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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.

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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

Non-Euclidean Geometries

Sunday, 23 February 2025 by Bruce Rawles

At the end of last month’s post we gave this example of non-Euclidean geometric art inspired by M. C. Escher’s pioneering graphics which explored various geometries and illusory perspectives.

Also noted in last month’s post, here’s a still image related to a scene in the movie “Inception” that had a physical implementation of an impossible (never-ending) staircase; our limited geometric perspectives can deceive us:

Penrose stairs in the movie "Inception"

Since there’s no point in “reinventing the wheel” I’ll quote from Wikipedia’s definition for non-Euclidean geometry, also since I’m a novice in that field, but an admirer of art and imagery inspired by that geometry that stretches our imaginations:

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

The principles below (from the next section in the Wikipedia page) reveal how the behavior of parallel and perpendicular lines differs in two common non-Euclidean realms; Hyperbolic Geometry and Elliptic Geometry: with a graphic demonstrating how if we don’t make certain Euclidean assumptions about space (e.g. parallel and perpendicular line behavior) such as the requirement that the sum of the angles of a triangle add up to 180°.

The essential difference between the metric geometries is the nature of parallel lines. Euclid‘s fifth postulate, the parallel postulate, is equivalent to Playfair’s postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l.

Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line (in the same plane):

  • In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.

  • In hyperbolic geometry, they diverge from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.

  • In elliptic geometry, the lines converge toward each other and intersect.

Here’s a simple graphic that gives a hint about the fundamentals of these mind-warping geometries that might be akin to warm-up exercises for four-dimensional (and beyond) geometries and polytopes, which we like to explore on this website now and then.

Comparison of geometries: Elliptic, Euclidean, Hyperbolic
By Cmglee – Own work, CC BY-SA 4.0, Link

By Cmglee – Own work, CC BY-SA 4.0, Link

Here’s an example of hyperbolic geometric art:

Rhombitriheptagonal tiling
Rhombitriheptagonal tiling – Parcly Taxel, Public domain, via Wikimedia Commons

Here’s an astronaut’s perspective example of how the sum of the angles of a non-Euclidean triangle can add up to more than 180° on the surface of a spheroid (our beloved planet):

Spherical geometry example: sum of angles exceeds 180 degrees
Spherical geometry example: sum of angles exceeds 180 degrees; Lars H. Rohwedder, Sarregouset https://en.wikipedia.org/wiki/File:Triangles_(spherical_geometry).jpg

For further exploration and many more wonderful and artistic images, I recommend this article “The Use of Non-Euclidean Geometry in Art” on the naiadseye blog …
and this post “Non-Euclidean Geometry Art August 4, 2014″ on “Harrison Hartle’s Art/Music/Theater F200 blog.”

The latter has this excellent TED talk video about “The beautiful math of coral” by Margaret Wertheim (2009); I first learned of this fascinating realm of modeling coral reefs (and many other “hyperbolic geometry” life forms with crochet hooks) from my colleague Libby M. in Oregon at a Geometers meeting about 20 years ago who had made and shared several knitted models inspired by this work:

Margaret Wertheim (in the video above) references the work of Froebel, an educational pioneer featured on this website. Froebel-inspired products are available from RedHen Books and Toys featuring unique, hard-to-find educational materials, toys and books such as Bradford Hansen-Smith’s circle folding videos, books and supplies.

In addition to the other affiliates on the GeometryCode.com website featured in the sidebar on every page that help support this “labor of love” website, please check out Ka Gold Sacred geometry jewelry by artist David Weitzman including wearable geometric art including Flower of Life, Seed of Life, Golden Spiral, Fruit of Life, Vesica Piscis, Star tetrahedron “Merkaba”, Fibonacci Whirling Squares Spiral “Phi”, Metatron’s Cube, Chambered Nautilus, Labyrinth, Torus Tube, Tetractys, Sri Yantra mandala, Tree of Life, and Hexagram (Star of David).

The Wolfram MathWorld page on Non-Euclidean Geometry also has many pertinent links and resources.

The Bridges organization has more related material on the subject of mathematical art.


In case you still need a 2025 calendar, there are 10 months left in the year as of the date of this post, and 2025 calendars are still available – as well as my over-quarter-century-old timeless classic reference book, Sacred Geometry Design Sourcebook – Universal Dimensional Patterns, heavily inspired by Escher, and other geometric luminaries.

Filed Under: 2D Geometries, 3D Geometries, 4D Geometries, Fractal Geometry, Platonic Solids, sacred geometry art, sacred geometry jewelry, sacred geometry news, sacred geometry toys, sacred geometry videos

M. C. Escher’s Geometry and Illusory Perspectives Revisited

Monday, 27 January 2025 by Bruce Rawles

Musing about how the interpretations of our physical senses are often misled by optical (and other) illusions, I was reflecting on being inspired decades ago by the classic M. C. Escher images in the book (The Graphic Work of M.C. Escher Introduced and Explained By the Artist Paperback – January 1, 1973
by M. C. Escher). The cover of this Escher book features an image of a small-stellated dodecahedron with cutouts in half of the lower portion of each triangular face of each stellating pentagonal pyramid permitting polychromatic reptile heads and limbs to protrude (Gravity, June 1952. Lithograph and Watercolour.) Many of Escher’s illustrations demonstrate these sensory deceptions – which can be beneficially used to inspire a willingness to see things from multiple (open-minded) perspectives –  such as images in his Impossible Constructions and Mathematical galleries and other classic images employing space-filling tessellation patterns and numerous instances of multiple simultaneous perspectives. In addition to an early introduction to Platonic Solids, Archimedean Solids, stellations, and more in the polyhedral realms, Escher’s pioneering visionary art seemed to suggest to me the value of exploring perspectives beyond  what our unreliable senses (and our misinterpreting minds) report and stoked a curiosity to explore realms beyond ordinary viewpoints. My other primary blog is devoted to these metaphysical explorations.

Examples which show the distortions of our sensory misinterpretations include these images (I haven’t copied the images due to copyright requirements, so please tap the links on this post to check out these example Escher gems (many more in these extensive archives: Selected Works by M.C. Escher):

  • Balcony. July 1945, Lithograph (a coastal condo scene where the center “bulges” out in fisheye lens fashion)
  • Three Spheres I. September 1945, Wood Engraving. (I had a vibrant fluorescent “black light” poster of these stacked spheres on my wall in high school)
  • Reptiles. March 1943, Lithograph (including a 3D dodecahedron and a 2D tiling (tesselation) pattern of tiled reptile shapes)
  • Up and Down. July 1947, Lithograph (staircase/balcony composition combining bird’s eye and worm’s eye views of the same scene blended into one)
  • House of Stairs. November 1951, Lithograph (an impossible realm of stairs crawling with multi-legged creatures that can crawl or spiral roll)
  • Relativity, July 1953, Woodcut (another impossible realm of staircases)
  • Sun and Moon. April, 1948, Woodcut, printed from four blocks (Solar/lunar motifs overlap with space-filling birds)
  • Double Planetoid. December 1949, Wood Engraving printed from four blocks (star tetrahedron made from an “organic” tetrahedron and a “man-made” tetrahedral world that interpenetrate but do not appear to touch)
  • Order and Chaos II. August 1955, Lithograph (featuring a small stellated dodecahedron with pentagrams on the pentagonal bases)
  • Three Worlds, December 1955, Lithograph (leafs float on the surface of a fish pond reflecting bare winter tree branches which also seem like roots); this image was used as the original album cover for the Beaver & Krause vinyl album “In A Wild Sanctuary” which you can hear here, combined with their more recent album, Gandharva (celestial musician). Fascinated by the idea of musically interesting harmonics and proportions, I made (using a FORTRAN program) a spreadsheet using the CDC3400 our high school had access to once a week showing resonant frequencies and harmonics of the Grand Gallery of the Great Pyramid at Giza, Egypt (see other internal references to the amazing geometry of this pyramid) and other accessible chambers. On two occasions, I visited Beaver’s Los Angeles recording studio in the early 1970s and planned to join him to record electronic (Moog analog synthesier) music in the Great Pyramid at Giza, Egypt in fall 1973, but this trip never materialized, as the Middle East political situation then was hardly conducive to travel. In November 1992 (19 years later) I was able to explore the interior of the Great Pyramid and record some flute music by Gregg Braden (his website) with my wife Nancy and 18 others on a tour led by Braden.
  • Print Gallery, May 1956, Lithograph (a recursive image of an art gallery where the interior columns become exterior posts of an overhanging roofline); this is the image that reminded me of non-Euclidean Geometry and revisiting Escher’s art for this post.

Here is a close-up of Escher-inspired tiles that adorn the front yards of at least two local homes in Green Valley, AZ; photographed on nearby neighborhood walks during the past 18 months.

interlocking Escher-inspired reptile tiles

For further exploration, check out:

  • the official M. C. Escher  website
  • the M. C. Escher Wikipedia page
  • Wolfram MathWorld references to Escher, such as Penrose Stairway, Freemish Crate, Impossible Figure, etc.
  • Lessons in Duality and Symmetry from M.C. Escher
  • The Influence of the Perspectives of M.C. Escher … just for starters!

The end of the last article above reminded me that there’s a scene in the movie “Inception” (a fave flick) that had a physical implementation of one of these impossible (never-ending) staircases. featured in popular Escher image:

Escher’s art also explored Non-Euclidean Geometries, which will be featuring in next month’s post. The art below reflects an example of an Escher-inspired Non-Euclidean geometry.

In case you still need a 2025 calendar, there are 11 months left in the year as of the date of this post, and 2025 calendars are still available – as well as my over-quarter-century-old timeless classic reference book, Sacred Geometry Design Sourcebook – Universal Dimensional Patterns, heavily inspired by Escher, and other geometric luminaries.

Filed Under: 2D Geometries, 3D Geometries, Platonic Solids, sacred geometry art, sacred geometry books, sacred geometry news

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Recent GeometryCode Posts

  • What really makes “sacred geometry” sacred: beyond symbols to source
  • Vandorn Hinnant – 45 Years of Dreaming with Open Eyes – Exhibition Catalogue
  • Encyclopedia of Polyhedra By George W. Hart … and other geometric gems
  • Non-Euclidean Geometries
  • M. C. Escher’s Geometry and Illusory Perspectives Revisited

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The New Invisible College - A Gateway to the Mystical Tradition of the West


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