sacred geometry news

Spirals, Ferns, and Universal Unfolding Motifs

Thursday, 1 December 2022 by Bruce Rawles

While I try not to obsess about geometric forms, I do observe when certain motifs appear more frequently in various locales. When we visited San Miguel de Allende, Mexico earlier this year, I noticed an abundance of small stellated dodecahedra (and similar shapes) particularly evident in metal lamps, yet also in other art forms as well; each pentagonal pyramidal stellation arm encodes the golden ratio. …

Having just returned from a second visit to New Zealand, it seemed that one of the most ubiquitous geometric shapes we saw on this trip was the spiral, which appears in Maori jewelry, carving, tattoos, art, and culture, connecting the natural world with the art of the indigenous people of Aotearoa. Below I made a mirrored tiling pattern from an image of an unfurling silver fern frond which exhibits an excellent example of fractal geometry.

Wikipedia says “The koru is a spiral shape based on the appearance of a new unfurling silver fern frond. It is an integral symbol in Māori art, carving, and tattooing, where it symbolizes new life, growth, strength, and peace.”

Reflecting intuitively on our spiral “journey” through life like the homeward trajectory Dorothy embarks upon in the Wizard of Oz when she follows the yellow brick road, we find context in the cyclical, yet ever-evolving, ever-unfolding awareness with each revolution of the spiral. In three dimensions, we have the double helix of the DNA molecule in everything we typically call life in this world, embodying the golden ratio in each pentagonal molecule appendaged to the helix and also in the (Fibonacci proportion) 34/21 Angstrom ratio of each complete turn of the helix. Every year, each season has familiar patterns and signs, yet there is a much larger progression we discover when we “zoom out” a bit from our myopic, provincial perspectives. The Chambered Nautilus shell has a pleasing uniformity as one spins the shell around, since each accretion uses the same angular template as the preceding one, and builds upon the strength of sameness throughout.

Here are a few more spirals from our most recent NZ trip; enjoy!

New Zealand: Napier museum; Maori wood carvings with numerous spiral forms

Auckland Airport ceiling lamp fixture — Auckland NZ Airport ceiling lamp fixture

Auckland NZ Airport ceilng art: spiral fish school

New Plymouth, NZ art gallery staircase; spiral flourishes

Napier NZ boardwalk art with overlapping complementary spirals

In memoriam: William Goldstein

Friday, 14 October 2022 by Bruce Rawles

I recently learned of the passing of William Goldstein (a.k.a. Will Gold), who left us on August 11, 2022. William was a fellow geometer, researcher, historian, mystic, and much more. It was only fairly recently (September 2021) that I discovered his work, our many common interests, and favorite books while exploring his extensive, fascinating writings, which are a great complement to my more mathematical leanings. I began a brief stint as his webmaster, and I was hoping to share many recorded conversations between William and me with my GeometryCode readership, but that was not to be.

Rather than have this important body of work disappear, I offered his family the opportunity to keep the site going so that others might benefit from his insights and helpful resources, including an extensive library of unique and interesting books, most of which are still available. The newly relocated website (formerly Chartres-sacred-geometry.com) is now live at Chartres.GeometryCode.com. It is my hope that my readers will avail themselves of his thoughtful and insightful writing and excellent book (and other media) recommendations of value for geometry enthusiasts, particularly the in-depth cultural and historical underpinnings behind so many important discoveries and real human progress.

Here is the prior article I wrote about William’s site, which I would recommend reading as well.

Thank you, William, and to your family as well!

The Golden Icosahedron and other interesting Golden Ratio appearances

Wednesday, 31 August 2022 by Bruce Rawles

How did I miss this very cool, interesting polyhedron? Glad to have just discovered this fascinating shape (with lovely rotating animation) on this post by “RobertLovesPi” here, with this commentary:

The Platonic Icosahedron has twenty faces which are equilateral triangles. In the Golden Icosahedron, twelve of those triangles (the yellow ones) have been replaced by acute, isosceles triangles with a leg:base ratio which is the Golden Ratio.

Here are a few other appearances of the golden ratio (a.k.a. golden proportion, a.k.a. golden number, a.k.a. golden section, a.k.a. extreme and mean ratio, a.k.a. divine proportion, a.k.a. Phi (ø) or Tau (τ) proportion, a.k.a. sacred cut, etc., and other synonyms for the ubiquitous (1 + √5)/2 ≈ 1.6180339887… number):

Golden Triangle (featured 12 times in the Golden Icosahedron above) and in every pentagram and regular decagon with lines connecting vertices to its center.
Golden Rectangle
Pentagram
Pentagon
Lots of other instances throughout this website!

This unexpected golden ratio appearance first made my acquaintance at a sacred geometry conference hosted by Luke Gatto a few decades ago in Boulder, Colorado, and I shared it many years later here in 2011:

How to graphically derive the Golden Ratio using an equilateral triangle inscribed in a circle

and resurfaces here on a Quora post – and I’m copying the entire account verbatim; thanks, Johann!

Johann Holzel
Author has 4K answers and 1.1M answer views
Some ratio has to satisfy a/b = (a+b)/a. The fact that this ratio should have such a simple continued fraction, involves √5 and therefore pentagons, has a special relation between its negation and inverse, etc., these must have all been surprising developments when each was first discovered (in many cases, by ancient Greek geometers—although apparently it, or √5, was also the first number proven to be irrational, which was even more surprising).
But nowadays, you learn most of those things together, and are immediately shown how they fit together, so there’s not much room for surprise.
And most appearances in nature make sense once you think about them. For example, there are cases where the most irrational logarithmic spiral is obviously the best to use, and the fact that it can be closely approximated by the arithmetically-simple Fibonacci spiral makes it unsurprising that evolution would hit on it multiple times.
(In fact, most claimed appearances in nature or in art are either not even close to ϕ at all, or are just boring cases of an artist who heard great art is supposed to be full of golden ratios so they followed what they thought they were supposed to do.)
So the most unexpected places are probably the ones that come from amateurs. And probably the most famous of these is the equilateral triangle inscribed in a circle, just because of how deceptively simple it is.
A 1983 article in American Mathematical Monthly consisted of basically this picture, accompanied by the word “Behold!”
A and B are the midpoints of two sides; extend the line from A to B out to the circle and the intersection is C.
The ratio of AB to BC is obviously somewhere around 1.6-ish just from visual inspection. So is the ratio of AC to AB.
But in fact, they’re identical, which means they’re both exactly ϕ, the golden ratio.
Why?
The golden ratio doesn’t just show up in triangles unless you put it there (e.g., the golden triangle is specifically designed as the isosceles triangle with the self-similar bisection), and that isn’t the case here.
This was discovered by George Odom, an amateur mathematician, who was in correspondence with famous geometer H.S.M. Coxeter. Coxeter wanted to see how many readers could work out why ϕ shows up here.
To get the answer, you have to notice that this chord looks like it can be used to construct a regular pentagon. Then you can complete the construction, and prove it’s exact via the chord theorem. And then there’s no mystery left—unlike triangles, pentagons are full of the golden ratio. (Coxeter just gave the constructed pentagon as the answer.)
That’s deeper than the kind of thing you’d expect from what looks like a simple recreational mathematics problem. And even after you get it, it’s still almost magical that something so with no obvious pentagons, or self-similar folds, or anything like that, gives you ϕ, so simply.

Reflections on the Sacred: Geometry, Mind, Non-duality

Sunday, 5 June 2022 by Bruce Rawles

Over the years, I’ve been asked by many: “What Makes Sacred Geometry sacred?” to which I usually respond with something like this: Either everything is sacred … or nothing is sacred. While that terse response may seem to be evading the question, I’ve realized more and more deeply that it’s true, and if we gradually (with help from the Inner Inspiration we all can employ) shift our perception from the ephemeral to the eternal then our view of everyone and everything becomes kinder, more compassionate, tolerant and forgiving of the minutiae of the fleeting phantasms of form and the cosmically temporary. If truth doesn’t change, our task must be to seek and find what is true and cannot change; the laws of mathematics and geometry seem worthy candidates for possible means of finding the unalterably true. However, for something to be unequivocally true, it must apply to everyone and everything. The discipline of the scientific mind requires that we keep looking if we find exceptions to our postulates and theorems and persist until we find the safety, security, and serenity of what can never change. William Shakespeare penned lovely prose about the ideal of looking ever inward to the mind for what doesn’t alter or change when it (the mind) finds the alterable:

Sonnet 116: Let me not to the marriage of true minds
BY WILLIAM SHAKESPEARE

Let me not to the marriage of true minds
Admit impediments. Love is not love
Which alters when it alteration finds,
Or bends with the remover to remove.
O no! it is an ever-fixed mark
That looks on tempests and is never shaken;
It is the star to every wand’ring bark,
Whose worth’s unknown, although his height be taken.
Love’s not Time’s fool, though rosy lips and cheeks
Within his bending sickle’s compass come;
Love alters not with his brief hours and weeks,
But bears it out even to the edge of doom.
If this be error and upon me prov’d,
I never writ, nor no man ever lov’d.

Seeking what is true for all, we realize we are all in the “same boat” either expressing kindness or in one way or another asking for it. Seen from this perspective, our true kinship – our real relationship – to everyone and everything merits kindness in response to expressions or requests for the same. That seems like a very serviceable and practical definition of sacredness – generalized to encompass all, sacredness reflects the perfection beyond the forms our senses and the mind’s machinery of interpretation seem to establish as real. We rarely question the vast number of assumptions we make every second about the world that seems vulgar and profane or at the very least disappointing in its impermanence.

A few dictionary definitions of the word sacred (most closely aligned with how I think of the word) are “revered, inviolable, inviolate, unimpeachable, invulnerable, untouchable, inalienable, protected, defended, secure, safe, and unthreatened.” These adjectives would suggest that nothing in this material world – where every form that appears eventually disappears – would qualify for sacredness; no exemption for the changed, changing, and changeable from decay, dissolution, and death. Yet while the laws of physics (particularly the Second Law of Thermodynamics) would suggest that nothing in spacetime is forever, we certainly can use our experiences in the world of space and time to guide us (even if largely as counterexamples) to find what is endless.

Our experience of sacredness – which I would equate with the purely innocent eternal nature of our being and all that remains in mind after we have completely forgiven ourselves and everyone for everything – depends completely on the thought system with which one views everything! If we look for what unites us, what is intrinsically the same for all – regardless of all the polarizing temporary attributes that seem to define our ephemeral personas and certainly our bodies – we find the sacred, innocent undifferentiated awareness from which all creation derives and to which we all belong.

What prompted this metaphysical musing? My computer calendar showed me that a decade ago today, I published my second book in print form, followed by a Kindle edition a few months later. I was also reflecting fondly on some related anniversaries:

10 years ago today, The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation became available in print
15 years ago (2007), I “rediscovered” the pure non-dual metaphysics of A Course In Miracles by way of Gary Renard‘s first book: The Disappearance of the Universe: Straight Talk about Illusions, Past Lives, Religion, Sex, Politics, and the Miracles of Forgiveness and shortly thereafter started ACIMblog.com and started a weekly study group (now online)
25 years ago (1997) I launched my first book, Sacred Geometry Design Sourcebook – Universal Dimensional Patterns
30 years ago (1992), having “rediscovered” what has become known as sacred geometry via fellow mystical explorer and contemporary, Gregg Braden at a lecture in my mother’s living room in Livermore, California a year before – we (my now wife, Nancy, and I) accompanied Gregg (21 of us) to Egypt to explore the Great Pyramid of Giza and numerous sites up and down the Nile River
50 years ago (1972) I graduated from Livermore High School

Thanks to all of you for many decades of support, collaboration, and shared enthusiasm for finding what is true! Here are some excerpts from the June 2012 bulletin that provide details about my second book, which combined interests in both sacred geometry and non-dual metaphysics. The book “The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation became available 10 years ago today: on June 5, 2012.

	Learn more about The Geometry Code book
	Free sample chapter from The Geometry Code book with free bulletin subscription
	Preface by the author from The Geometry Code book
	Table of Contents from The Geometry Code book
	Acknowledgments for The Geometry Code book
	Endorsements for The Geometry Code book

About the book: The Geometry Code

While I was completing work on Sacred Geometry Design Sourcebook, (completed in 1997), I was already working on another book, which was ready on June 5th, 2012. Rather than reinvent the wheel, I think the Preface I wrote explains how the book came about and gives a good “big picture” of what it is about. I’ve uploaded the Table of Contents, Endorsements from friends and colleagues, Acknowledgments, a sample chapter, and more.

TurboSquid 3D geometric models

Tuesday, 31 May 2022 by Bruce Rawles

Fellow geometer, 3D animator, techno-nerd, mystic, and life-long friend, Dave Van Dyke alerted me to some examples of very cool 3-dimensional models using TurboSquid (3D Models for Professionals) a couple of days ago. The links below connect with a small set of the many examples of 3-d geometric models – some free, others inexpensive – all gems, especially considering the “labor of love” efforts put into these detailed models! Going beyond geometric constraints, you have orders of magnitude more to explore! They can be downloaded and adapted wherever your imagination might lead! Here are some samples to pique your interest!