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Iridescent Platonic Solids: a quick AI generation experiment

Saturday, 1 November 2025 by Bruce Rawles

I just tried a quick experiment with ChatGPT to see what it would generate with the prompts below. When we visited Oaxaca, Mexico, a decade or two ago, we were inspired by the colorful carved and painted “alebrijes”. We brought home a few souvenirs, so I started with this whimsical idea. The response was great in terms of making colorful animal figurines – they look like they might be illuminated with ultraviolet light … but the only animal that could slightly be considered to be even partially constructed from a Platonic solid was the hindquarters of the bull:

ChatGPT experiment: rainbow colored alebrijes

Then I tried eliminating the animal element but adding iridescence; the iridescent quality of the polygon surfaces was superb in this regard, just the quality I was envisioning. However, I wanted the 5 Platonic Solids, and the AI query got 4 out of 5 right, with the icosahedron duplicated (lower center and lower right) and the octahedron replaced with what looks like a hexagonal bipyramid in the upper right of the 6 polyhedra.

ChatGPT experiment: platonic solids with iridescent gradient faces; icosahedron replaced with hexagonal bipyramid

I gave it another go, and the iridescence mostly went away (although still nice rainbow gradient surfaces, now with black edges), and the bipyramid corrected to an octahedron.

ChatGPT experiment: platonic solids with rainbow gradient faces; icosahedron duplicated

I think I might dust off MegaPOV (or whatever might be a good ray-tracing app for the Macintosh) and see if I can get an AI app to generate the code needed as input to a more conventional ray-tracing app to see if I can coax a bit more mathematical precision.

Any suggestions, anyone?

 

Filed Under: 3D Geometries, AI-generated geometric art, Platonic Solids, sacred geometry apps, sacred geometry art, sacred geometry toys

Details about contents of Sacred Geometry Design Sourcebook and instructions for creating the graphics therein

Saturday, 19 July 2025 by Bruce Rawles

I’m glad I reviewed some of my oldest posts, as this post (originally published January 30, 2007) needed resurrecting and updating! I will leave the details from 18 years ago mostly intact, so if you want to dive in, scroll down to the “HOW-TO” INFO STARTS HERE heading for lots of “how to” info on many of the pages to make optimum use of my first book, Sacred Geometry Design Sourcebook!


Here’s another post from my blog archives that details the contents of specific pages of my Sacred Geometry Design Sourcebook, as well as instructions on how to replicate many of the images in the book.

30Jan2007 09:17 AM Filed in: Geometry-Math
Today’s post is an email conversation I had with a visitor to the intent.com website – our first website, which has changed hands several times since the 1990s! – that I probably could have written a decade ago, so this might be useful to those of you who already have purchased a copy of the book (and thanks to all of you who have supported me in this way over the years!) There are a few other goodies on my site map that might also be of interest.

Hi Julian:

Many thanks for your inquiry. I got so involved with the reply that I’ve decided to post it on my blog for today, as well as in this email; I’m grateful that you asked the question, so now I can answer it for the world.

On 30 Jan 2007, at 6:54 AM, Julian wrote:

Dear Bruce:

I was reading your web pages about your book SACRED GEOMETRY DESIGN SOURCEBOOK, and I wondered if you could tell me whether the book provides the reader with details of how to draw the 2-D patterns for themselves or whether they are just templates without such instruction.

The simplest answer is “yes and no”, depending on which of the 1300+ images you’re referring to.


The detailed answer is – here goes! – that I cover some of the philosophy and underlying math, concepts, and archetypal ideas at the beginning of the book, and provide a
generous assortment of references in the back of the book. For the remainder (majority of the 256 pages), I give the images as much room as possible so that not only can the patterns be photocopied easily (I went with spiral binding just for that reason), so that the image quality would be as high as possible for an 8.5″ by 11″ format.

“HOW-TO” INFO STARTS HERE:

Some of the illustrations give step-by-step procedures (in graphical form, assuming some basic familiarity with how to use a compass and straightedge, such as:

page 44 (showing how “unit cells” for the tiling patterns can be used to create an enormously expanded variety of additional patterns by recognizing how the space-filling shapes can be varied; this applies to the “unit cell” examples on pages 16-43,

page 45 (showing how each of the patterns on pages 16-43 can be embellished with fractal or other inscribed detail for each of the polygons for an infinite (literally!) variety of possible variations, which also apply to the majority of the remaining images in the book; page 237 gives a 3D example of this),

pages 46-84 have either explicit (most of these pages) or simple to observe implicit “unit cells” which show how these can be created in a great variety of ways,

page 84 (Pentagon Rotation Grid) gives the 73.2% proportion crucial to the exact construction of this pattern,

page 85 (Genesis of the Seed of Life) shows the step-by-step “compass only” construction of this important and universal pattern,

page 86-95 (variations on Seed of Life and Flower of Life) show how once the Seed of Life is constructed, many other patterns can be easily derived – Flower of Life, Hexagonal Grid, 2nd Harmonic Overlay which is used in Mika Feinberg’s beautiful LightSOURCE screensaver animation (no longer available); see my links (resources) page, Tree of Life, Fruit of Life, Heart and Ankh matrices, recursions, Metatron’s Cube, etc.,

page 95: since the Dodecahedron is the most complicated shape to derive from Metatron’s cube, the top center illustration on this page shows which vertices are used to create the “dodecahedron 2D shadow” with small circles highlighted in the larger image,

page 96: The general instructions for creating Nested Inscribed Polygons appear on this page,

pages 100-101-102: other examples of the crucial proportions needed to create these images either by hand or with a computer graphics program; numerous pages provide these instructions in the text without detracting from the space given to the images,

pages 104-105, 110-113, 123-124, 128, 130, 138-139, 144-145, 155, 163, 166, 171, 176-180, 182-187: all have instructions and details on how to create the images,

page 146: very detailed step-by-step instructions for inscribing a pentagon within a circle,

page 147: very detailed step-by-step instructions for inscribing a pentagon starting from one side of a given length,

page 156: very detailed step-by-step instructions for creating a golden rectangle (including “whirling squares and more),

page 156: very detailed step-by-step instructions for dividing a line by the golden ratio,

page 157: very detailed mathematical information about golden ratio progressions and powers, illustrated graphically,

page 188: shows how the Parthenon at the Acropolis in Athens, Greece, incorporates the golden ratio,

page 189: an amazing amount of data on this page about the Great Pyramid at Giza, Egypt showing phi (golden ratio) and pi proportions, and the proportions of the so-called “King’s Chamber” although the so-called “sarcophagus” (granite box) within is too large to fit through the only passage leading into that chamber, which violates the generally accepted funerary rite theory,

page 190: the classic “Measure of Man” (Vitruvian Man) by Leonardo daVinci, copied around the world, showing the golden ratio proportions in the human body,

page 191: the only known CAD drawing adaptation (to my knowledge) of Leonardo’s “Ideal Church” sketch,

page 192: detailed specifics about the Shoemaker’s Knife of Archimedes, giving several variations all showing the mathematical principle,

page 194: details of the geometry of the classic 1991 Barbury Castle, England crop circle formation,

page 195 and page 52: the details of the geometry encoded in the Sri Yantra (a classic Hindu mandala) and the cross section of the Great Pyramid at Giza, Egypt (also the “squaring the circle” conundrum), both with a 1-Phi-Square Root of Phi triangle which has a 51 degree, 51 minute slope,

page 196: numerous details common to the 5 Platonic and 13 Archimedean Solids,

pages 197-214; numerous details specific to the 5 Platonic and 13 Archimedean Solids, including how the insphere/intersphere/circumsphere/side length proportions are calculated,

pages 215-224: similar details specific to the 4 Kepler-Poinsot solids, the Star Tetrahedron (a.k.a. Stella Octangula) and related polyhedra,

pages 225-229: numerous graphics showing how the Platonic Solids relate to each other in a myriad of fascinating (awe-inspiring, really!) ways,

pages 230-235: numerous relationships between 3D polyhedra with 5-sided symmetry and the “shadow” they cast on a decagon (10-sided polygon), with construction details,

page 236: step-by-step instructions on creating an accurate drawing of the Icosahedron and Dodecahedron starting from a Golden Rectangle,

pages 237-255: generous appendices for hands-on explorers of all ages and levels of experience, including tables and charts of regular polygon angles, apothem, radius and side ratios and areas, radius ratios by coordination number for Ionic Chemical Bonding (which relates to properties of materials at the molecular and planetary levels), Fibonacci Numbers, Perfect Right Triangles (when I put the book together, I wasn’t aware of the Phi-1/Phi-Square Root of 3 Right Triangle that Mike Green of British Columbia introduced me to), Prime Numbers, extensive tables with all sorts of data on the Platonic and Archimedean Solids (useful for a variety of purposes, including model construction and computer simulation and animation, a map of planet Earth showing superimposed Platonic Solid Vertex Latitudes and (example) Longitudes, a fun stereogram with 6 Small Stellated Dodecahedra (there, I gave the clue  a short bio of myself, 3 pages of bibliography (more on my books and links pages and blog), and unique graphical index to all the illustrations in the book. Whew! I’d almost forgotten how much I packed into this labor of love over a decade ago!

Many of the 1300+ images are somewhat self-explanatory graphically (especially if you have created the basic shapes like the Seed of Life, Golden Rectangle, and a few others by hand with compass and straight-edge, which I highly recommend for anyone, as mentor Keith Critchlow so aptly reminded me when reviewing my original manuscript)…

… and of course, if you find re-creating one of the patterns challenging, I’d be happy to answer other questions via email so that I can share with other enthusiasts; thanks for asking!

Filed Under: 2D Geometries, 3D Geometries, Fractal Geometry, sacred geometry art, sacred geometry books

What really makes “sacred geometry” sacred: beyond symbols to source

Saturday, 10 May 2025 by Bruce Rawles

I’m occasionally asked what makes “sacred geometry” sacred. A short answer I often give is something like: “Either everything is sacred, or nothing is sacred, depending on one’s perspective.”

From the perspective of the all-encompassing, inclusive non-dual Identity that we all share that transcends the personal, we’re all one in the same essence: our eternal transpersonal nature embraces and encompasses all creation and is a reflection of Perfect Oneness, the sacred unchanging innocence of the changeless, timeless, spaceless, formless Unity. Non-duality is a term that attempts to convey the idea of this sacred reality, which transcends ideas, words, and symbols. However, ideas, words, and symbols can, at best, be properly understood – merely reflect some aspects of the quintessential experience of our steadfast source.

It is only when we harbor nightmarish grievances against parts of our Self that seem split off and separate – the profane belief we could be apart from ourselves, each other and our Creator – that we seem to exist in a state of perpetual war, or at least a less than peaceful ceasefire amid the challenges of an uncertain, lonely and fearful world. The world of duality, conflict, polarization, contrast and differences is our default experience in this world, making it seem like the only alternative, yet there is a part of our mind – however deeply buried – that remains connected to Source, an abiding awareness that remembers our inseparability from the eternal perfection of creation, despite our challenge to awaken that possibility, let alone make that our every day experience.

As the Greek philosopher Plato taught, there is a deeper reality beyond the forms that appear to us in the material world. Elsewhere on this website, there are numerous references to the five regular polyhedra credited to Plato called the Platonic Solids, building blocks of three-dimensional geometry and primary to countless forms in our space-time dimensional world. Plato suggested that these 3D geometric symbols represented five elements: tetrahedron (fire), hexahedron or cube (earth), octahedron (air), icosahedron (water), and dodecahedron (ether or spirit).

Plato also was credited with inexorably leading his students back to the un-manifest realm of mind (source) rather than become muddled and confused by the seemingly endless variations and permutations of form (symbol) and his upward pointing finger in the “School of Athens” painting reminds us that understanding begins with returning our mind to cause (source) rather than obsessing over effect (symbol.) The “back to top” symbol on this website is the zoomed-in hand of Plato pointing up, reminding us to get “above the battleground” of duality before trying to contemplate or comprehend the paradoxes of our dualistic world.

That vertical orientation of Plato’s upward-pointed finger symbolizes the inevitable (and joyous) return to oneness. It represents the undoing of the fragmentation of dualistic thinking expressed in our lives as “we/they” and “us vs. them” polarizations, as well as the myriad forms in which we compare two or more things for the purpose of maintaining our belief in separation. One can look at anything in the world in terms of intrinsic oneness, with duality or multiplicity being merely an “optical delusion of consciousness” as Albert Einstein put it; in that state of mind, everything becomes an opportunity to heal our minds of warring interests and perspectives as we forgive ourselves for investing in interpretations that are merely silly mistakes and amusing errors, easily correctible from the proper vantage point … above the world of dividing symbols.

Plato is also famous for his Allegory of the Cave, which draws one’s attention to the metaphysical symbolism of light and shadow. The reality we don’t see in our imprisoned mental state – where we are fascinated by a world of sensory and material distractions, diversions, and deceptions – is the light of day that casts shadows on the wall of the metaphoric cave of mindlessness. We’re identified with shadows (symbols) instead of the greater reality of the shared light that is our common identity (source) and we’re so inured and adjusted to this extremely deficient (mis)perception that we don’t want to be reminded that there is a real alternative to our dark and dismal self-imposed prison in the labyrinthine cavernous abyss of duality.

We could look at any number of geometric forms for examples of these two perspectives – two completely mutually exclusive thought systems – that reflect our journey back to source using symbol as a starting point, and returning us to our shared essence as the ultimate destination.

“… the destination is at the source.” – Michael Hedges, Road to Return (Lyrics)

The graphic below symbolizes two identical circles in five arbitrary stages of “returning to source” (unity) from our perception of duality. The circle is a symbol of perfection as there is no point on the circle that is further or closer to the center than any other. All symbols are inherently limited by the dualistic limitations of our language and the geometry of space (and time) itself, so this illustration is no exception. Below are comments on the five “stages” of the overlap and reunification of what never was separate in truth, but only in our shadowy self-imprisoned “cave thinking” of apartness.

  1. We symbolically seem apart, and our “sphere of awareness” is limited to a separate self, and we don’t believe we share anything with anyone else; this is the state of perpetual war that is fundamental to all dualistic thinking.
  2. The barely overlapping circles depict that there is now a glimmer of realization that we’re not alone. We’re all “fighting the same hard battle,” but we gain hope, solace, and inspiration from the occasional breakthrough moments when we realize we do indeed share the same interest in “going home” to the Oneness we never left in Truth. We’re mostly “asleep,” dreaming of being exiled from the peace of true connection, but now we have instants of sanity that motivate us to consider our thoughts of interconnectedness and sameness as vital to our real progress.
  3. This “midpoint” example geometry is known as the “vesica piscis” (and other names like mandorla or almond), where the center of one circle touches the circumference of the other. This might represent a halfway point in our journey from separate interests to shared interests, from insanity to sanity, from duality to unity. There are many references to the vesica piscis on this website.
  4. The fourth image brings to mind moments just before or after a total eclipse; the antithesis of the second phase, this would be akin to having a mind that now defaults to seeing shared interests instead of separate interests, where we choose sanity most of the time instead of on rare occasions.
  5. The fifth stage of total overlap requires explanation in that the two circles have “melted into one,” and there are no differences detectable. Because there aren’t any obvious examples in our everyday world, this symbol merely attempts to represent a state where time and space have been transcended. In the eclipse example, light is diminished by one shape occulting (hiding) another, yet in the mental metaphoric realm, the opposite is true: the unlimited light is joined and becomes greater than the sum of its parts because there are no longer separate parts! This graphic is woefully inadequate in expressing the idea of a “oneness joined as one,” but our dualistic minds have a tough time with that, too! Not to worry!

The important “take-home lesson” of this example is that—in keeping with the Platonic idea that symbols are not sources—as we expand our definition of Self (as a circle of infinite diameter would embrace all), the idea of an “other” self (equally expanded to embrace totality) would be indistinguishable and resolve all differences, conflicts, polarities, uncertainties, loneliness, and fear. For more about non-duality, my other blog may be helpful, particularly this post.

“As nothingness cannot be pictured, so there is no symbol for totality.” (ACIM, T-27.III.5:1)

circle overlap symbols (sacred geometry)

Filed Under: 2D Geometries, sacred geometry art, sacred geometry metaphysics

Vandorn Hinnant – 45 Years of Dreaming with Open Eyes – Exhibition Catalogue

Friday, 4 April 2025 by Bruce Rawles

For many years, I’ve admired the geometric art of geometer colleague and artist Vandorn Hinnant; this month’s post features a pdf file of a retrospective of four and a half decades of his geometry-inspired artistry. Enjoy! Be sure to visit his websites LightWeavings.com and VandornHinnant.com for much more info.

Vandorn Hinnant – 45 Years of Dreaming with Open Eyes – Exhibition Catalogue

Download the pdf here.

The Title/Cover page provides the details of the live exhibit which runs from March 20 through June 28, 2025 at the Mary G. Hardin Center for the Cultural Arts, 501 Broad Street, Gadsden, Alabama.

Here are just a few examples of his inspired/inspiring art:

Vandorn Hinnant: "Times Golden Arrow” a.k.a. "Golden Rectangle Mandala” ©2010
Vandorn Hinnant: “Times Golden Arrow” a.k.a. “Golden Rectangle Mandala” ©2010
Vandorn Hinnant: The Truest Eye a.k.a. Root Three Spiraling Fractal Trigons ©1997
Vandorn Hinnant: The Truest Eye a.k.a. Root Three Spiraling Fractal Trigons ©1997
Vandorn Hinnant: Ark ©1997
Vandorn Hinnant: Ark ©1997
Vandorn Hinnant: Before We Knew - From The MOTHER of Everything ©2002
Vandorn Hinnant: Before We Knew – From The MOTHER of Everything ©2002

Filed Under: 2D Geometries, golden ratio, sacred geometry art, sacred geometry news

Encyclopedia of Polyhedra By George W. Hart … and other geometric gems

Sunday, 16 March 2025 by Bruce Rawles

Apples and Oranges by George W. Hart https://www.georgehart.com/Applorng.html
Apples and Oranges by George W. Hart
https://www.georgehart.com/Applorng.html

It’s time to revisit the work of other polyhedra enthusiasts. Of particular note (having crossed my digital path again recently) is George W. Hart who has shared some excellent resources along these lines (faces, vertices, etc.) for decades. Here’s one gold mine to explore: Virtual Polyhedra: The Encyclopedia of Polyhedra By George W. Hart. Here is part of the table of contents and highly recommended if you want to explore the genres and categories of polyhedra; lots of great imagery, rotatable 3D models (rotate them with your mouse), facts and details:

  • Platonic Solids (Regular Convex Polyhedra) Background List of models
  • Kepler-Poinsot Polyhedra (Regular NonConvex Polyhedra)Background List of models
  • Archimedean Polyhedra (Semi-Regular Convex Polyhedra) Background List of models
  • Prisms and Anti-Prisms Background List of models
  • Archimedean Duals Background List of models
  • Quasi-Regular Polyhedra Background List of Models
  • Johnson Solids (the remaining convex polyhedra with regular faces) Background List of models
  • Pyramids, Dipyramids, and Trapezohedra Background List of models
  • Compound Polyhedra — Introduction Background List of models
  • Stellated Polyhedra — Introduction Background List of models
  • Compounds of Cubes Background List of models
  • Convex Deltahedra Background List of models
  • Zonohedra Background List of models
  • Uniform Polyhedra Background List of models
  • Uniform Compounds of Uniform Polyhedra Background List of models
  • Stellations of the Icosahedron Background List of models
  • Stellations of the Rhombic Triacontahedron Background List of models

Perhaps the most well-known and ubiquitous polyhedral shapes have fold-up models (and related patterns) for the first 3 categories above in Sacred Geometry Design Sourcebook (SGDS pages 196-220) and also on this website for the 5 Platonic Solids and 13 Archimedean Solids as well as Illustrations of Platonic And Archimedean Solids, Math Tables for Platonic And Archimedean Solids (and much more about polyhedra on this website.

compound of five cubes by George W. Hart
compound of five cubes by George W. Hart

If you exhaust this generous resource (or somehow overlook a few gems) here are more of his contributions – the videos are particularly helpful if you want to replicate his constructions:

  • The Pavilion of Polyhedreality
  • George’s Instagram page with some excellent 3D geometric sculptures
  • Mathematical Impressions: The Golden Ratio (short video)
  • Hyperboloids (short video)
  • Little Zonohedral Library (short video; my favorite of these which also features our mutual friend Russell Towle who showed me his 3D zonohedral models (such as Rhombic Spirallohedra)
    years ago in Dutch Flat, California)
  • Ceci n’est pas une lampe (fun, clever, short video)
  • Birdland (another clever short video)
  • Seven Slide-Together Constructions (another interesting short video)
  • … and MANY more!
Stained Glass Ball By George W. Hart https://www.georgehart.com/stained_glass.html
Stained Glass Ball By George W. Hart
https://www.georgehart.com/stained_glass.html

Filed Under: 2D Geometries, 3D Geometries, applications, Archimedean Solids, golden ratio, Platonic Solids, polyhedra, sacred geometry art

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