Archive for the ‘Geometry’ Category

Macrocosmic Harmony

The best temples serve as bridges between human scale and the macrocosm. When Beverly Spicer sent me the following image, something clicked and I understood the human body, the Sphinx and the Pyramids in a new way. I’m not sure which book this black-and-white geometric analysis of Pharaoh Rameses II was scanned from—it appears to be mirrored all over the Internet.


The nemes headcloth has distinctive, straight-edged folds, which imply an intersection point above the head. Nested equilateral triangles locate the pharaoh’s third eye appropriately as an all-seeing-eye in the center of the triangle (like the eye of providence on the dollar bill).

That reminds me of an image I made recently. Here is essentially what I wrote about it on Facebook:

Beauty is experienced in the world, love in the body, and truth in the mind. The eye of horus in the center symbolizes the view of any observer. The transcendentals are ontologically one and thus they are convertible: e.g., where there is truth, there is beauty and love also. The transcendentals are irreducible qualities of awareness. The circle represents the seeming boundary between your interior sensations, thoughts, and perceptions and your exterior body, mind, and world.


Just as the transcendental qualities of awareness (love, beauty, truth) are ontologically one, so too are the trinity of data types you experience. How far from your awareness are your sensations, perceptions or thoughts? Examine your experience and you will see that there is no distance.

Do the thoughts running through your mind take place in a fundamentally different place than the sensations of your body or your perceptions of the world? They are all convertible because all this data is experienced by awareness and yet we have no trouble identifying which channel the information flows through. There is a fundamental and irreducible trinity to awareness.

I now see this ontological trinity as the fountainhead of relationships such as Isis/Horus/Osiris, Mary/Jesus/Holy-Spirit, and even Neutron/Proton/Electron.

Following a hunch I wanted to see if the Sphinx’s nemes followed the same pattern. I discovered that the pattern is similar but is also subtly different. The Sphinx’s nemes doesn’t imply nested triangles but the uraeus symbolizing the pharaoh’s pineal gland is located in the center of the vesica piscis as indicated by the crossed red lines. Incidentally, I described how to draw this image step-by-step in the latest installment of my Photoshop User Magazine column, Beyond Photoshop.


Zooming out, the G2 pyramid also appears to fit the pattern.


It clicked for me when I realized that the geometric template for the Sphinx is the same as the geometric template for the Great Pyramid that I published in my book Quantification. The edge of the inscribed equilateral triangle measures 555.5 feet in the Great Pyramid.


In this way the House of the Temple (it was never a tomb) is a macrocosm of the Body carved on the Sphinx. Zooming out once again, the Engraved Hourglass Nebula is a macrocosm of that House. As above, so below.


The above image wonderfully illustrates Mircea Eliade’s homology, Body—House—Cosmos, from his book “The Sacred and the Profane” (1959).

A big synchronicity for me is that I learned about this three-level holographic homology not from reading Eliade but from reading Beverly Spicer’s excellent book and PhD thesis from the School of Architecture at the University of Texas called, “The Ka’bah: Rhythms of Culture, Faith and Physiology,” which she was kind enough to send me. Remember Beverly was the person who sent me the black-and-white Rameses II image? That whole sequence of events was quite illuminating!

Speaking of illumination, did you know that the human eye is most sensitive to a wavelength of 5555 Ångströms, which forms the center point of our visual perception? At other frequencies, more radiant intensity is required to achieve the same luminous intensity.

See how the colors are strongest in the middle of the color ramp and fade away as they approach the infrared and ultraviolet ends of the spectrum? 5555Å is the central wavelength.

5555 Ångströms

5555 Ångströms

The peak luminous intensity of sunlight also has a wavelength of 555.5 nanometers (5555 Ångströms). This should be no surprise because our eyes evolved, or perhaps were designed to view the light of our particular Sun. The beauty of repeating 5’s harmonizing our visual perception, 5 symbolizing life, is not lost on me.

After reading a stimulating article Joe Dubs linked on Facebook entitled, “The Pythagorean Relationship Between Pi, e, and Phi” by Peter Felicetti, I realized I already knew the content, but I hadn’t considered the following geometry algebraically before. One can arrange Phi, Pi and e as an almost perfect Pythagorean triangle, but there is a tiny gap that I labeled “The Mystery” at the end of my article on Graham Hancock’s site.

This mystery is drawn to scale

This mystery is drawn to scale

When I saw it algebraically, a few more puzzle pieces fit together and it is no longer a mystery to me but a harmonious work of rational design from a very high level.

Tech note: I realize that some of my detailed graphics might be hard to see. I recommend pressing Command+ on the Mac or Ctrl+ on Windows several times to enlarge the content displayed in your browser. Use Command/Ctrl- to make it smaller later on.

The Quadrivium

The Quadrivium

The same gap that we hear between twelve just perfect fifths and seven octaves (known as the Cycle of Fifths or the Pythagorean comma), the relationship of transcendental constants arranged in a Pythagorean triangle, and the orbital mechanics of our planet are all related because we are living in a mathematical construct. This was known long ago.

“All is number.” -Pythagoras

The above image expresses the quadrivium: number, geometry, music, and astronomy. In other words, number in itself, number in space, number in time, and number in spacetime.

I knew the Solar Year also factors in the Earth’s equatorial circumference—the following equation was published by John Michell and Robin Heath in The Lost Science of Measuring the Earth. I made this illustration commemorating this important relationship, one that essentially defines the foot.


Speaking of Michell and Heath, they were the authors who deciphered the 5:12:13 triangle and its connection with the lunar cycle. When I was researching extending the Pythagorean theorem into three dimensions, I came across a diagram connecting the 5:12:13 and 3:4:5 triangles. That, and studying Gary Osborn’s work, inspired me to put together this image showing how dynamics related to the Earth, Moon and Sun emerge from something as fundamental to geometry as the first two Pythagorean triples.

Number in space and spacetime

Number in space and spacetime

Earth’s orbital circumference has certain clear resonances with our measurement systems.


Bringing together what I wrote about in Cubits, in my book Taking Measure: Explorations in Number, Architecture and Consciousness, and from my recent experience participating in Graham Hancock’s forums when I was author of the month, these pieces now fit together with greater clarity.



Michael Kuser wrote something on my wall that inspired the following image. The first part shows the combined diagonals of the cubes bounding Earth and Moon compared to their average distance from the Sun are proportioned as the foot is to the mile (99.3%).

Lawrence Edlund discovered that the average distance of the Earth to the Sun compared to the distance light travels in one revolution of the Earth around the Sun are proportioned as the inch is to the mile (99.9%).

I put two and two together and graphically show how our units of measure resonate with our local macrocosm.


In “Rotations” you see more proof that our system is a product of base-10 design. The Moon’s orbital period with respect to the fixed stars is 27.3 Earth days. Earth actually rotates 366.3 times per year relative to the stars because the orbital motion of the Earth around the Sun causes roughly one additional rotation of the planet on its own axis. Here is what Knight and Butler had to say about this phenomenon in Before the Pyramids (p240), where they describe how the megalithic system is based on the division of a circle into 366 units:

“In a solar year the Sun rises 365 times but, during the same time, a star will have risen 366 times. It sounds odd but it’s true. Each day according to the rising of a star (a sidereal day), is 23 hours 56 minutes 4 seconds in length, whereas a mean solar day is 24 hours in length. That leaves a discrepancy of 236 seconds, which over the course of a year amounts to another 24 hours. It is part of the clockwork mechanism of our solar system that there are different sorts of years, dependent on what one is observing. Our megalithic and pre-megalithic ancestors in Britain focused on the number of times a star rose in a year, and the result was 366 times [366.3 with one more digit of accuracy].”



Tonight I realized that the product of these rotations, 27.3 x 366.3 = 9999.99. What are the chances? I can’t begin to calculate the astronomical odds against such a numeric symmetry if our planet and its natural satellite were formed by random processes. Note that this product is independent of units and calculated solely on number of Earth rotations relative to our Moon and Sun.

Peter Plichta’s book God’s Secret Formula: Deciphering the Riddle of the Universe and the Prime Number Code taught me that 1/27.3 ≈ 0.03663 and 1/366.3 ≈ 0.00273, which you can easily verify with a calculator.

The ratio of the polar diameter of the Earth to the polar diameter of the Moon is 3.663 and vice-versa, the reciprocal ratio is 0.273 with more than 99.9% accuracy. This means the Earth and Moon system’s rotations and sizes are related in a reciprocal mathematical relationship.

Of course this is another example of what I have called “The Mystery of 273,” in this page out of Quantification.

The Mystery of 273 (update)

The Mystery of 273 (update)

Is what we call “conscious thought” incredibly rare in the grand scheme of things, relegated as it is to 2-3mm of neocortex on the surface of our brains?

We and all other organisms on this planet live in a thin biospherical shell 2-3 miles thick. The Earth orbits the Sun in the sweet spot where oceans are kept liquid and the planet retains a transparent atmosphere so perfect for astronomy and our survival. A planetary magnetic field and ozone layer shields us from the charged particles continually pouring out of the Sun, along with many variables conductive to life being fortuitously fine-tuned.

Our system orbits the center of the Milky Way galaxy in another sweet spot, a circular band where products of stellar fusion such as elements heavier than helium are plentiful enough, but star formation isn’t still so active that we are bombarded out of existence. In this light, consciousness appears to be a gossamer membrane, a rare and exquisite miracle in all the cold vastness of space. This membrane/shell structure is another Body—House—Cosmos homology if you think about it.


The following comparison highlights another possibility. The neuronal structures within our brains, roads (which evolved organically around Paris rather than through city planning as in much of North America), and the large-scale structure of the universe (simulated on the Max Planck supercomputer) all exhibit the same specific branching, filamentous pattern. Come to think of it, so too do mushroom mycelial mats, and maps of the entire Internet. What these disparate phenomena have in common is that they are all information processing networks.



If we view Paris from the level of cars, then roads don’t seem to be self-aware. If we view the large-scale structure of the universe from the vantage point of supercomputer simulations computed on Earth, it doesn’t seem self-aware. If we were able to view a neural structure from the point of view of electrons it probably wouldn’t seem self aware to them either, just a kind of electrical network.

However, when we use our own neural structure to introspect, then awareness is self-evident. It’s really a question of which way you are looking. If the universe looked inward, then it presumably would be self-evident that it is self-aware, including all its in-formation including suns, planets, cities, roads, people, neurons, electrons, etc.

The amazing thing is we experience this same self-awareness at our level. We are a microcosm of larger wholes and have microcosms within. Perhaps cells are also self-aware, coordinating the activities at their tiny scale. The organism of which they are a part wouldn’t seem to be self-aware to them at all. Talk about all that would seem like mere philosophy.


This is yet another example of the Body—House—Cosmos homology. The human built environment accessed by roads, our “House,” is a macrocosm of the neurons in our Body, and the large-scale Cosmos is a macrocosm of our House. It is the same form, harmonizing and coordinating information at vastly different scales.

The universe itself looks like it might literally be “the mind of God,” a self-aware matrix in the process of creating complex in-formation. Awareness is not rare, but the ground of being. There has only ever been, is, or ever will be awareness being aware of itself.

“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 in The Observer (1931).

That said, our limited manifestation of consciousness might be rare and in a way, precious. We live at a golden scale in the macrocosm.



However, our awareness doesn’t require human minds anymore than the Sun requires a candle for illumination.


Everything is sacred, everything is precious.

Recent Articles

I’ve recently written a few articles outside this site that you might enjoy. I was Graham Hancock’s author of the month back in September 2014 and I wrote “Measuring the Mystery” for that occasion, which features several images from my book Quantification that hint at many intriguing mysteries.


Ancient Origins published an article I wrote in November 2014 entitled, “Mathematical Encoding in the Great Pyramid” which goes into the multi-layered encoding in the pyramid, exposing the mathematical nature of reality.


I was pleased to be included in a select group of writers included in Ancient Origin’s second year anniversary edition ebook, Unraveling the Mysteries of Ancient Artifacts, where I wrote an article entitled, “The Mystery of Obelisks.” You can download this fascinating ebook by subscribing for free. The mystery in the title refers to the obelisks’ precise positions on the Earth.


I wrote a step-by-step tutorial entitled, “Drawing in Photoshop” in Photoshop User Magazine explaining how to draw the following image, which I discovered encodes the sacred geometry of the Sphinx and the Pyramid. I have been writing a column called “Beyond Photoshop” in this print magazine since 2007, but this is one of my rare articles that crosses over into SIPS territory.


I also discussed how this image came about in Macrocosmic Harmony. This image got the most traction of any I’ve ever posted on my Facebook wall, with over 500 shares to date. I think that is because the image shows the simple beauty of how sacred geometry is the basis of form.


As above, so below.


In no particular order, here are some new images I’ve created recently. I posted these on my Facebook wall but they quickly got lost in the ephemeral feed. This blog is a much better repository for this research.

It turns out that the following 6:5 relationship in the Pyramid of Khafre is 99.9% accurate. The discovery was made by Peter Casperson in his book Den Matematiska Nyckeln Till Guds Namn (which translates from Swedish as “The Mathmatical Key to God’s Name”) published in 2004. The illustration is my own. The 6:5 relationship ties in with what I’ve written before in Taking Measure, the Snowflake and the Flower, and Cubits, among other blog posts.

The Pyramid of Khafre encodes 6:5 proportions

The Pyramid of Khafre encodes 6:5 proportions

The outer perimeter of the egg’s cross-section is 33,333 feet assuming the eggshell has thickness of ~5 feet at the scale of the Great Pyramid. I made a similar oval image a few years ago called The Megalithic Egg but it was based on the 3:4:5 triangle instead.

The circumference of the egg based on the geometry of the Great Pyramid is 33333 feet

The egg formed by the elevation of the Great Pyramid has a circumference of 33333 feet

“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.” -Aristotle

I made a round image (with Photoshop’s polar filter) of moai on Easter Island looking into the center, and then felt inspired to have them look at the Orion nebula. The 15 moai fit the stages of the alchemical process and I took the stage names from a fantastic novel by Katherine Neville called The Eight that I was re-reading. The philosopher’s stone naturally fits in the center as the final goal of al-kemi. I was playing with the swirling colors as seen above in the nebula or below in the chemicals of the alchemist’s cauldron. Are we looking into the heavens or deep within? They end up in the same place: awareness.

The Great Work

The Great Work

Mostly I have been working on paying projects at lately with little time for SIPS, although Dennis Fetcho interviewed me yesterday and we talked about SIPS and Quantification.

Here is a full accounting of our solar system:

Inspired by Don Barone on Graham Hancock's forum

Inspired by Don Barone on Graham Hancock’s forum

Is this the basis of the Fahrenheit scale, why water boils at 212°?


It’s interesting also that 33°F = .55555…°C

Here’s a discovery I made some time ago but don’t think I’ve posted it in the blog before. The Governor’s mansion at the top of Victoria Peak overlooking Hong Kong is 3333.3 km from Borobudur. Amazing distance between city mountains: sacred and mundane!


The Eye of Horus symbolizes the view of an observer, as an all-seeing-eye.

The Eye of Horus

The Eye of Horus

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, The Assayer, 1623

The World Wide Web


The traditional Buddhist retreat lasts for 3 years, 3 months and 3 days. Make sense?



I was Graham Hancock’s author of the month for September 2014 and wrote the following article called Measuring the Mystery.

I wrote an article called Mathematical Encoding in the Great Pyramid on


Here is the universal accounting, according to cosmologists. They actually give slightly different values but I looked at the uncertainty in the measurements and think the following percentages are the canonical values, as they fit within the tolerances of the measured data:


We can’t measure dark matter or dark energy. They pass through ordinary matter without interacting. Cosmologists needed something to explain why galaxies and galactic clusters don’t fly apart so they invented “dark matter”. Then they needed something to explain why expansion is accelerating so they invented “dark energy”. These quantities are more of a measure of our ignorance. However, regardless of whether dark energy and dark matter turn out to be like epicycles in earlier cosmological models, the math needed to account for our observations is balanced using the above breakdown, and I find that very interesting.

Discovered by Kevin McMahon

Discovered by Kevin McMahon

Urban Asplund discovered that the distance from the “Pyramids of the North” at Old Uppsala, Sweden is 3333.33 miles (99.99%) to the Obelisk of Axum, Ethiopia (which was stolen and moved to Rome for most of the 20th century and returned and re-erected in Ethiopia in the 21st century).


“The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and…there is no rational explanation for it.” -Eugene Wigner, nobel laureate


“How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?” -Albert Einstein

“Everything we call real is made of things that cannot be regarded as real.” -Niels Bohr






“The most beautiful system of the sun, planets and comets could only proceed from the counsel and dominion of an intelligent and powerful Being.” -Introduction to the Principia by Isaac Newton

Here is the proof, chocolate is a very close cousin to happiness. You knew that already!




There is actually a minimum and a maximum temperature. Absolute zero is the coldest possible temperature where there is absolutely no movement. The temperature of the big bang has been calculated as 33 orders of magnitude hotter than absolute zero, as measured in degrees Kelvin. At this maximum temperature all four forces (gravity, electromagnetism, strong and weak) merge into one force and the fabric of spacetime itself is thought to dissolve into nothingness.


Quantum foam (also referred to as space-time foam) is a concept in quantum mechanics devised by John Wheeler in 1955. The foam is supposed to be conceptualized as the foundation of the fabric of the universe. This “foam” is the smallest measurable length, representing the granularity of spacetime itself. The human eye is 33 orders of magnitude larger than the quantum foam.


The gap in quality

The gap in quality

Thanks to Joe Dubs for the factorial connections!

Thanks to Joe Dubs for the factorial connections!

John Michell discovered that 10 times the perimeter of the blue hexagon equals the mean distance from Earth to Moon (99.4%). Even more accurately, that’s 6 x 60 x 666 miles (99.6%).






“Several words derive from Amun via the Greek form, Ammon, such as ammonia and ammonite. The Romans called the ammonium chloride they collected from deposits near the Temple of Jupiter Amun in ancient Libya sal ammoniacus (salt of Amun) because of proximity to the nearby temple.” The Colossoi of Memnon stand at the entrance to what was once the Mortuary Temple of Amenhotep III. The name Amenhotep means “Amun is satisfied.” The name Amun meant something like “the hidden one” or “invisible”. -wikipedia




The Brandaris connection was discovered by Shannon van den Heuvel:



The royal Egyptian cubit is approximately this same distance, .5236 meters or 1.718 feet (99.4%). I think it is precisely expressed as (6/5)^3 feet or 1.728 feet exactly (see Cubits).

Peter Stanbury discovered the following relationships of the proton to electron mass ratio and the fine structure constant’s relationships to pi. I added the neutron to electron mass ratio and the boxed equation. Taken together, we see how pi, phi and mathematical ratio are fundamental to matter. All is number.




“Let no one ignorant of geometry enter.” -Engraved on the door of Plato’s Academy

The molecular weight, specific heat capacity and temperature of air control how fast sound waves propagate through the atmosphere. At 3.3°C the speed of sound in air is 333.3 m/s and if the wavelength is 1/3 of a meter, the resulting frequency is exactly 1000 cycles per second, the center of human hearing. The meter, the second, base 10, and the qualities of our atmosphere conspire to produce this metrological sonic harmony.



from my book Quantification

from my book Quantification


I keep adding to this post, so check back again sometime.

Return top