute any number in the Fibonacci Series easily!
You can use phi to compute the nth number in the Fibonacci series (fn):
fn = Phi n / 5½
(This provides an estimate which always rounds to the correct Fibonacci number.)
The DNA spiral is a Golden Section
The DNA molecule, the program for all life, is based on the golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.
34 and 21, of course, are numbers in the Fibonacci series and their ratio, 1.6190476 closely approximates phi, 1.6180339.
Living organisms grow and growth can be shown to evolve in fractal cycles. The golden section produces growth in form with self-similarity of form from one stage to the next stage. A gnomon is a portion of a figure which has been added to another figure so that the whole is of the same shape as the smaller figure (Huntley, 1970,169). The concept of the gnomon illustrates the fact that the ratio represented by the spiral constant is the only growth ratio by which a new unit can grow in proportion to the old unit and still retain the same shape. This is exactly the process of growth in a spiral shell. As noted by D’Arcy Thompson (1961, 179):
In the growth of a shell we can conceive no simpler law than this, namely that it shall widen and lengthen in the same unvarying proportions: and this simplest of laws is that which Nature tends to follow. The shell, like the creature within it, grows in size but does not change its shape; and the existence of this constant relativity of growth, or constant similarity of form, is of the essence. and may be made the basis of a definition, of the equiangular spiral.
B-DNA has spirals in phi proportions
Insight on B-DNA proportions .
The DNA cross-section is based on Phi
A cross-sectional view from the top of the DNA double helix forms a decagon:
A decagon is in essence two pentagons, with one rotated by 36 degrees from the other, so each spiral of the double helix must trace out the shape of a pentagon.
The ratio of the diagonal of a pentagon to its side is Phi to 1. So, no matter which way you look at it, even in its smallest element, DNA, and life, is constructed using phi and the golden section!
The Phi-cycle of the Planets:
Even the orbital radii of the planets are related to PHI as is the Astronomical Unit, the radius distance from the Sun to the Earth. We will look at that in a moment, but the purpose of this research is to find the metric that will unify the physics of the cosmos.
Mercury is the starting point.
Mercury = 1
Perihelion = 2- ½ (Ö2+1) = 0.792893
Aphelion = ½ (Ö2+1) = 1.207107
Start with a circle for Mercury’s mean orbital distance, and then construct a square with sides of the same length. The midpoint between the radius of the circle and the diagonal of the square is Mercury’s aphelion (A), the outside point of its orbit. The same distance towards the Sun is Mercury’s perihelion (P), the inside point of its orbit.
John N. Harris, M.A. has done an amazing re-analysis of the solar system based on logarhythmic and phi-based proportions. His log spiral form with phi-proportions is illustrated beautifully in the following graphic.2
This new work fits perfectly with the work I started 28 years ago that I recorded in three notebooks filled with calculations in order to find a universal metric that tied everything together. We will soon see the same type of geometrical framework applied to the atomic world find that it is only a matter of scale and that the universe is self-similar in all of its parts and appears like a divine plan, a cosmic program, or geometrical construction of the highest order.
The Speed of Light
Viewing the planetary distances, another relationship emerges that suggests a deeper significance to the velocity of light with respect to planetary orbits. Taking the value of PHI as 1.6180339 and the orbital distance to Mercury in English miles and converting the speed of light to miles per minute, we get the following calculation:
PHI x 2 = 3.236678
Mercury’s mean distance from sun = 36,270,000 miles
Md/c in minutes = 3.24 minutes
Setting Mercury to 1 so that Mercury becomes the new standard of astronomical units, i.e. sun to Mercury = 1 AU, we then calculate the relative distances to outer planets.
Venus = 1.86859426 Au’s
1.86859426 x 3.24 = 6.055 and Vd/c = 6.0637 minutes
Earth = 2.583325718 x 3.24 = 8.3699 and Ed/c = 8.38308 minutes
Mars = 3.936156024 x 3.24 = 12.753 and Md/c = 12.7731
Considering margins of error in these planetary ratios, there is a fairly close match between figures to indicate that the speed of light and the golden section are related to the harmonics of the universe. 3

AU Ratios Expressed in Miles:

The Golden Ratio and the fine structure constant:

The following calculations were supplied to me by theoretical physicist Tony Bermanseder from his study of the dimensions in the Great Pyramid of Cheops. There are some who feel that the dimensions and ratios of the pyramid represent a record of an advanced science in stone.
Tony is the author of QR (Quantum Relativity) Theory and has worked out the mathematics that can unify the Quantum Theory with the Theory of Relativity. Please have patience with the mathematical calculations as in further sections I am going to bring more geometry and calculations to demonstrate our universe is indeed a cosmos and reveals an underlying design.
We proceed to closer examine the seeming arbitrariness of J. Iuliano’s work to derive those seemingly ad hoc approximations to the face-pyramid angle and a possible connection between the transcendental p and the fine structure constant Alpha.
We apply our own process of QR’s ‘reverse engineering’ to show the fine structure connection previously discussed in the Alpha-Variation paper.
I refer all readers to that paper for the background data.
Much of Quantum Relativity and its superbrane parameters is based on the fine structuring of fundamental constants, such as Planck’s Constant h=lps/2Re.c³ and Re=10¹⁰.lps/360 as the classical Electron Radius, related to the Compton Radius in the proportionality given by Alpha as Alpha=Re/Rcompton, Rcompton relating the de Broglie matter wave momentum via p=h/l.
The basic QR parameter is the magnetic permeability constant mo=120p/c defining Maxwell’s Constant moeo=1/c² in dimensionless (say superconductive form for the Action-Law of Action=Charge² and for unitary resistance in the Quantum Hall effect, Josephson Junctions and the Conductance Quantum 2e²/h).
The fourfoldedness of the Unified Field of QR (UFoQR) also defines mo=[8p/360][Ne*/Re], where N is a counter for the magnetocharge e*=lps/hc=2Re.c².
This allows determination of N=2700/c³ in unified action units and independent of experimental measurement.
Because N=2700.e/{[ec].c²}=2700.{e/c²}/[ec], with [ec] the unified magnetic monopole-mass in superbrane space and the Planck-Length LPdefining that superbrane space via the Planck-Length-Oscillation: OLP={e/c²}=LPSqrt(Alpha); N MUST be the decisive and precise wormhole-perimeter lps=2700/(27x10²⁴) as the Weyl-Geodesic assuming its linearised form at this threshold between the metricated field equations of General relativity and its nonmetricated encompassement and quantum substructure described by QR.
This linearization of superstring class I as the Planck-Boson so is physically-metrically manifested in the heterotic superstring class HE(8x8) as the Ng Van Dam macroquantisation of the Planck-Scale then further amplified by the Electron Radius and its associations with the atomic and subatomic realms of the Compton-de Broglie scenarios.
So this linearization from the preBig Bang (preinstanton and preinflaton) epochs of the circularized ‘higher’ superstring dimensions unfolded the scale-invariance of angular displacement.
This unfolding then crystallizes the 360 factor as a degree-measure, which is NOT arbitrary, but associated with the QR definition for the approximation for the transcendental number p, defining circularity.
This we shall now discuss in some detail.
Draw a circle radius R for x-axis diameter given by coordinates (-R,0) & (R,0) and y-axis diameter given by coordinates (0,-R) & (0,R).
Draw the line segment (R,0)-(0,R) with length Sqrt[2].R and midpoint (R/2,R/2) for the half segment length RR1*=Sqrt[2].R/2.
Now project this midpoint radially to the circle’s perimeter as point R1 to form the angle (45+45/2)=67.5 degrees=ORR1.
This defines the half angle f1=45/2 degrees as the angle R1*RR1 in cosf1=RR1*/RR1 and sin(2f1) =RR1*/OR=RR1* for the unit square OR=1=unit radius.
So sin(2f1)=RR1.cosf1.
Generally, a polygonal line-segment keeps doubling from a square with 4 sides to an octagon with 8 to a hexadecagon with 16 and so on.
The generalized length for the line-segment is R.Rn=sin[2fn]/cos[fn].
The perimeter for the circle then sums the line segments in n and approaches 2p as n gets larger and larger.
Archimedean Circumscription/Inscription approximation for transcendental p then is:
p=2^[n+1].sin{45 degrees/2^[n-1]}/cos{45 degrees/2^n}
For n=1; the approximation is: p(1)=3.061467458…
For n=2; the approximation is: p(2)=3.121445152…
For n=3; the approximation is: p(3)=3.13654849…
For n=14; the approximation is: p(14)=3.141592653…correct to 8 decimal places.
Now express the trigonometric functions as expanded series:
sinx=x-x³/3!+x⁵/5!-…..in odd terms for x and
cosx=1-x²/2!+x⁴/4!-….in even terms for x to form the limit
lim(n->oo){2^[n+1].[45.2^[1-n]-45³/3!.(2^[n-1])³+45⁵/5!.(2^[n-1])⁵-..]/[1–45².2^[2n]/2!+45⁴.2^[4n]/4!-….]
=lim(n->oo){2^[n+1].45.2^[1-n]}=2².45=180.
lim(n->oo){1/n}=0, therefore all the trigonometric terms of the form1/2^n vanish.
So we have indeed justified J. Iuliano’s bold assumption to simply form the identity 360=2p or p=180 to freely interchange radian measure with the angular degrees.
Of course this works only for the demetricated superbrane scenarios and requires thus a dimensionless setting for the spacetimes as the ontological foundation for mensuration physics.
But the 360 factor in the Electron Radius definition now becomes the necessary circularization for the wormhole perimeter lps.
Precisely 10¹⁰ wormhole perimeters fit into the linearised and metricated macroquantised wormhole perimeter represented by 2p.Re AS 360.Re circularized.
This also unitizes our four folded UFoQR in 8p/360 as the A/4 factor in Stephen Hawking’s Black Hole event horizon entropy counter for the information mapped surface area as hologram, measured in Planck-Areas.
In short the entire quantum geometric structures described by QR fall into place.
J.Iuliano’s pyramid angle now is justifiably ‘reverse engineered’.
Recalling our Fine Structure Constant derivations; we have the Universal Wave function approximation for AlphaB=1/137.0470721 and the little gamma approximation as Alphag=1/137.0471224.
Using the Iuliano formulations, we then compute the pyramid angle for those two cases as:
arc cosB{-137.0470721/10⁴}=90.78524646 degrees or (16p+1.584501463) radians=51.84998392 radians and representing 51 degrees 50' 59.94" as the pyramid angle.
arc cosg{-137.0471224/10⁴}=90.78524675 degrees or
(16p+1.584501468) radians=51.84998393 radians and representing 51 degrees 50' 59.94".
Both values represent J. Iuliano’s calculation to that precision in his pyramid face-angle of (51.849982814). 4 Tony B.
The great genius Leonardo Da Vinci observed the golden section in nature and incorporated it in his remarkable paintings.
Addendum:
I have consulted my notebook for early calculations on the speed of light and PHI and have found the following which will be elaborated on in subsequent sections.
A circle, divided by a decagon, has a base which is a phi-ratio of the circle. If we consider time to be one turn of the circle, then we can relate time to PHI. 360/10 = 36 degrees or the phi angle. Log 5 (Golden Spiral logarithmic expansion factor) equals 100,000. The base of time in this schema is 1/6 of a circle X 10. The speed of light is given as 186,282 mps and when converted to arc minutes per second (divided by 1.1516264), it equals 1.6175558 (which is a division by Log 5). 360/phi equals 222.49224 degrees of our time circle. This is a difference of .9865509 which is accounted for by the time difference in a year circle which is .9863003. In other words, there is a deep relationship between our measurement of time and the golden section. Is that why the golden section is found so pervasively in the process of growth?

How do Fractals Contribute to Beauty? was originally published in Extreme Life Goals on Medium, where people are continuing the conversation by highlighting and responding to this story.

How do Fractals Contribute to Beauty?

What is a fractal?

Why are fractals important?

Why do we hear so much about fractals now?

Fractality Relationships

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