Archive for the ‘Geometry’ Category

New Website

In the course of identifying patterns in art, architecture, urban design & the cosmos through the lens of Secrets in Plain Sight, what I find always seems to boil down to sacred geometry.

What is Sacred Geometry?

There is no American geometry, no Russian geometry, no Chinese geometry because geometry is universal. There is only one geometry.

One quickly realizes that simplicity and geometry are the language of timelessness and universality. -Paul Rand (1914-1996)

Ancient sages and modern physicists say that the universe is pure geometry. Sacred geometry is the practice of creating microcosms of universal geometry. As such, sacred geometry is a mysterious mirror reflecting the exterior universe in your interior universe.

Experiencing geometry in this way naturally feels sacred. It is not religious or something to over-intellectualize, but the sacredness of geometry is definitely something to experience.

Sphinx Geometry

As an example of sacred geometry, for millennia geometers have been trying to reconcile the square and the circle, by length or area, trying to bring these two penultimate symbols of straight and curved, rational and transcendental, known and unknowable together.

Even though it was proven that “squaring the circle” exactly is impossible, it didn’t stop geometers from trying to find approximations because the drive to square the circle is part of the human soul. It is the desire to reconcile opposites, to find balance within duality and bring order from chaos.

Squaring the circle

How does Sacred Geometry differ from the Geometry you learned in School?

The geometry you may have learned in school is concerned with axioms, theorems, proofs, tests, grades and so on. This approach to geometry has been a particularly effective means to stimulate rational thinking for more than two thousand years.

“Almost from the time of its writing and lasting almost to the present, Euclid’s ‘The Elements’ has exerted a continuous and major influence on human affairs. It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century. It is sometimes said that, next to the Bible, The Elements may be the most translated, published, and studied of all the books produced in the Western world.” -Bartel Leendert van der Waerden, 1903-1996

Tip: See for a 21st century approach to learning Euclidean geometry.

sacred geometry diagram

Sacred geometry has entirely different foci—the experiential and philosophical aspects of geometry. I find it amazing that we can explore the universe within using a pencil, paper, straightedge and compass. Why are these same tools so effective in exploring and creating in the world, and what does this say about the nature of reality? Where do the universe-within and the universe-without meet?

Sacred geometry is a particularly effective means of stimulating trans-rational thinking—the phenomenological world beyond rationality. Trans-rational thinkers retain their left hemisphere’s capacity to analyze, yet simultaneously treasure their right hemisphere’s capacity to synthesize and see interconnections between microcosm and macrocosm.

Creativity is just connecting things. When you ask creative people how they did something, they feel a little guilty because they didn’t really do it, they just saw something. It seemed obvious to them after a while. That’s because they were able to connect experiences they’ve had and synthesize new things. -Steve Jobs, 1955-2011

Sacred geometry hasn’t been taught much in the last two thousand years, perhaps because its practical value isn’t obvious. However, there is more to reality—and more to living life to its fullest—beyond obvious practical considerations. Geometry is an elegant vehicle that has the potential to move you to experience the exquisite sacredness of existence, which can be powerfully healing and transformative.

Do I have to remember anything I learned in School to explore Sacred Geometry?

The short answer is not really. Of course there is a basic language of points, lines, circles, angles, etc. that is helpful to recall. Prior knowledge of classroom geometry only enhances the experience but is by no means necessary. What you need to know about the mechanics of drawing sacred geometry can be learned in 15 minutes—it’s very simple and anyone can do it. There are no tests, grades, theorems or proofs. The purpose of drawing with the traditional instruments is to lead you to an experience, an experience shaped by the perfection inherent in the geometry, guided by your own higher wisdom.

New Website

Metatron's cube

I built a new website called Sacred Geometry Academy ( to initiate small groups in the universal language of sacred geometry in the course of intensive weekend workshops. This is the most effective way that I have found to ground the message of Secrets in Plain Sight in one’s own life. Personally experiencing sacred geometry in a small group can be a most powerful and transformative experience. The first such workshop was a great success and you can read what the participants wrote about their experiences here.

If your budget allows I invite you to make time to attend one of these events:

  • Access extraordinary states of mind simply by drawing with pencil, straightedge & compass and adorning what you’ve drawn with natural materials.
  • Move beyond talk into life-changing transpersonal experiences in a safe space where you can connect with your own inner-healing intelligence.
  • See how everything around you is in a very real way interconnected by perceiving all-pervasive patterns of geometric order surrounding our lives.
  • Learn to properly draw with compass and straightedge even if you have never drawn before. Everyone can do it!
  • Enjoy a deeply rewarding sensation of satisfied completion with every drawing.
  • Use the traditional tools of the Tibetan Masters—chak-purs.
  • Interact with the archetypal forces operating just below the psyche’s surface.
  • Harness the power of mandala to move beyond the thought stream of the mind.
  • Feel group connectedness when co-creating a large mandala.
  • Honour the impermanence of all things through a letting go / gratitude session.
  • Give yourself time, space and permission to set aside the outside world for one weekend and explore geometry in a way that can facilitate a reconnection with your essential centre.

I’ve had to close comments on this site due to increasing wordpress comment-spam. If you have any questions or comments please feel free to contact me directly.

The Metre

The metre (spelled meter in United States) was introduced by the First French Republic in 1799. It was originally defined to be one ten millionth of the distance (10,000 km) between the North Pole and the Equator through Paris.


Since then the metre has been defined variously as the length of certain prototype objects, the wavelength of a specified light spectra, and since 1983 in terms of the speed of light, which is itself defined by the metre and the second.

The first metre, Paris; Image Ken Eckert CC BY-SA 4.0

The first metre, Paris; Image Ken Eckert CC BY-SA 4.0

I discovered that the metre exposes something amazing about the absolute size of the Earth and the Moon. The Earth’s polar diameter is 10,000 times the square root of the divine proportion (aka golden ratio, approximately 1.618033989) in kilometres. The combined polar diameters of Earth and Moon are 10,000 times the divine proportion in kilometres. My book The Divine Proportion delves deeply into this amazing ratio which is symbolized by the Greek letter Ф.


The reason the above correlations are only 99.97% accurate is because when GPS satellites became operational in the 1990’s we learned that one ten millionth of the distance between the North Pole and the Equator is not exactly 1 m as extrapolated at the end of the 18th century, but is actually 1.000197 m.

In 1973, John Michell published in The View Over Atlantis that the Great Pyramid’s slope of 51°51′ encodes the proportions of the Earth and Moon. The base of the pyramid in the following diagram corresponds with the Earth’s equator. The apex of the pyramid coincides with the center of the Moon when the Moon is drawn tangent to the Earth as shown. The sole surviving ancient wonder of the world therefore encodes the proportions of Earth and Moon in its aeons-old architecture.


I discovered that the Great Pyramid’s location on our planet is divinely proportioned with respect to the longest possible distance on land, based on the global sea level since the last ice age.

Published in The Divine Proportion in miles

Published in The Divine Proportion in miles

In my Secrets In Plain Sight Volume 1 film, I pointed out Webb’s discovery that the Great Pyramid’s latitude (29.979245.8°N) precisely encodes the speed of light (299792458 metres per second).

The Great Pyramid's latitude encodes the speed of light

The Great Pyramid’s latitude encodes the speed of light

The metre is officially defined as, “the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second,” and the latitude 29.979245.8°N makes me wonder if the pyramid was originally designed as a shining mountain of metrology? The metre might have merely been rediscovered in relatively recent times (circa 1799) as an obvious method of rationalizing the circumference of the Earth.

The fact that the light speed latitude and the Ф point along the longest distance on land should coincide is nothing short of miraculous.

Author Gary Osborn discovered that the location of the so-called granite “coffer” within the “King’s Chamber” inside the Great Pyramid is the focus of a golden spiral (based on the divine proportion). This explains why the King’s Chamber is offset with respect to the centerline of the pyramid. Read on to see how this supports the notion that the coffer may have been used as an resonation chamber for the human body.

Gary Osborn ©2005

Gary Osborn ©2005

I speculate that they put salt water in the coffer and the ancient initiate floated in the warm dark like people do in modern isolation tanks. It would be the ultimate place for the human body to resonate with the pyramid mountain, the Earth’s landmass and the Earth-Moon system.

Consider also that the human body is divinely proportioned with respect to the scale of all things. We are literally the bridge between macrocosm and microcosm.

Flinders Petrie meticulously surveyed the Great Pyramid in the 19th century. See page 27 of The Pyramids and Temples of Gizeh for Petrie’s measurements of the King’s Chamber. Petrie measured everything in inches but if we convert to metres, something amazing pops out—the King’s Chamber encodes divine proportions directly in metres.

The dimensions of the King's Chamber encode the divine proportion in metres

The dimensions of the King’s Chamber encode the divine proportion in metres

The base circumference of the King’s Chamber is (2 x 5.24) + (2 x 10.47) = 31.42 m. 10π is approximately 31.42. Thanks to Leo Nolette for spotting this connection. So the King’s Chamber encodes the ratio of a circle’s circumference to its diameter in metres and of course base 10.

I have created numerous illustrations of the Great Pyramid’s mysteries over the years and published many of them in my books. This one shows how the pyramid “squares the circle” and how the mathematical constant e (Euler’s number) is encoded by comparing the pyramid with an inscribed equilateral triangle.

This one shows how the human body resonates with the divine proportion and the pyramid form.


This one shows how the Sphinx was located in proximity to the G2 pyramid (neighboring the Great Pyramid) to purposefully encode a single geometric template based on the vesica piscis and the divine proportion. This setup creates a sympathetic resonance as above, so below.

“Through the phenomenon of cosmocization…the human becomes microcosm of the house or temple, just as the house or temple becomes microcosm of the cosmos. The inverse is true as well: the cosmos is macrocosm to the house, the house is macrocosm to the human, and the human is, therefore, in profound interrelationship with the cosmos.” -Beverly White Spicer

Cosmocization is what I referred to in the cover of my most recent book, Tripartite.


In 1999, Osborn showed the correspondence between what I’m now calling the Cosmocization Chamber (aka King’s) and the third eye at the center of the human brain.

Gary Osborn ©1999

Gary Osborn ©1999

Osborn later connected the physiology of the third eye to the ancient Eye of Horus symbol, which he called the Gate of God.


The third eye is the mind’s eye, the Gate of God, the locus of enlightenment. This reminds me of a quote from my essay on Light.

“If we consider the word light as a euphemism for the word knowledge, our definition of the word light is then clarified. We then find that light is knowledge and we are all on the path of true knowledge.

Now, knowledge as listed in the dictionary comes from the French root word knowen, to know; (1) having understanding gained by actual experience, or practical experience; (2) the state of being aware of something or of having information; (3) the act of understanding: clear perception of truth; (4) something learned and kept in mind.

If we interchange the word knowledge for the word light we have the true Rosicrucian meaning…If light is knowledge, and knowledge is born intuitively, then the “Path of True Light” is not outward but inward.” –Ronald Cannata (1971)

The Great Pyramid elegantly encodes great truth using the universal language of proportion and metrology. The way that it does so on multiple simultaneous levels is truly illuminating. We still have much to learn from this secret in plain sight.

Location, Location, Location

It has long been known that the three most important factors determining real estate value are: location, location, location. I notice that this aphorism is true not only for residential and commercial properties, but also somewhat unexpectedly for sites having great symbolic and/or spiritual significance.

My research reveals a previously unknown system of non-monetary valuation transcending any particular culture, tradition, or time period that takes the entire Earth and the built environment into account.

Repdigits and transcendental mathematical constants are the keys to locating sites in this system. For example, consider the fact that the Western Wall in Jerusalem is 666.6 nautical miles from the Kaaba in Mecca. These two are the most sacred sites in Judaism and Islam.

'Sacred Stones' from Quantification

‘Sacred Stones’ from Quantification

Why nautical miles? There are 6x6x600 nautical miles in the meridian circumference of the Earth, revealing more repdigits in the unit used to do the encoding.

The nautical mile was originally defined as one arc minute and the metric system was originally defined by dividing the arc length from the pole to the equator into 10 million parts called metres. So one degree is 10,000,000 m / 90° = 111,111.111… m and one arc minute is 111,111.111… / 60′ = 1851.85185… m.

The nautical mile was defined as exactly 1852 m by international convention, so they rounded it off to the nearest meter. This obfuscates two very interesting patterns. The repeating numbers use the same digits of the Great Pyramid’s slope angle of 51.85° (51°51′) and the following:

1851.85185… yards = 5555.555… feet = 66,666.666… inches.

There is also an unusual link between nautical miles and meters:

666.6 nautical miles = 1234543.2 meters
6666.6 nautical miles = 123456543.2 meters

The black stone in the Kaaba is 5055005 yards from the Mahabodhi Temple in India, the location of the original tree under which the Buddha became enlightened.

'Sages' from Quantification

‘Sages’ from Quantification

The Buddha apparently died 333,333 feet from the place where he was born.


The distance between the center of the world’s largest Buddhist meditation center (capacity 1 million meditators) and the third eye of the world’s largest Buddha statue is exactly e million (2,718,282) yards. The transcendental constant e is Euler’s number, used in finance, engineering and many scientific disciplines.

300,000 Buddha statues adorn the Dhammakaya Center and the yard is a unit equal to 3 feet. The ‘Great Retreat’ in Buddhism correspondingly lasts 3 years, 3 months and 3 days.


According to the Bible, Jesus lived 33.3 years and performed 33 miracles. Did King David, father of Solomon, really rule for 33 years and did Jacob really have 33 children or are these repdigits actually codes?

The Kaaba, the stone cut temple of Ellora in India and the largest religious complex in the world at Angkor Wat not only form a great circle alignment but the distances between them also encode 33° and π.


Long ago, the Western Wall in Jerusalem formed the foundation of Solomon’s Temple. Solomon’s Temple was 33.33° from Stonehenge.


The Dome of the Rock stands on the site of Solomon’s Temple and is 3333.33 km from the center of the symbolic Place de la Nation of Paris.


In the 19th century, one of Cleopatra’s Needles was moved exactly 3333.33 km from the harbor wall in Alexandria to where it now stands in London. Likewise the Ishtar Gate of Babylon was moved 3333 km to the Pergamon Museum in Berlin.


The Eiffel Tower stands 3033.33 km from the site of the former Lighthouse of Alexandria, Egypt. In turn the Lighthouse was 3333.33 km from the standing stones of Carnac, France. The oldest standing obelisk in the world, in Heliopolis Egypt, is in a great circle alignment with the Lighthouse of Alexandria, Egypt and the replica Lighthouse of Alexandria, Virginia (333 feet high), officially titled, “The George Washington Masonic National Memorial.”


The center of the Eiffel Tower is exactly e million (2,718,282) feet from the center of the Cathedral of La Sagrada Familia in Barcelona, which is topped by solar discs. The Sun disc’s circumference is e million miles (99.99%).


The world’s tallest building is currently the Burj Khalifa in Dubai. It is e thousand feet high and 1633333 meters from the Kaaba. The total land area of the League of Arab States equals 13333333 square kilometers (99.9997% accurate).

'Arabic Connections' from Quantification

‘Arabic Connections’ from Quantification

A higher structure known as the Kingdom Tower is currently under construction in Jeddah, Saudi Arabia. It will be 3303 feet high when completed. Funding was just secured to complete the tower, probably by 2020.

The Kingdom Tower will be 33.3° from the Golden Temple in India, which the most sacred shrine in Sikhism.


The center of the Kingdom Tower of Jeddah is 3,333,333 inches from the word Allah (in Arabic calligraphy) atop the world’s highest and largest clock tower in Mecca, steps from the Kaaba. The Kingdom Tower is also oriented so one leg points at the Kaaba and the Abraj Al-Bait Tower.


The vector connecting the word Allah on the clock tower terminates at the center of the equilateral triangular void within the Kingdom Tower, symbolizing the Eye of Providence, or Allah in this case.

Eye of Providence from Jewish cemetery - Image Aw58 CC BY-SA 3.0

Eye of Providence from Jewish cemetery – Image Aw58 CC BY-SA 3.0

Sometimes important sites are related by repdigits within cities. For example, consider Tehran, Iran. The Tower of Freedom is 55,555.5 feet from the center of the Mausoleum of the Ayatollah Khomeini. Interestingly, this distance is also equal to 666,666 inches and both complexes ironically have similar plans. The Tower was built in 1971 to honor the Shah and later renamed Azadi, meaning Freedom, after the Revolution of 1979 that ousted the Shah. The architect of the tower was a Baha’i who was driven out of the country when religious pluralism was rescinded in the Revolution of 1979, led by the Ayatollah Khomeini. It is therefore highly unlikely that there was a conspiracy to locate the Mausoleum of the Ayatollah in a repdigit relation with the Shah’s tower. The encoding transcends culture and religion, a secret in plain sight.


I have thus presented you with many such secrets in plain sight, many of which hinge on different units of measure. These units of measure are ultimately based on, and have surprising resonances with the dimensions of the Earth and the Moon. For example, the Imperial system of inches, feet, yards, furlongs and miles has clear resonances that are far from accidental. There are 63360 inches in a mile (5280 feet/mile x 12 inches/foot). There are 63360 furlongs in the Earth’s diameter. The circumference of a circle encompassing the Earth and the Moon measures 63360 miles.


John Michell discovered that the combined Moon & Earth radii measure 1x2x3x4x5x6x7 miles. The arc length around one-quarter of the circle circumscribing the Earth and Moon is 8x9x10x11 miles, which is also curiously equal to the Earth’s diameter.

Read the essay “Yards” in my Anthology for more info on the resonances of that particular unit.

The Imperial system is not unique in having resonances with the Earth and the Moon. The Metric system does also. The combined diameters of the Earth and the Moon are equal to 10,000 times Ф km (99.97% accurate). The symbol Ф represents the Divine Proportion. The Earth’s polar diameter is 10,000 times the square root of Ф km. The Earth and Moon are divinely proportioned with respect to each other as shown by the triangle and the diagram showing two intersecting golden rectangles tangent to the Earth below.


In fact the Imperial and Metric systems are related by the divine proportion, or nearly so (see the tiny half percent gap on the upper right).


So our systems of units resonate in multiple fascinating ways with the structure of our planet and its natural satellite. Here are some surprising repdigit patterns in our units, revealed by a hexagon inscribed within the Earth.


I do not believe the Moon was made by aliens or is made of cheese, nor do I believe the Earth is flat or that sea-monsters patrol its edges. Please read David Mathisen’s well-written article, “The Invisible Kraken: Evidence that the Earth is not Flat” and think about it if you are a “Flat Earth Truther.” Do not contact me with flat earth theories or conspiracies about NASA or scientists in general, or about people suppressing flat earth theories or the theorists themselves.

“Belief is the death of intelligence.” -Robert Anton Wilson

So what do I ascribe to the meaning of location, location, location? Although you might initially think that these structures were sited according to a grand human conspiracy, the great antiquity of many of these examples stretches this theory well beyond its rational breaking point. The standard materialist view would have physical agents coordinating all of this, so aliens are the next most likely suspects. However, consider their motives.

If aliens encoded repdigits and transcendental mathematical constants both in our units of measure and in the locations of many significant structures, they must have had for thousands of years, and continue to have up to the present time, intimate knowledge about what our species considers significant. If we could put aside our primate/primal fear of people who aren’t from around here—aliens—we might consider that these agents know us very well, so well perhaps, that they must not be alien at all. They might know us better than we know ourselves.

Why would anyone bother to encode all of this—what could they possibly hope to accomplish with such a massive effort? Clearly the encoding transcends any particular culture, ethnic group, religion or belief system, spans a great deal of time, and reveals a high level of intelligence. As such it appears to be a universal message to humanity, one that has been there in plain sight all along, waiting for discovery. This has only been possible for about a decade, since GPS satellite data has been publicly accessible through computer interfaces such as Google Earth. Now that the genie is out of the bottle, what does it mean?

Encoding messages in the locations of significant structures is a non-destructive, gentle approach that reminds me of the low-impact of crop circles. Information is conveyed but it remains entirely up to us to assign meaning, if and when we are able to do so and curious enough to do it.

I think we have to work this out backwards, in other words, “by their fruits we shall know them.” The fruits have to be knowledge and perhaps human enlightenment. So who would want to educate and enlighten us?

We can rule out God, unless you define God as a being who transcends religion. We can rule out aliens too, unless they aren’t actually alien and have transcended their own selfish agendas.

More likely are higher selves and/or spiritual beings, but that requires one to consider trans-material reality—things beyond that which we can readily weigh and measure.

It’s amazing to me, but scientific materialists who believe that everything is made of atoms tend to see no contradiction that everything came from nothing in the Big Bang. Scientific materialists also believe in dark matter and dark energy. Neither dark matter nor dark energy has ever or probably can ever be detected, and yet we are supposed to believe these substances are 96% the mass of the universe, because that figure makes the math in the standard model of physics work out. That is like saying invisible pink unicorns are 96% the mass of the Universe, an unfalsifiable (you can’t ever prove that it is false) and therefore unscientific premise. Time to rethink the standard model?

“According to the standard Big Bang narrative, space and time did not exist prior to the Big Bang. Therefore, the “cause” of the Big Bang could not have existed within space and time. In that case, it existed outside space and time, hence was dimensionless, hence was immaterial, hence was mental! The very thing that scientific materialism claims cannot exist — independent, causal mind — is the only possible, plausible candidate for explaining the Big Bang.” –Hyperreason by Mike Hockney

In addition, we can rule out dark matter and dark energy as immaterial encoders (assuming they even exist) because these universal forces or fields presumably don’t “know” anything, including what is significant to our particular species.

The encoders are minds possibly imagined by a singular mind (the Universe itself) with an intelligence that chooses to encode repdigits and transcendental mathematics in the locations of our significant sites for some teleological purpose. I suspect that this purpose is to teach us eventually who we are.

Information processing networks

Information processing networks

“Let us pretend that there was a machine, which was constructed in such a way as to give rise to thinking, sensing and having perceptions. You could imagine it expanded in size (while retaining the same proportions) so that you could go inside it, like going into a mill. On this assumption, your tour inside it would show you the working parts pushing each other, but never anything which would explain a perception.” -Gottfried Wilhelm von Leibnitz

Perhaps the first thing to realize is that more is going on here than just atoms. Quantum physicists have known for a century that atoms are really 99.9999999999% empty space, but we have yet to come to terms with that information.

99.9999999999…% = 100%.

The only possible descriptions of sub-atomic particles are mathematical descriptions because that is precisely what they are: mathematics.

Our Pater & Mater (Latin: Father & Mother) are Pattern and Matter

Our Pater & Mater (Latin: Father & Mother) are Pattern and Matter

“If you only knew the magnificence of the 3, 6 and 9, then you would have the key to the universe.” -Nicola Tesla

This is the key: the universe is 100% mathematics, the universal language. There is no objective “substance” out there. However, if all is number, then who are you?



Saturn's Patterns

Saturn’s Patterns

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 begin 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

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