Archive for the ‘Molecules’ Category

Hidden Universal Symmetry


For millennia philosophers, mathematicians, architects, artists and mystics have been interested in the divine proportion, but its understanding has been highly fragmented. My recent work attempts to show how the divine proportion is a hidden universal symmetry worthy of comprehensive study and understanding.

Euclid’s book Elements (300 BCE) contains the first known written definition of the divine proportion, something he called the mean and extreme ratio:

“A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.”

Euclid’s mean and extreme ratio is better known today as the golden mean, golden ratio or divine proportion. These names refer to the act of cutting a line at one uniquely important point. Point C cuts the line AB at the divine proportion if and only if the whole line (α+β) is in proportion to the longer segment (α) in exactly the same proportion as the longer segment (α) is to the shorter segment (β).


The Greek letter Ф (phi) is used to denote the divine proportion. Ф is an irrational number whose decimal approximation continues forever without repeating. In fact, the divine proportion turns out to be “the most irrational number” of all. [1]

“Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold, the second we may name a precious jewel.” -Johannes Kepler (1571-1630)

Scientists are just recently starting to catch on to the divine proportion’s role in patterning the universe.

In 2010 an international team of scientists discovered the divine proportion in the quantum world. Researchers created a chain, one atom wide, of linked magnetic atoms and tuned the system to the “Schrödinger’s cat state” of quantum uncertainty [2].

They discovered that in that state the frequencies of the atoms’ spins resonated in the golden ratio. The team’s leader Dr. Radu Coldea said, “It reflects a beautiful property of the quantum system—a hidden symmetry.” The story about their discovery was the most-visited story in 2010 on EurekAlert, the global source for science news. I will save my speculations about why this was so until a bit later in this article.

In 2015, by combining advanced knowledge of mathematics and physics, Tamar Friedmann and C. R. Hagen discovered that the energy levels of a hydrogen atom are based on π, the ratio of a circle’s circumference to its diameter. [3] They showed that the modern quantum-mechanical description of hydrogen’s atomic energy levels is mathematically equivalent to John Wallis’ historic mathematical formula for π (published in 1656).

My opinion of the age old debate is that mathematics appears to be discovered rather than invented.

The Wallis formula is an infinite product equivalent to the area of a square compared to the area of an inscribed circle (4/π). I find it especially interesting that 4/π is 99.8% equal to the square root of the golden ratio. Matter and energy are thus divinely proportioned.

From my book The Divine Proportion: [4]

A two-dimensional representation is the most effective way to perceive the divine proportion’s essential quality of self-similarity. The golden rectangle is just such a two-dimensional figure (the golden triangle is another) where the ratio of the longer side to the shorter side is equal to Ф divided by 1, which reduces to Ф.


On the right side of the figure, the shorter side is subdivided again such that what was formerly the smaller part is now the larger, namely the side with the length of 1. The ratio of the new larger segment (1) to the next smallest segment equals 1 divided by 1/Ф, which also reduces to Ф. The subdivision continues in this way, always reducing to Ф.

The golden rectangle implies a spiral, composed of circular arc segments centered on the corner of each square. The smallest rectangle at the end of the spiral is also a golden rectangle, so the process of divine subdivision can continue infinitely inward. The divine proportion is always the same and does not depend on the length of the line being divided.

“Like God, the Divine Proportion is always similar to itself.” –Luca Pacioli (1447–1517)

Spiral galaxies and cyclones (hurricanes) naturally approximate the form of golden spirals.

Images from The Divine Proportion

Images from The Divine Proportion

The fact that the same form appears at such vastly different scales and contexts reveals a universal pattern. I think the reason we see this pattern is that it is built from a simple “recurrence relation”—an equation that recursively defines a sequence. Once one or more initial terms are given, each further term of the sequence is defined as a function of the preceding terms.

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, …) is an example of a recurrence relation where each subsequent number is the sum of the previous two. You can create arbitrary recurrence relations like the Fibonacci sequence from any two initial terms. Taking the ratio of any two adjacent terms approximates the golden ratio: the higher the value of the terms, the better the approximation.


I recently created a geometric recurrence relation with the 3:4:5 Pythagorean right triangle. I made this image by simply iterating the triangle and the squares made from its edges. It’s amazing that it has a natural, fractal quality and that the spirals form discrete approximations of golden spirals.

Fractal 3:4:5 triangle and squares

Fractal 3:4:5 triangle and squares

I suspect that a cyclone is formed by a recurrence relation of air pressures. A simple mathematical sequence that builds on itself results in this form. The same is likely to be true of stars in spiral galaxies in terms of solar wind pressures.

The divine proportion turns up at the stellar scale. In 2015 a team of astrophysicists discovered that “the brightness of some stars pulsates at primary and secondary frequencies whose ratios are near the golden mean.” [5]

How could it be that the quantum effect mentioned with “Schrödinger’s cat state” atoms appears also at a cosmological scale? Read on to understand that the divine proportion patterns matter at all scales.

I discovered that the divine proportion shows up on the planetary scale.

From my book Secrets In Plain Sight: Leonardo da Vinci: [6]

The combined polar diameters of Earth and Moon (16185.6 km) divided by the equatorial diameter of the Earth (12756.2 km) is 99.8% equal to √Ф. The size of our planet and its natural satellite fit this elegantly simple geometry.

“Simplicity is the ultimate sophistication.” –Leonardo da Vinci

The circumference of the circle is equal to Ф x π and the perimeter of the square is equal to 4 x √Ф. Therefore, the circumference is 99.9% equal to the perimeter. It is impossible to exactly square the circle because one can never quite rationalize π.

“It is the mark of an instructed mind to rest satisfied with the degree of precision which the nature of the subject admits and not to seek exactness when only an approximation of the truth is possible.”

At the scale of cities, a few architects mysteriously encode the divine proportion in the their culturally significant structures without publicizing that knowledge.

From The Divine Proportion:

The CN Tower in Toronto is the tallest structure in the Western hemisphere at 553.33 meters in height and was the tallest structure in the world when built in 1976. The glass floor and outdoor observation deck in the Main Pod are 341.99 meters (1122 feet) in height. 553.33 / 341.99 ≈ Φ (99.99%).

Le Corbusier, the lead architect of the United Nations Secretariat building, previously developed a proportional system used for building design called the Modulor, which was based on human measurements, the divine proportion, and Fibonacci numbers. [7]

“Mathematics is the majestic structure conceived by man to grant him comprehension of the universe.” -Le Corbusier

At the molecular scale, the code of life itself is divinely proportioned.

From The Divine Proportion:

The view at the top of the illustration shows that the structure’s axis is arranged as a decagon, with 1 as the radius and Ф dimensioned across the molecule. The double helix’s relationship of major to minor grooves is also divinely proportioned. The overall pitch of the spiral compared to its diameter is likewise an expression of Ф (99.9%).

DNA is organized into nucleotide triplets, called codons. Geneticist Jean-Claude Perez discovered that the frequency in which codons appear in the human genome are strongly linked to Ф. [8]

The code of life is a multidimensional embodiment of the divine proportion.

Leonardo da Vinci showed how the human body is structured by the divine proportion.

In the book I explore whether the mass appeal of Da Vinci’s art could be due to his practice of encoding the divine proportion into his works. After all, his dozen or so paintings are the most visited, most discussed and most expensive art in the world.

From Secrets In Plain Sight: Leonardo da Vinci:

Leonardo depicted a man in both a circle and a square, based on the ancient canon of proportion found in the writings of Vitruvius…The primary divisions in the golden rectangles point to the navel and heart. The navel is the center of the physical body…The heart is the center of the energetic body, with three chakras above and three below…The golden rectangle anchored on the page itself points to the idealized man’s third eye.

In a recent study, plastic surgeons surveyed 1682 people to discover their aesthetic preferences for navel position on the human abdomen. “When comparing navel position on the same body, the majority of participants choose the one with the navel relocated according to the golden ratio (i.e., 1.618).” [9]

It appears that people prefer the proportions of the idealized body that Da Vinci depicted over 500 years ago. I assume this is because the human body is structured using the divine proportion.

From Secrets In Plain Sight: Leonardo da Vinci:

In Da Vinci’s painting “St. John the Baptist” (c.1513), he uses the golden rectangle to point out the prophet’s third eye and his finger pointing to heaven…The spiral convergences on the prophet’s third eye. On the right image, the secondary vertical division locates the upward pointing index finger and dramatically verifies the analysis.

The thesis of my book on Da Vinci is that he consistently pointed with the divine proportion’s divisions to physical and illuminated third-eyes.

“Adrian Bejan, professor of mechanical engineering at Duke’s Pratt School of Engineering, thinks he knows why the golden ratio pops up everywhere: the eyes scan an image the fastest when it is shaped as a golden-ratio rectangle.” [10]

The divine proportion must therefore be built into our visual cortex.

The pineal gland is the physical manifestation of the third-eye chakra. A fascinating article from states the following:

In 2004, Dr. David Klein, Chief of the Section on Neuroendocrinology at the NIH, discovered that the interior of the pineal gland has retinal tissue composed of rods and cones (photoreceptors)…just like the eye, and is even wired into the visual cortex of the brain. He also noted that it even has vitreous fluid in it. Within this context, it would seem that the scientific community may need more courage to follow this evidence wherever it may lead in the pursuit of scientific truth, especially where it concerns the pineal gland’s role as a human sensory organ, and to forego old and outdated theories.

“The photoreceptors of the retina strongly resemble the cells of the pineal gland.” -Dr. David Klein [11]

Albrecht Durer may have learned perspective and the divine proportion from Da Vinci or Luca Pacioli, who at one time lived with Da Vinci. After all, Pacioli and Da Vinci published a book together called De Divina Proportione (1509). Here is what Pacioli wrote in the book’s Introduction:

“A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching on a very secret science.” [italics by the author]

The science of the divine proportion had to be secret in their day because these Renaissance luminaries could have easily been burned at the stake for the implications of this science. I believe Da Vinci saw the divine not just in a transcendent heaven (following the church dogma), but through this science saw the divine immanently in the human body and in the world. If this is really true, then one doesn’t need the church hierarchy to “see God”—only the mind’s eye is required.

More ink has been spilled on Albrecht Durer’s “Melancholia I” than any other engraving. [12] Inspired by Franz Deckwitz’s study (1979), I found related but different geometry in my geometric analysis:

Albrecht Durer's Melancholia I

Albrecht Durer’s Melancholia I

I see the eyes of the melancholy genius staring directly at the third-eye of the cherub, who is surrounded by a “magical” pentagram. The pentagram traces the angle of the ladder and the horizon and is grounded by the square that it sits on. Again we are directed to a third-eye, this time of a divine cherub.

Pentagrams resonate with the divine proportion. Perhaps that is their magic?

Michelangelo also encoded the divine proportion in the world’s second most famous work of art after Da Vinci’s Mona Lisa, which also encodes it (shown in my book).

In a 2015 paper entitled “More than a neuroanatomical representation in The Creation of Adam by Michelangelo Buonarroti, a representation of the Golden Ratio,” [13] the co-authors build on Dr. Frank Lynn Meshberger’s 1990 discovery that, “The Creation of Adam represented in the Sistine Chapel shows God surrounded by a drape that has the shape of what he believed to be the sagittal section of a human brain.” [14]

The co-authors demonstrate that the gap where Adam receives the divine spark is precisely and fittingly pointed to using the divine proportion.

From Secrets In Plain Sight: Leonardo da Vinci:

With this geometry you can see that Adam’s body gracefully follows the arc and the primary horizontal division in both golden rectangles share the same datum line. This datum line leads from Adam’s eye straight into the heart of God, which is the pineal gland, or third eye, according to Michelangelo’s neuroanatomical representation [revealed by Meshberger]. The divine proportion and the third eye are literally the bridge to the infinite, or “divine realm.”

Michelangelo could have easily been burned at the stake for suggesting that the route to God is within, rather than through the official social hierarchy of God-representatives…Once you see its structure you being to understand why it is so powerful and sense Michelangelo’s humanistic intent of encoding such a secret in plain sight.

“The noblest pleasure is the joy of understanding.” – Leonardo da Vinci

I believe that the divine proportion works on us because it is part of our bodies, our DNA, and even our atoms, as this article has outlined. For this reason, we tend to resonate with—and preferentially choose—just about anything that is proportioned with the golden ratio, even if we are not conscious of it.

In the macrocosm, the divine proportion structures cyclones in the atmosphere, governs the proportions of the Earth-Moon system, patterns the pulsing of stars, and continuously orders spiral galaxies (my book The Divine Proportion has many more examples).

This hidden universal symmetry operates at all scales.

The charge radius of the proton is 8.775 x 10-16 m, which is 16 orders of magnitude smaller than our human scale. The entire universe has a radius of approximately 14 billion parsecs (4.3 x 1026 m), which is 26 orders of magnitude larger than the human scale. The full gamut of scale is thus 16 + 26, or 42 orders of magnitude from a proton to the entire universe. Amazingly, 42 / 26 is 99.8% equal to Ф, so it logically follows that the human scale is the divine measure of all things!

I see a rebirth of this ancient knowledge—a New Renaissance—starting to happen right now. However, it is going to require a major conceptual revolution. The time is ripe to scientifically understand this profound hidden universal symmetry and apprehend the fundamental mathematical reality of the universe. The “substance” of the universe is number—geometry—music, the universal language.

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth. —Galileo Galilei (1623)

What use is language to anything but consciousness?

“I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything we talk about, everything that we regard as existing, postulates consciousness.” -Max Planck (1931), who was awarded the Nobel prize for his discovery of energy quanta.

Scott Onstott

This article also appears on




“Water is the driving force of all nature.” -Leonardo da Vinci

Where there is water, there is life. That is what is so exciting about finding water on other planets, the possibility of extraterrestrial life.

Suomi NPP space telescope took this in 2012. Image credit: NASA/NOAA/GSFC/Suomi NPP/VIIRS/Norman Kuring

Suomi NPP space telescope took this in 2012. Image credit: NASA/NOAA/GSFC/Suomi NPP/VIIRS/Norman Kuring

Our planet’s surface is covered in mostly water. Our cells are made of mostly water. Seawater and blood are chemically similar. “[Seawater] has even been used in transfusions in emergency situations when blood or plasma wasn’t available and the patient needed to have his blood volume maintained.” –Source

I looked into the structure of water, and was surprised to find that scientists aren’t certain of the geometry of the water molecule. “A recent review listed 46 distinct models, so indirectly indicating their lack of success in quantitatively reproducing the properties of real water.” –Source

The most popular water model (SPC/E) lists the distance between the hydrogen and oxygen atoms at exactly 1 Å (1 angstrom = 10^-10 m). The angle between the hydrogen “arms” (shown as theta in the chart below) varies from 102.7° to 113.24° across the 46 models.



One of the most complex models, Six-site, has the angle at exactly 108°, well within the limits of the rest of the models. Knowing what I do about sacred geometry and the divine proportion, I’d like to propose my own water model. Here it is:

My water model

My water model

The beautiful thing about my model is that water is based on the pentagon/pentagram, the symbol of life. I wrote about this in my Snowflake and a Flower post back in 2011, “5 is the number of life.”

Furthermore, when water freezes, it spontaneously orders itself in hexagonal structures. I wrote, “6 is the number of structure.” Water is its own 6/5 relation, no flower needed.

You can see the hexagonal structure at the molecular scale…

The molecular structure of water ice. Image by P99am CC BY-SA 3.0

The molecular structure of water ice. Image by P99am CC BY-SA 3.0

…and much larger at the scale of snowflakes:


A question I posted on Facebook is, “How does each arm of a snowflake ‘know’ to grow itself in symmetry with the other arms? The arms are very far away from each other, atomically speaking. There must be a field that causes the spontaneous order to emerge.”

Some say it is sound that structures snowflakes but all you hear in a snowfall is silence. Some say it is the environmental conditions of pressure, temperature, humidity, etc that allows the crystals to grow exactly as they do. That is some “environment” or “music” to vary itself with such granularity such that every snowflake grown is unique.

My pond in winter

My pond in winter

Every snowflake appears to me like a crop circle design it is silent brilliance. If geometry were a song or a language then a snowfall would be a symphony or a work of treasured literature. If only we had the ears to hear or eyes to decode the geometric language, what we might learn!

“God geometrizes continually.” -Plato

There are 333 million cubic miles of water on Earth, an interesting volume. Of that only 0.003% is fresh, unpolluted water.

Fir0002/Flagstaffotos CC BY-NC

Fir0002/Flagstaffotos CC BY-NC

“You are not a drop in the ocean. You are the entire ocean in a drop.” -Rumi


There was an amazing crop circle that appeared on July 23rd, 2011 in Wiltshire UK:


Image source

Look closely, the crop circle represents the molecule Melatonin:Melatonin

Image courtesy Sbrools under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Thank you Jeremy Stride (who deciphered Code144 of Coral Castle) for bringing this correlation to my attention with a post on Facebook. My mind is blown for many reasons.

The core of this molecule is a pentagonal ring bonded to a hexagonal ring. See my post on The Snowflake and Flower to understand what I see as the significance of this fact.

The structure is similar to what we see in theobromine and serotonin, or chocolate and happiness, respectively.


Last night myself and a few others had a fascinating late night conversation with the brilliant Bruce Lipton at Hollyhock on Cortes island where I live. Bruce told us about recent research regarding the amazing molecule Melatonin which he likened to our body’s own “chloroplast”.

Bruce went on to discuss the fact that we are literally eating ourselves to death because the body doesn’t really need much food, as strange as that might sound. He said that we might live 140 years if we ate far less. If we sit in the sun for a time each day (while being mindful of our own vibration), our bodies might just get what they need through the alchemy of Melatonin.

“Not only does this amazing hormone counteract stress, fight off viruses and bacteria, improve the quality of sleep, minimize the symptoms of jet lag, and regulate biological rhythms, it may even help protect against cancer, reduce the risk of heart disease, and play a role in how long we live.” -from the book Melatonin by Reiter and Robinson

Consider the case of Prahlad Jani. This Indian sadhu claims to have lived without food and water for 71 years (he’s 82 now). He has been extensively tested in sealed laboratory conditions and defies all scientific attempts to explain how he can survive for weeks without food and water or using the toilet. Read about him in BBC and ABC articles. The Discovery Channel even made a documentary about him.

There are even websites about the benefits of sungazing for nourishment and health. I remember reading about a Catholic Saint in the Autobiography of a Yogi who hadn’t eaten for many years, getting all of her nourishment directly from the Sun.

Melatonin is secreted into the blood by the pineal gland, the tiny “pinecone” at the center of the human brain associated with states of enlightenment. Consider Genesis 32:

And he said, Thy name shall be called no more Jacob, but Israel: for as a prince hast thou power with God and with men, and hast prevailed.

And Jacob asked him, and said, Tell me, I pray thee, thy name. And he said, Wherefore is it that thou dost ask after my name? And he blessed him there.

And Jacob called the name of the place Peniel [pineal?]: for I have seen God face to face, and my life is preserved.

And as he passed over Penuel [pineal?] the sun rose upon him, and he halted upon his thigh. -Genesis Ch 32:28-31

This definitely puts a new spin on all the sun worship I’ve been uncovering in my own work. These synchronicities have blown my mind (again).


Image courtesy of NASA

From Melatonin to Melanin

The pineal gland secretes melatonin which activates the pituitary gland to release melanocyte stimulating hormone. It is in the melanocytes that melanin is produced. –Source

Melanin is a biopolymer and a neuropeptide. In the early 1970s, researchers found melanin to be an organic semiconductor (Science, vol 183, 853-855 (1974)). Studies revealed that melanin acted as an electrical threshold switch, emitting a flash of light—electroluminescence—when it switched. Though the findings were published, these findings largely were overlooked.

Melanin influences neural activity and mediates the conduction of radiation, light, heat and kinetic energy. As such, it is the subject of intense interest in biotech research and development, most notably in organic electronics. –Source

Black skin is high in melatonin and it looks like this is an advantage for both health and enlightenment:

“The incidence of roentgenologically visible pineal gland calcification is approximately twice as common in American whites as in blacks, a difference that is very striking after age 40. Comparison of this finding with reports in the literature shows that the incidence of pineal gland calcification is slightly higher in American blacks than in indigenous Africans, probably due to racial mixture among the American blacks we studied.” –Source

I’ve come across some evidence that Melatonin can even act as a radioprotective agent (to reverse the damage caused by the Fukushima disaster). Enlightenment or bust!

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