The Visual Spectrum of Light

Most people know the familiar colors of the rainbow and the visual spectrum of light as red, orange, yellow, green, blue, indigo and violet. This is often referred to by the acronym “ROYGBIV”:

Have you ever noticed though that the individual ROYGBIV colors we readily distinguish are not evenly dispersed across the spectrum? Some colors are tightly grouped within a small range of wavelengths and frequencies. Other colors spread across a wider band. Keep that thought.

Chakras and the Human Body

The concept of Chakras originated in the early traditions of Buddhism and Hinduism. The Sanskrit word “chakra” means “disk” or “wheel,” and refers to energy centers of the body, each corresponding to specific nerve centers and major organs. Early Sanskrit texts refer to them in meditative visualizations and also as physical entities within the body. In Hindu thought, there are seven primary chakras. These run from the base of the spine to the crown of the head. They are are often depicted in an image like the one below:

Note that the seven colors associated with the seven chakras correspond directly to the seven ROYGBIV colors of the rainbow and visible spectrum. The chakras appear evenly spaced on this diagram, but their actual positions on the human body vary in spacing. Keep that thought too.

The Golden Ratio Connection to Light, the Body and the Chakras

The amazing connection in these concepts came to Kalana Dido when he began to “connect the dots” in the pattern of spacing that was common to both the visible spectrum and the chakras of the human body.

The visible spectrum extends from the lower red frequencies of about 380 to 450 nanometers to the higher ultra-violet frequencies of about 625-700 nanometers. Kalana searched for a pattern to explain the uneven spacing in the spectrum and actual positions of the chakras on the human body. He noticed that the breaks that best defined the colors fell at alternating gaps of about 26 and 42 nanometers. Curiously, he noted that these numbers, which are double the numbers 13 and 21 of the Fibonacci sequence, have a ratio of about 1.615. This is a variance of only 0.16% from the golden ratio. He revised his model ever so slightly and found that it matched even better using golden ratio proportioned gaps of 26.18 and 42.36 nanometers.

He created this illustration of his concept. Note that the lines between the golden ratio-based circles indicate the primary color breaks, not what is within the circles themselves:

Note how the alternating gaps of the ROYGBIV colors of the spectrum align very closely with the positions of the chakras on the body:

• Red – Root (Muladhara) chakra, at the base of the spine. (Trust, belonging)
• Orange – Sacral(Swadhishthana) chakra, located just below the navel. (Sexuality, creativity)
• Yellow – Solar plexus (Manipura) chakra, located at the solar plexus. (Wisdom, power)
• Green – Heart (Anahata) chakra, located at the heart (Love, healing, relationships)
• Blue – Throat (Vishuddha) chakra, located at the throat (Communication, expression, truth)
• Indigo – Third Eye (Ajna) chakra, located between the eyes (Awareness, intention, purpose)
• Violet – Crown (Sahastrara) chakra, located at the top of the head (Spirituality, connection to the divine)

Note too how the seven ROYGBIV colors of the visible spectrum are spaced according to same alternating gaps of 26.18 and 42.36 nanometers. These numbers have relative ratios of 1 and 1.618. Oddly enough, these numbers also beautifully mirror the square and cube of the golden ratio, 2.618 and 4.236.

It has long been clear that the colors of the spectrum correspond to the colors of the chakras. It’s amazing though to connect the varied spacing gaps of the ROYGBIV colors in the spectrum to the same spacing in the chakras. To find that the spacing is related to the golden ratio is beyond amazing. What led the ancients of Hinduism to pick these particular positions to define the chakras? By contrast, various Buddhist traditions use 4, 5, 7 or 10 chakras. The 5 Buddhist chakras are sequenced from the base of spine to crown with colors of green, yellow, blue, red and white, which do not even correspond to the visible spectrum of light. It begs this question: Did the ancients of Sanskrit and Hindu tradition had a much deeper insight into the nature of energy, light and the human body than Western tradition may give them credit for?

Testing the model for validity and accuracy

I credit Kalana with much ingenuity and insight in finding these fascinating relationships, and thank him for bringing to my attention. I’m always excited to learn about new appearances and applications of the Golden Ratio. I’m also just as passionate about using my auditing background and analytic skills to assure that what is published on GoldenNumber.net is as accurate as possible. Kalana’s model fits the chakra positions on David’s classic male body above. It also fits the visible spectrum illustration quite well, but we put the model to the test.

Analysis of alternative spacings

Kalana tested his model with base wavelengths of 432 to 438 nanometers, and spacing gaps ranging from 25 to 27 nanometers. In doing so he found that the optimal spacing gap for alignment to the ROYGBIV colors occurred with a base of 438 nanometers and alternating gaps based on 26.2 and 42.3 nanometers. Below is a sample of his analysis:

Analysis of correlation to stated color wavelengths

To test the theoretically derived wavelengths from Kalana’s model, I compared the wavelengths for each color in his model to the stated wavelengths for each color of the visible spectrum as listed on Wikipedia:

I then created the graph below to show how the theoretical wavelengths in Kalana’s model compared to the Wikipeida color ranges:

The chart reveals that all of the theoretical wavelengths derived in Kalana’s model (in blue) fall cleanly between the actual low (in red) and high (in green) wavelengths for each respective color, with one exception. His model has blue at 506.54 nanometers while Wikipedia shows the actual color range for this color, which its calls cyan, has a range of 485 to 500 nanometers. This is a slight variation indeed of only 1.3% above the maximum, and is likely not that noticeable to the eye.

Analysis of spectrum images to actual spectral photos

In my analysis of Kalana’s work, it became very clear, and very frustrating, to find that there were many variations in images showing the visible color spectrum. Further research led me to learn that the spectrum of colors generated by RGB color monitors do not correspond exactly to the spectrums of sunlight that appear in nature. RGB monitors create 16,777,216 colors, but this is all based on all the mathematical combinations of 256 variations of red, blue and green. This realization just about led us to abandon the work on this model altogether. I suggested though that we use the nature spectrum of sunlight from rainbows rather than the various RGB renditions and artist renditions of spectrums. Kalana developed a new set of models based on photos of rainbows, and it still corroborated the results.

The image below applies Kalana’s model to the statue of David again. It uses three actual photos of rainbows, as shown in the inset at the lower right, as the source for the color spectrum:

Analysis of impact of individual body variations

Another potential challenge to the model was that the body proportions of individual people are of course not identical. To put this to the test, Kalana found a set of models from a swim suit site, and created yet another set of images. Despite the small range of individual differences, the results were still consistent and compelling. The images below compare a sample of individual bodies to actual photos of rainbows. Click on any image for the full size version:

Background

As a background to this story, Kalana Dido is interested in colors and their effects on the human body. He wanted to see if there was a correspondence between the visible light spectrum and the chakras. He aligned the chakras and the spectrum by shifting and resizing the images with the ratios in proportion. It was then that he noticed that the symmetries took shape.

He then measured the gaps and noticed that it was very close to the golden ratio. By shifting only 1 or 2 pixels with the chakras aligned to the spectrum, the golden ratio appeared everywhere in only two proportions. At this first stage, he noticed a spacing of 26nm and a second at 42nm, with its ratio of 1.615…, as being very close to the golden ratio of 1.618…

Kalana then wanted to solve the question of the exact value of the base by taking the reference spectrum and calibrating the wavelength of the yellow. He did this by taking 10 bases every 0.2 nm from 25 to 27 nm. After having establishing a corresponding set of yellows, he oriented his work to the 26.2nm base which seemed to best correspond visually to a real yellow. Soon thereafter, he made the connection between 26.2nm and 2.618 (phi squared) and 42.36nm and 4.236 (phi cubed).

Kalana isn’t taking these findings too seriously without additional scientific confirmation. He has a level of certainty though that the chakras are distributed according to a pattern of yxxyyx from base to crown, where y=Φx and Φ represents the golden ratio.

Conclusions

Kalana and I both regard this as a fascinating insight. It allows us to understand the association of several concepts. It appears to be yet another appearance of the golden ratio in nature, with its correlations to the human body, the seven key colors of the visible spectrum of light and the seven chakras of Hindu and Sanskrit traditions. It is not presented as a proven scientific concept or body of scientific evidence.

We hope that this presentation of Kalana’s original research and findings may lead to work by others. Key concepts to take away include:

1. The research reveals an apparent alignment of colors and positions in the chakras and the visible spectrum of light based on actual photos of rainbows and corresponding visible spectrum images.
2. The research demonstrates apparent golden ratio relationships in that alignment, with a base gap of approximately 26 nanometers, and with 42 nanometers as its golden multiple (26 * 1.618 = 42.07).
3. Testing was conducted to find the most accurate value for the gap in the wavelengths by comparison of results in 0.2 nm steps from 25nm to 27nm.
4. It was determined that the 26.2nm base gap produces the most consistent results.
5. The model then uses Φ²x10nm (2.618 nm) and Φ³ x10 (42.36 nm) as the golden ratio-based theoretical measures of the gaps. 
6. The model uses a scale of 10^-10 meters for x and y, where in the illustrations x = Φ² and y = Φ³.
7. A base of 438nm was chosen as the reference point based on best fit. The exact wavelengths of the colors cannot be defined without knowing the base. It could vary from 438nm slightly, but this was not found to impact the results significantly.
8. We acknowledge that the connection of the gaps in nanometers of 26.18 and 42.36, and the association of those numbers to the square and cube of the golden ratio of 2.618 and 4.236 is interesting and convenient, but unexpected and not necessary to the model. Nanometers are of course just one unit of measure, and the actual digits would be different if measured in inches or some other unit of measure. France is credited with the invention of the meter, which in 1790 was defined as one ten-millionth of the distance between the North Pole and the Equator through Paris. There is thus no obvious or implied reason for this connection in these mirrored digits other than coincidence … or a grander cosmic plan that is beyond our comprehension.

An invitation to continue the research

We invite anyone to test these results, further refine them and to share any new findings with Kalana and I.

You may also want to try this: In his research, Kalana made two sections of string in golden ratio proportion and used them to measure his own body. He found his own chakras fit the model as well. We’ll just invite you to take your own measurements and share your results in the comments below.

The post Light, the Human Body, Chakras and the Golden Ratio appeared first on The Golden Ratio: Phi, 1.618.

The Nautilus shell if often associated with the golden ratio. There is a fair amount of confusion, misinformation and controversy though over whether the graceful spiral curve of the nautilus shell is based on this golden proportion. Some say yes, but offer no proof at all. Some show examples of spirals, but incorrectly assume that every equi-angular spiral in nature is a golden spiral. Several university math professors say no, but they only compared the nautilus spiral to the spiral created from a golden rectangle. Another university professor says no, but only measured height and width of the entire shell. Let’s look at this objectively and solve this mystery and debate.

The Golden Spiral constructed from a Golden Rectangle is NOT a Nautilus Spiral.

A traditional Golden Spiral is formed by the nesting of Golden Rectangles with a Golden Rectangle. This resulting Golden Spiral is often associated with the Nautilus spiral, but incorrectly because the two spirals are clearly very different.

A Golden Spiral created from a Golden Rectangle expands in dimension by the Golden Ratio with every quarter, or 90 degree, turn of the spiral. This can be constructed by starting with a golden rectangle with a height to width ratio of 1.618. The rectangle is then divided to create a square and a smaller golden rectangle. This process is repeated to arrive at a center point, as shown below:

The golden spiral is then constructed by creating an arc that touches the points at which each of these golden rectangles are divided into a square and a smaller golden rectangle.

You can find images of nautilus shells and spirals all over the Internet that are labeled as golden ratios and golden spirals, but this golden spiral constructed from a golden rectangle is nothing at all like the spiral of the nautilus shell, as shown below. This had led many to say that the Nautilus shell has nothing to do with the golden ratio.

Is there more than one way to create a golden spiral?

There is, however, more than one way to create spirals with golden ratio proportions of 1.618 in their dimensions. The traditional golden spiral (aka Fibonacci spiral) expands the width of each section by the golden ratio with every quarter (90 degree) turn. Below, however, is another golden spiral that expands with golden ratio proportions with every full 180 degree rotation. Note how it expands much more gradually. The golden ratio proportions are indicated by the red and blue golden ratio grid lines provided by PhiMatrix software.

The center/vortex of the above spiral increases to a width of 1 at point A. The half rotation of 180 degrees to point B expands the width of the spiral to 1.618, the golden ratio. Continue another half turn of 180 degrees to point C to complete the full rotation of 360 degrees. The width of the spiral from the center is now 2.618, which is the golden ratio (phi) squared. The golden ratio lines in red indicate how another full rotation expands the length from the vortex by phi squared, from phi to phi cubed. And so the pattern of expansion continues. This Golden Spiral based on a 180 degree rotation is a much better fit to the Nautilus Spiral.

A golden mean gauge seems to match the spirals of some Nautilus shells, so is that the answer?

If you measure a Nautilus shell with a golden mean gauge, you may find that the gauge isn’t far off the distance from the inner spiral on one side of the center point to the outer spirals on the other side.

Does this explain its association with the golden ratio?  Let’s explore a little further.

Another spiral variation may relate the Nautilus spiral to phi

Let’s continue to explore that fit of a slightly different variation on a golden spiral. Rather than seeking a golden ratio from the spiral’s center point, let’s try measuring the dimensions and expansion rate formed by these three points:

• Point 1 – The outside point of any spiral of the nautilus shell
• Point 2 – The first inside spiral at one full rotation (360 degrees) from Point 1
• Point 3 – The second inside spiral found at two-and-a-half rotations (900 degrees) from Point 1.

As illustrated in the Nautilus shell below, the distance from Point 1 to Point 2 divided by the distance from Point 2 to Point 3 is quite close to a golden ratio for the complete rotation of the Nautilus spiral shown below.  This is indicated by the golden ratio ruler below, which has a golden ratio point at the division between the blue and white sections. When the blue section has a length of 1, the white section has a length of 1.618, for a total length of 2.618.

Using this approach, the actual spiral expansion rates for the above Nautilus shell, taken every 30 degrees of rotation were:   1.572, 1.589, 1.607, 1.621, 1.627, 1.622, 1.616, 1.573, 1.551, 1.545, 1.550 and 1.573.  This averages to 1.587, a 1.9% variance from 1.618.  This is not exactly a golden ratio, but then it’s not hard to see why it would appear to be one.

The two golden spirals we’ve identified then look like this:

The image below has the “golden ratio to opposite spiral” overlayed in red on a nautilus shell spiral. As you can see, the fit is fairly good for the first three full rotations from the center point. Beyond that point, this particular nautilus shell begins to show a slightly more gradual and open curve than this golden spiral. All in all though, its relationship to a golden ratio spiral is becoming more apparent.

Below is a photo of another nautilus shell. It has the same general pattern in that its spiral curve conforms fairly closely to a the “golden ratio to opposite spiral” for the first three rotations, but this one has a tighter curve than the golden ratio spiral in its final outward spiral.

Spiral growth rates from the center point

Let’s take another look at the spirals of the Nautilus based on the center point. If we measure the actual dimensions of the above Nautilus shell, we find that its expansion rate with each rotation from its center point can be as low as 2.58.  This is slightly less than 2.618, Phi squared, as in the idealized golden spiral above. Expansion rates in this same shell ranged to 2.9. Rates over 3 were observed in other shells. Note how the expansion rate varies for any given Nautilus as you rotate the shell, as illustrated below:

Measurements made using PhiMatrix software

So, we see that not every nautilus spiral is created equal, nor is it created with complete perfection. Just as with the human form, nautilus shells have variations and imperfections in their shapes and the conformity of their dimensions an ideal spiral using either of the two methods shown here. So while many inaccurate claims have been made regarding both its existence and non-existence, we see that the Nautilus spiral can exhibit dimensions whose proportions come close to phi. You’d likely have to search quite a few beaches to find a Nautilus shell whose spiral fits any of these phi-based spirals perfectly, and may never find one.  We can see though that the visual appearance of dimensions come close to phi proportions, and understand why this has lead many people to associate it with the golden ratio, and to view it as one of the most beautiful spirals in nature.

So what do you think? Is the Nautilus spiral related to the golden ratio or not? Share your thoughts below.

See the Spirals page for more information on spirals in nature.

References:

Following are comments by three Ph.D.s in mathematicians who say that the Nautilus has no relationship to the golden ratio. This is true with respect to the classic golden spiral, but misses the fact that there is more than one way to construct a spiral with golden ratio proportions.

The Man of Numbers – In search of Leonardo Fibonacci by Dr. Keith Devlin (page 64) – “Unfortunately, the belief that the Nautilus shell has the form of the Golden Spiral is another of those false beliefs about Euclid’s number. To be sure, the Nautilus shell is a spiral, and it is moderately close to spiraling by a constant angle, but that angle is not the Golden Ratio. Not even close. So there is no connection. And that is why this topic is tucked away at the end of this book!”

Replicator Constructions by Dr. George Hart – “My goal here was to comment on the common misconception that the nautilus has a golden ratio spiral. A real nautilus doesn’t. This is what a nautilus shell would look like if it were based on a golden spiral. I built it in halves on a raft, then glued the halves together. I’m quite happy with the final result.  There’s a video explaining more about it here.”

The Golden Ratio—A Contrary Viewpoint by Dr. Clement Falbo (page 127) – “The nautilus is definitely not in the shape of the golden ratio. Anyone with access to such a shell can see immediately that the ratio is somewhere round 4 to 3. In 1999, I measured shells of Nautilus pompilius, the chambered nautilus, in the collection at the California Academy of Sciences in San Francisco. The measurements were taken to the nearest millimeter, which gives them error bars of ±1 mm. The ratios ranged from 1.24 to 1.43, and the average was 1.33, not
phi (which is approximately 1.618). Using Markowsky’s ±2% allowance for
to be as small as 1.59, we see that 1.33 is quite far from this expanded value of
phi. It seems highly unlikely that there exists any nautilus shell that is within 2% of the golden ratio, and even if one were to be found, I think it would be rare rather than typical.”

Note: A special thanks go to Oliver Brady for his astute analysis of this article, which led to improvements in its clarity and accuracy.

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The human body illustrates the Golden Section or Divine Proportion. We’ll use the golden ratio building blocks developed on the Life page again for each line segment:

The Divine Proportion in the Body

• The white line is the body’s height.
• The blue line, a golden section of the white line, defines the distance from the head to the finger tips.
• The yellow line, a golden section of the blue line, defines the distance from the head to the navel and the elbows.
• The green line, a golden section of the yellow line, defines the distance from the head to the pectorals and inside top of the arms, the width of the shoulders, the length of the forearm and the shin bone.
• The magenta line, a golden section of the green line, defines the distance from the head to the base of the skull and the width of the abdomen. The sectioned portions of the magenta line determine the position of the nose and the hairline.

Although not shown, the golden section of the magenta line (also the short section of the green line) defines the width of the head and half the width of the chest and the hips.

The Human Body is based on patterns of 5, which is the basis for Phi as well

Another interesting relationship of golden section to the design of the human body is that there are:

• 5 appendages to the torso, in the arms, leg and head.
• 5 appendages on each of these, in the fingers and toes
• 5 openings on the face.
• 5 sense organs for sight, sound, touch, taste and smell.

The golden section in turn, is also based on 5, as the number phi, or 1.6180339…, is computed using 5’s, as follows:

5 ^ .5 * .5 + .5 = Phi

In this mathematical construction “5 ^ .5” means “5 raised to the 1/2 power,” which is the square root of 5, which is then multiplied by .5 and to which .5 is then added.

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Dr. Stephen Marquardt.

The design of life is based on a “Golden Ruler™, a ruler composed of golden ratios within golden ratios. It has long been known that the Golden Section, (also Golden Ratio or Divine Proportion), appears in certain proportions of living organisms.  In 1997, I began to wonder how universal this was and how far it could be applied.  I took a golden section of a line not just once, but many times, to form what I like to call the “golden ruler,” shown as follows:

First, take a line and divide it so that the proportion of (B) to (A) is the same as the proportion of (C) to (B):

Divide the line again and again in the same way:

Combine the segments to create a measuring stick or “Golden Ruler™:”

This appears in the proportions of many life forms:

You can take the process another step further by creating a “Golden Grid,” taking it from one dimension to two dimensions:

This approach can be used to explain the physical proportions of many life forms, but the most beautiful application is found in humans:

It appears that this “golden ruler” is a common, if not universal, metric in the design of living organisms.

Try it on the Do It Yourself page!

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Population growth is also related to the Fibonacci series.  In 1202, Leonardo Fibonacci investigated the question of how fast rabbits could breed under ideal circumstances.  Here is the question that he posed:

The Fibonacci series can be used to predict urban populations

It appears that the Fibonacci series can even be used to predict populations of major cities, as shown by the relationships of various U.S. urban areas in 1970:

Method 1 takes the population of the largest city and divides it again and again by phi.  Method 2 takes the population of each successive city and divides it by phi.

Source: http://www-personal.umich.edu/~sarhaus/

Multicellular organisms

In biology, once an egg is fertilized, it divides and multiplies in count until it reaches a point at which the ratio of the succeeding number of cells to the previous number of cells is phi (1.618 …).

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Ideal facial proportions are universal regardless of race, sex and age, and are based on the phi ratio of 1.618. For example, if the width of the face from cheek to cheek is 10 inches, then the length of the face from the top of the head to the bottom of the chin should be 16.18 inches to be in ideal proportion.  Deviations from this ideal can result in health problems.  Corrective procedures that return the face to this ideal can improve health.

People with longer than ideal faces tend to have breathing problems

Studies indicate that people who have long faces tend to have problems breathing through their nose as their sinus cavities tend to be narrow, inhibiting airflow.  Consequently, these people tend to breathe through their mouth, a contributing factor for some people who snore or who suffer from sleep apnea.  Mouth breathers also tend to have narrow mouths and crooked teeth.  Orthodontic appliances can widen the face and mouth, which in turn widens the sinus cavities, allowing easier breathing through the nose.

People with shorter than ideal faces tend to have jaw problems

People with shorter faces tend to have abnormal jaw development which causes excessive pressure on the jaw joint.  Many people with short faces suffer from headachesbecause their jaws are positioned in a manner that can restrict blood flow to the brain.  Stress can further aggravate this problem because people subconsciously clench and grind teeth.  Orthodontic appliances can lengthen the face, which helps relieve pressure on the jaw joints.

Ideal	Long face Syndrome	Short face syndrome

Orthodontic appliances can adjust facial proportions and improve health

These insights and illustrations were provided by Yosh Jefferson, DMD, MAGD, a general dentist in Mount Holly, NJ and founder of FACES, an organization dedicated to the study and treatment of facial esthetics and jaw joint problems.  In his practice, Dr. Jefferson manipulates patients’ facial-skeletal structures to improve overall health and hasdemonstrated that changing a patient’s facial structure can produce dramatic aesthetic results similar to plastic surgery without one incision.  Visit Dr. Jefferson’s site athttp://www.facialbeauty.org.

According to Dr. Jefferson, “The further away a person’s face is from the ideal proportions and profile, the more likely that person will have certain medical problems.  In fact, when I see a patient for the first time, I can predict with great accuracy the medical problems from which they suffer, especially sinus problems and migraines.”

Dentists who specialize in an emerging field called functional orthodontics can improve facial beauty, and in some cases, overall health, without the cost, pain, and recovery time and potential risks associated with plastic surgery.  Using orthodontic appliances that change the shape of the mouth by manipulating the upper or lower jaw, Jefferson and other functional orthodontic practitioners can enhance beauty and eliminate such chronic health problems as sinus infections, headaches, earaches, breathing difficulties and jaw disorders.

Reference:  “Unraveling the mystery of facial beauty and its biologic significance,” Yosh Jefferson, DMD,  MAGD, IBO, Journal of General Orthodontics, Volume 7, Number 2, June 1996

See also http://www.agd.org/consumer/topics/overallhealth/longface.html

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There are many examples of the Divine Proportion found throughout the design of the universe and everything in it, but let’s take a look at one of the most important things first: You!   We’ll need a few Golden Section / Fibonacci building blocks:

Each line is 1.618… times longer than the one before it. (Conversely, a section drawn at 0.618 (or 61.8%) of each line equals the length of the one before it.)

Let’s start with something simple. Take your hand off your keyboard or mouse and look at the proportions of your index finger.

Hold your hand up to the screen. Don’t be shy!

Each section of your index finger, from the tip to the base of the wrist, is larger than the preceding one by about the Fibonacci ratio of 1.618, also fitting the Fibonacci numbers 2, 3, 5 and 8.

By this scale, your fingernail is 1 unit in length.

Curiously enough, you also have 2 hands, each with 5 digits, and your 8 fingers are each comprised of 3 sections.  All Fibonacci numbers!

Here are other ways of expressing the same basic relationship in its connection to the golden ratio and golden spiral:

Source: www.creationresearches.com

The ratio of the forearm to hand is Phi

Your hand creates a golden section in relation to your arm, as the ratio of your forearm to your hand is also 1.618, the Divine Proportion.

Even your feet show phi

The foot has several proportions based on phi lines, including:

1. The middle of the arch of the foot
2. The widest part of the foot
3. The base of the toe line and big toe
4. The top of the toe line and base of the “index” toe

Note that not every individual has body dimensions in exact phi proportion but averages across populations tend towards phi and phi proportions are perceived as being the most natural or beautiful.

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B-DNA has spirals in phi proportions

DNA in the cell appears as a double-stranded helix referred to as B-DNA.This form of DNA has a two groove in its spirals, with a ratio of phi in the proportion of the major groove to the minor groove, or roughly 21 angstroms to 13 angstroms.

The DNA cross-section is based on Phi

Dr. Robert Langridge, known as the pioneer of molecular graphics, earned his Ph.D in Crystallography in 1957 with a dissertation on x-ray crystallographic, model building and computational studies of the structure of DNA. Over the next several decades he continued his work, which led to this 1985 photo of him:

In the photo is an image of the cross section of DNA, revealing a clear 10-sided geometry, the foundation of 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!

This decagon-based nature of the DNA cross section was independently recognized by Dr. Stephen Marquardt. His research on human facial attractiveness led him to discover the relationship of the golden ratio to both the decagon structures found in both DNA and in the “Marquardt Beauty Mask” he created which provides an archetype to understanding human facial beauty.

His work describes Phi in 2D to show the pentagon and decagon as it relates to DNA, and extends that to “Making the Mask” to reveal a:

“complex of forty two (42) “secondary” Golden Decagon Matrices, which are exactly the same shape as the Primary Golden Decagon Matrix but smaller by various multiples of phi, are mathematically and geometrically uniquely positioned in the Primary (or framework) Golden Decagon Matrix. It is these forty two (42) Secondary Golden Decagon Matrices which ultimately form the various components of the face.”

The transformation from decagon to the Beauty Mask is discussed on this site at “Facial Analysis and the Marquardt Beauty Mask” and on Dr. Marquardt’s site at BeautyAnalysis.com.

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Fibonacci numbers in plant spirals

Plants that are formed in spirals, such as pinecones, pineapples and sunflowers, illustrate Fibonacci numbers.  Many plants produce new branches in quantities that are based on Fibonacci numbers.

Fibonacci numbers in plant branching

Here a sunflower seed illustrates this principle as the number of clockwise spirals is 55 (marked in red, with every tenth one in white) and the number of counterclockwise spirals is 89 (marked in green, with every tenth one in white.)

Here a plant illustrates that each successive level of branches is often based on a progression through the Fibonacci series.

Fibonacci numbers in plant sections

You might expect symmetry in plants, but if you cut a fruit or vegetable you will often find that the number of sections is a Fibonacci number:

Bananas have 3

Apples have 5

Fibonacci numbers in flower petals

Many flowers have petals that total a number in, or very close to, the Fibonacci series:

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Below are just a few:

The eye, fins and tail all fall at golden sections of the length of a dolphin’s body. The dimensions of the dorsal fin are golden sections (yellow and green). The thickness of the dolphin’s tail section corresponds to same golden section of the line from head to tail.

The spiral growth of sea shells provide a simple, but beautiful, example.

Every key body feature of the angel fish falls at golden sections of its width and length. The nose, tail section, and centers of the fins of the angel fish fall at first (blue) golden sections. The second golden section (yellow) defines the indents on the dorsal and tail finds as well as the top of the body. The green section defines the marking around the eye and the magenta section defines the eye.

The eyes, beak, wing and key body markings of the penguin all fall at golden sections of its height.

All the key facial features of the tiger fall at golden sections of the lines defining the length and width of its face.

The body sections of an ant are defined by the golden sections of its length. Its leg sections are also golden sections of its length. 

The facial features of a koala bear show golden ratio proportions in the dimensions and positions of the eyes, nose and mouth in relation to the dimensions of the face.

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