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

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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

Non-Euclidean Geometries

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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.

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

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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):

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 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.

Plato’s Spirit, Carl Sagan’s Blue Dot, Contact, symbols and sources

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"Pale Blue Dot" Voyager 1 photo of earth (NASA/JPL-Caltech)
“Pale Blue Dot” Voyager 1 photo of Earth (NASA/JPL-Caltech)

“The Pale Blue Dot is a photograph of Earth taken Feb. 14, 1990, by NASA’s Voyager 1 at a distance of 3.7 billion miles (6 billion kilometers) from the Sun.”

*A year or so ago, a friend reminded me of this inspiring prose “The Pale Blue Dot” by Carl Sagan (copied below) inspired by the NASA image above… and the ever-increasing importance of finding and supporting the interests we ALL share on this tiny planetary sphere we call home:
“Look again at that dot. That’s here. That’s home. That’s us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every “superstar,” every “supreme leader,” every saint and sinner in the history of our species lived there-on a mote of dust suspended in a sunbeam.
The Earth is a very small stage in a vast cosmic arena. Think of the endless cruelties visited by the inhabitants of one corner of this pixel on the scarcely distinguishable inhabitants of some other corner, how frequent their misunderstandings, how eager they are to kill one another, how fervent their hatreds. Think of the rivers of blood spilled by all those generals and emperors so that, in glory and triumph, they could become the momentary masters of a fraction of a dot.
Our posturings, our imagined self-importance, the delusion that we have some privileged position in the Universe, are challenged by this point of pale light. Our planet is a lonely speck in the great enveloping cosmic dark. In our obscurity, in all this vastness, there is no hint that help will come from elsewhere to save us from ourselves.
The Earth is the only world known so far to harbor life. There is nowhere else, at least in the near future, to which our species could migrate. Visit, yes. Settle, not yet. Like it or not, for the moment the Earth is where we make our stand.
It has been said that astronomy is a humbling and character-building experience. There is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly with one another, and to preserve and cherish the pale blue dot, the only home we’ve ever known.”
― Carl Sagan, Pale Blue Dot: A Vision of the Human Future in Space

sphere inside dodecahedron (inside rotating circular hoops) frame from the movie "Contact" by Carl Sagan

Recalling a cameo role played by a dodecahedron – a symbol for “Ether” (or Spirit) according to Plato – in a major motion picture, I recently re-watched (and enjoyed from a very different perspective) a movie I hadn’t seen in decades, Contact, written by Carl Sagan and Ann Druyan, starring Jodie Foster and Matthew McConaughey. The two stars seemed to search for meaning via science and religion respectively. It would appear that neither can find what they are searching for in isolation, just as we miss the “big picture” when we exclude other grander perspectives, putting our tiny lives in context. The very touching scene (toward the end of the movie when her dodecahedral “craft” fulfills its mission of transcending our understanding of space and time) where Foster’s character is reunited with her father in perhaps “another dimension” suggests that only our limiting perspectives and beliefs needlessly constrain us.

I recall attending a lecture in San Francisco, California in the 1970s by Dr. Ramamurti S. Mishra who compared science to a blind man and religion to a lame man; they are far more effective when they teamed up by the lame man metaphorically riding on the blind man’s shoulders. We’re all – it would seem – like individuals trying to make sense of our isolated perspectives like the proverbial “Blind men and an elephant.”

The movie also featured cameo appearances (in terms of locations) by the (now defunct) Arecibo Observatory Radio Telescope – a sizable geometric structure! – and the Very Large Array (VLA) Radio Telescope (a two-hour drive from Albuquerque, 50 miles west of Socorro, New Mexico which my wife & I visited a couple of decades ago) consisting of movable large radio telescope dishes on railroad tracks, allowing the virtual aperture to be adjusted (or something like that; my recall of the science is a bit rusty.) Both of these facilities employ lots of geometric engineering!

What prompted watching the Contact movie again (via a YouTube sidebar suggestion) was this brilliant video clip by Sagan (from his Cosmos series) that related 2D to 3D to 4th-dimensional polytopes – “flatland” squares to cubes t0 HyperCube tesseracts – and inspired my pencil metaphor decades later.

Plato (who studied and reported Socrates) suggested that symbols are not the source they represent. So if geometric symbols (such as spheres, dodecahedra, and the like) merely represent the eternal ideas that outlast and transcend the specific ephemeral instances of the forms that point to the concepts, principles, laws, and ideas they represent, it would behoove us to keep looking for the reality behind and beyond the forms that are limited by the particular spaces and times they appear to inhabit. Mind envelops matter; the Source contains – but is not limited by – symbols, which are mere shadows of what they represent, as Plato’s Allegory of the Cave would suggest. The content of the mental universe is more profound than any examples in form could possibly convey.

If you scroll down on any page of this website, you will notice a symbolic hand with a forefinger pointing up; this image was cropped from Plato’s representation in “The School of Athens.” Read more about this in a post from earlier this year.

Circle as symbol: Inclusion or exclusion?

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The Introduction to this website (Sacred Geometry Introductory Tutorial) begins with a geometric symbol that is perhaps the simplest and yet most profound – the circle. Circular patterns and archetypes appear in countless variations in nature, art, architecture, engineering, and just about every area of life we can imagine. Here are a few familiar examples; I’m sure you can think of countless others, both microscopic, macroscopic (telescopic) and in between, in all dimensions of space, time, metaphor, and thought. We touched on the sphere (Music of the Spheres – The Planets) in a recent post; the 2-dimensional counterpart to the sphere is the ubiquitous circle and provides the foundation for it’s rotation into spheres of any number of dimensions – the quintessential template for polytopes beyond our imagination. Rather than getting boggled or bogged down with those mind-warping ideas, let’s return to the simplicity of the circle.

circle: eyecircle: full moon circle: cucumber cross-section circle: water droplet and concentric ripples circle: pipe cross-sections circle: generic clock face

All of these visual examples are reflections of a form reflecting an ideal; here is what appears in the Introduction to this website (slightly edited from the original):

The Sphere


(charcoal sketch of a sphere by Nancy Bolton-Rawles)

Starting with what may be the simplest and most perfect of forms, the sphere is an ultimate expression of unity, completeness, and integrity. No point of view is given greater or lesser importance; all points on the surface are equally accessible and regarded by the center from which all originate. All atoms, cells, seeds, planets, and globular star systems echo the spherical paradigm of total inclusion, acceptance, simultaneous potential and fruition, the macrocosm and microcosm.

The Circle

The circle is a two-dimensional shadow of the sphere, regarded throughout cultural history as an icon of the ineffable oneness; the indivisible fulfillment of the Universe. All other symbols and geometries reflect various aspects of the profound and consummate perfection of the circle, sphere, and other higher dimensional forms of these we might imagine.

The ratio of the circumference of a circle to its diameter, Pi, is the original transcendental and irrational number. (Pi equals about 3.14159265358979323846264338327950288419716939937511…) It cannot be expressed in terms of the ratio of two whole numbers, or in the language of sacred symbolism, the essence of the circle exists in a dimension that transcends linear rationality. Our holistic perspectives, feelings, and intuitions encompass the finite elements of the ideas within them, yet have a greater wisdom than can be expressed by those ideas alone.

When I was writing my first book (Sacred Geometry Design Sourcebook – Universal Dimensional Patterns), the cover art as well as a majority of the 1300 images (including corner “thumbnail” variations) feature circles, either directly or indirectly, so I made a graphical index, rather than a text-oriented one; the dots that dominate the first column (circles and spheres) in these 2 pages hint at the importance of these primal, archetypal, quintessential symbols: circles and spheres.

Sacred Geometry Design Sourcebook - graphical index - page 1 Sacred Geometry Design Sourcebook - graphical index - page 2

When I was writing my second book (The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation) I also used a “Fruit of Life” symbol (6 circles around an inner circle) to represent the interconnectedness and inseparability of all aspects of what is eternally true, reflected in the 7 Hermetic Laws of antiquity. These 7 circle-framed images (along with the other supporting graphics) are all set within a circle which intuitively reminds us of the infinite and eternal in our minds.

On page 62 of the Geometry Code book, I shared an idea that I find increasingly more helpful in both pragmatic and profound ways. Here is the graphic and supporting text for this portion of the chapter “Mentalism: 1st of the 7 Hermetic Laws”:

The Wikipedia definition of scapegoat is “Scapegoating is the practice of singling out one child, employee, member of a group of peers, ethnic or religious group, or country for unmerited negative treatment or blame.”

circle symbols: inclusive? or exclusive?

The biblical story of the scapegoat offers a very helpful clue. It employs the symbol of the circle as a mnemonic for practicing inclusion or exclusion. In historic accounts, the guilt (sins) of the community were projected onto a goat, which was then banished to the realms outside the inner flock – peripheral to the circle of what we’re willing to accept or allow in our minds. Everything within the circle is OK (innocent as lambs), and if you’re outside, tough luck, goat! The problem is that the very premise that divides wholeness into parts is flawed, and even the ‘good’ that remains begins to be suspect because of the inherent lack of trust. The circle closes in and eventually strangles the dualis- tic split mind into oblivion; it becomes a singularity and winks out; but not without a lot of grief and struggle!

Seeing ‘others’ as ‘out there’ throws away the gifts they bring to us each ʻpresentʼ moment that we lucidly bring their symbolic contribution as mirrors of self back into mind. This works for both seemingly positive and negative experiences, until ultimately we realize that any so-called good experience (e.g. pleasure) in the world is no better or worse than any so-called bad experience (e.g. pain) in the world, because those experiences are all projections of a massive dream that we’ve made up to reinforce the belief in separation. Ego can use any excuse to squander the gifts by assigning ‘otherness’ to others and losing the meaning that the intrinsic connection – always in mind – can bring.

Sometimes it’s blatant when we intentionally erect an inner wall and make a fence around those we ‘like’ or agree with and those we’re not so fond of. This has nothing to do with behavior since it would be impossible to physically assist or care for everybody (let alone all the animal, vegetable, and mineral ‘life’ on just this third spheroid helically spinning around our favorite star). However, it has everything to do with inwardly caring for all, equally. This is an egalitarian ideal we probably all pay lip service to in various ways, but then often find ourselves annoyed with certain ‘others’ or internally profiling a group (whether a species, family, region, nation, or galaxy) and then suddenly – Whoops! – there’s that nasty we/they paradigm in action again, begging for our self-forgiveness.

The profound helpfulness of these circle symbols is evident when we consider that a circle of infinite scope – total inclusion – has no one left out, no one excluded, no one denied, and all equally vital to the whole; each mind embracing this idea IS the whole seen holographically. When, on the other hand, we go with the polarizing default of inner/outer, we/they, us/them, no matter where the line is drawn – or whatever shape, for that matter, a circle is just a placeholder for any exclusionary perspective – we’re always at war with the alleged other in our mind. No peace is possible when the foundation for peace (total inclusion) is denied.

The pragmatic benefit of identifying with the “infinite circle” that includes all of us is that – with considerable practice! – we can train our minds to realize that we can truly afford to “Be kind, for everyone you meet is fighting a hard battle.” Even if there’s no instant transformation/manifestation in the surreal world of our space-time dreams, there is always an opportunity to forgive ourselves for dreaming of separation by seeing our shared interest of peace and benefitting from a tranquil mind in the process, regardless of what seems to be happening around us!

Since (in the USA at least) this is an election year, I try to remind myself that we only get to vote every year or so, but we vote for either the infinite (transpersonal) circle of embracing everyone in our gracious consideration – or the finite polarizing (personal) circle which leaves some out; our scapegoats outside the circle become our projected “goat-scape” yet the problem of our dualistic dueling dream has never left our mind.