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sacred geometry jewelry

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

Geometric gemstones

Friday, 6 December 2024 by Bruce Rawles

triangular geometric gemstone: Hashnu trioptic blastRecently a friend (Rick Fuller) showed me his gemstone grinding equipment including two grinders; one with a caliper-style readout to set grinding angles and another with a digital readout. My experience in this area is negligible, but the designs of cut gemstones seem profoundly connected to 3-dimensional solid geometry and all the laws, theorems, and insights gleaned over centuries about geometric proportions and more. Geometry is everywhere, and the geometric (polygonal) facets of countless gemstones are an excellent example.

Thinking about both the microscopic and macroscopic geometries involved and how this might be optimized in this endeavor, I did a quick online search for “gemstones cut to angles that match atomic lattice structure” and found this interesting AI-generated response:

Gemstones and Lattice Structure

Based on the search results, Corundum (Al2O3) is a crystalline mineral that exhibits a hexagonal crystal system. Its lattice parameters are well-defined, and the angles between its crystal axes are fixed at 90° and 120°.

In the context of gemstone cutting, Corundum is notable for its unique properties. Rubies and sapphires, which are varieties of Corundum, can be cut to specific angles to enhance their optical properties and brilliance. The cutting angles are designed to match the atomic lattice structure of the mineral, taking into account the hexagonal symmetry of the crystal.

For example, the refractive indices and birefringence of Corundum can be optimized by cutting the gemstone at specific angles, such as:

  • The “ideal cut” for rubies and sapphires, which involves cutting the stone with a pavilion angle of around 40-41° and a crown angle of around 25-30°. This angle combination minimizes internal reflections and maximizes brilliance.
  • The “step-cut” or “tabby cut,” which involves cutting the stone with a series of flat, parallel surfaces. This cut style is often used for sapphires and takes advantage of Corundum’s hexagonal symmetry to reduce birefringence and improve color distribution.

While other gemstones may not have the same level of precision in their cutting angles, Corundum’s unique properties make it an ideal candidate for cutting to angles that match its atomic lattice structure. This attention to detail can result in more vibrant colors, higher brilliance, and a more pleasing visual appearance.

Searching for related topics led to these quasi-random resources:

  • Hashnu Stones & Gems (the site listed in the book my friend showed me; the first to photos are from his book)
  • Gem Stone Geometry (Ceylon Natural Gems)
  • Gemstone Encylopedia (315 Gemstones from Actinolite to Zunyite)
  • Geometric Shape Gemstones (Etsy)
  • Solid geometry (Wolfram Mathworld)
  • Euclidean geometry (an excellent and extensive article)

… and since we’re on the topic of geometric jewelry, check out Sacred geometry jewelry by artist David Weitzman (Ka-Gold) … Related Content: 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, Hexagram (Star of David).

pentagonal geometric gemstone

Filed Under: 3D Geometries, Numbers and Proportions, polyhedra, sacred geometry jewelry

GeometryCode.com Annual Survey (December 2023)

Friday, 1 December 2023 by Bruce Rawles

Thanks to all of you for your continued and ongoing support of this labor of love website!

It has been quite a while since we’ve had a reader survey or poll! Here are the results of the last one from a couple of decades ago:

 

Here’s an opportunity to share your geometric interests, passions, curiosities, favorite topics, and whatever you think will be helpful, useful, and fun!

We’ll replicate this survey each year, provide results annually, and hopefully guide our content to where you find existing and unexplored subjects!

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sacred, mystical or symbolic geometry
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books on sacred geometry & related topics
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geometric art, graphics, jewelry, decor
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geometric apparel, printed items, wares
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geometric tools, toys, products
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geometry interviews, conversations
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geometry in the natural world, biology, mineralogy
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geometry in man-made world, architecture, technology
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interconnectedness & related topics
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proportions, numbers (e.g. golden ratio, Fibonacci Series)
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fractals, Mandelbrot Set, etc.
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tilings, tesselations, space-filling geometries
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hyperdimensional geometries
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physics, quantum theory, metaphysics
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macrocosmic geometry (e.g. planets, galaxies)
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microcosmic geometry (e.g. crystals, molecules)
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primal 3D geometric forms (e.g. polyhedrons: Platonic, Archimedean Solids, etc.)
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geometry and sound, music, acoustics
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geometry videos, tutorials
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geometric construction techniques and tips
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Learn about Sacred Geometry Design Sourcebook – Universal Dimensional Patterns by Bruce Rawles
Learn about The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation by Bruce Rawles
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Filed Under: 2D Geometries, 3D Geometries, 4D Geometries, applications, Archimedean Solids, audio, coloring books for adults, Fibonacci Numbers, Fractal Geometry, golden ratio, Hermetic Laws, modern physics, Numbers and Proportions, Platonic Solids, polyhedra, sacred geometry animation, sacred geometry apps, sacred geometry architecture, sacred geometry art, sacred geometry books, sacred geometry calendars, sacred geometry coloring books, sacred geometry interviews, sacred geometry jewelry, sacred geometry physics, sacred geometry toys, sacred geometry videos

Geometric Lamps … and Coloring Pages by HYBYCOZO / COZO

Monday, 24 January 2022 by Bruce Rawles

A while ago, geometer colleagues alerted me to some remarkable geometric lamps by COZO (many using classic Platonic Solid foundations) with elaborate patterns on each face, making for a lovely immersive light experience as these patterns are projected onto walls, ceiling, and floor. Here’s an example of a Dodecahedron Lamp with an inset Flower of Life pattern in each of the 12 pentagonal faces:

Dodecahedron Lamp with an inset Flower of Life pattern in each of the 12 pentagonal faces

Returning to the HYBYCOZO website, I discovered that they also offer nice collections of geometric coloring pages – both free and paid options. Here is an example of one of the free patterns in the free collection. The HYBYCOZO coloring book features some fine examples of tesselations, Islamic-inspired patterns, a bit of non-Euclidean geometry, and more.

These patterns are a good complement (with minimal overlap) to the wide variety of geometric patterns in Sacred Geometry Design Sourcebook (SGDS) Universal Dimensional Patterns. This timeless resource/reference book of geometric patterns has been assisting and inspiring geometric enthusiasts since 1997.

Filed Under: 3D Geometries, Platonic Solids, polyhedra, sacred geometry art, sacred geometry jewelry

3D Hexagonal … and other … jewelry

Tuesday, 5 September 2017 by Bruce Rawles

I recently learned about some exquisite jewelry from Rebecca Li; such as her “SIGNATURE SIX SENSES TALISMAN” series:

“My mission is to help us touch Mindfulness, and I do that through objects of modern art–because art is magical; I believe it has the power to take us to the present moment.”

… and her additional commentary about what inspired her to make these jewelry gems:

My inspiration for designing Six Senses Talisman Collection:

    Why the hexagon?

When I was thinking about designing a piece of pendant talisman, I knew I wanted to use a gemstone as the main element, but I struggled to come up with a unique shape that really resonated with me. Then an interesting thing happened to me. I started to see hexagons everywhere unexpectedly, to the point, I couldn’t ignore them anymore. I took it as a sign; I just knew I should use hexagons as my building block in my talisman design.

    Why hexagonal crystal points?

This particular shape is the result of one hexagon rotating in space. Imaging a hexagon standing up with one diagonal being the center line, rotating this hexagon 60 degrees six times and connecting those points together then we get this hexagonal crystal point. My message here is six. We say intuition is our sixth sense. I feel like this 3D shape is the perfect interpretation of my message.

    Hidden Flower of Life and Seed of Life:

I see the hexagon as the minimalist interpretation of Flower of Life as well as Seed of Life. The hexagon contains those two powerful ancient sacred geometries.

Design elements in her jewelry:

  • Natural healing gemstones
  • Solid gold
  • Sacred Geometry
  • Modern minimalist style

More info here.

Rebecca Li - jewelry example: SIGNATURE SIX SENSES TALISMAN

Filed Under: 3D Geometries, sacred geometry jewelry, sacred geometry news

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Sacred geometry Jewelry Sacred geometry jewelry by artist David Weitzman Ka-Gold Related Content: 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, Hexagram (Star of David).
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