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Geometric gemstones

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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:

… 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

Sacred Geometry in the documentary “Leonardo da Vinci” by Ken Burns

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Leonardo da Vinci (Ken Burns PBS documentary) screen snap of geometric shadow illustration

This evening we finished watching the excellent 2-part documentary “Leonardo da Vinci” by Ken Burns (and others) which – like everything Ken touches – was wonderful. No surprise, and, of course, highly recommended. Please support PBS and your favorite public/community/independent media, too!

Initially, I thought I would use my default browser (Brave) to find some still images from the documentary, and using this search term: “sacred geometry in Ken Burns Leonardo documentary” got this interesting – not bad! – AI-generated response:

In the Ken Burns documentary about Leonardo da Vinci, sacred geometry is explored as a key aspect of da Vinci’s work and thought process. Da Vinci’s use of geometric shapes and proportions in his art and designs is highlighted as a way to understand the underlying structure of the universe.

  • Geometric Lines and Proportions: The documentary shows how da Vinci used geometric lines and proportions in his paintings, such as the Mona Lisa and The Last Supper, to create a sense of balance and harmony. These geometric elements are also seen in his designs for machines and buildings.
  • The Vitruvian Man: Da Vinci’s famous drawing, the Vitruvian Man, is featured in the documentary as an example of his use of sacred geometry. The drawing depicts a nude male figure inscribed within a circle and square, demonstrating the mathematical proportions of the human body.
  • Nature and Geometry: The documentary explores how da Vinci saw geometry as a way to understand the natural world. He believed that the patterns and shapes found in nature, such as the spiral of a shell or the branching of trees, were evidence of a deeper geometric order.
  • Da Vinci’s Notebooks: The documentary features da Vinci’s notebooks, which contain his thoughts on geometry, proportion, and the natural world. These notebooks show how da Vinci saw geometry as a way to understand the underlying structure of the universe.

Overall, the documentary presents sacred geometry as a central aspect of da Vinci’s work and thought process, and shows how it influenced his art, designs, and understanding of the natural world.

Check it out. The topic of sacred geometry – although not referred to by that name – makes a cameo appearance in this 2-part series among Leonardo’s wide range of mathematical and scientific interests which, in turn, are just a small portion of the depth and breadth of the realms explored by this amazing historical figure. I appreciated how this “Renaissance” icon appeared to embrace anything and everything as a classroom for for generalizing a deeper understanding, as Egyptologist John Anthony West told us was the goal in ancient Egypt, rather than encouraging specialization.

(The image at the top of this post is a screen snap from the documentary. There are way too many amazing illustrations, so this is just one of the numerous featured in this program.)

In addition to the Vitruvian Man, this PBS program also showed a generous assortment of da Vinci’s geometric and geometrically-oriented mechanical drawings of both natural and man-made objects.

When I was putting together my first book, I was inspired by the geometry of one of da Vinci’s sketches of an “ideal church” and made this (Claris CAD) image from that concept (page 191 from Sacred Geometry Design Sourcebook):

Sacred Geometry Design Sourcebook (page191) - Ideal Church (adapted from a sketch by Leonardo da Vinci)

Some of the less obvious geometries in the “Vitruvian Man” (a.k.a. “Measure of Man”) by da Vinci – in addition to the more widely known phi proportion: toes-to-navel and navel to top of head – are revealed on page 190 of the SGDS book:

Sacred Geometry Design Sourcebook (page 190) Leonardo da Vinci: Vitruvian Man (Measure of Man) illustration showing sacred geometry proportions

Enjoy!

Fractal and Geometric Animations

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Here are some interesting geometric animations… Lots more to explore!

A few example animations found using sacred geometry as a search term on Giphy:

Fractal (Octahedra) Polyhedra with Sierpiński triangle faces via GIPHY

A Flower of Life variation with animated rainbow colors via GIPHY

Fractal (Dodecahedra) Polyhedra with pentagon faces via GIPHY

A couple of example animations found using polyhedra as a search term on Giphy:

A Rhombic Triacontahedron appears when the animated mesh reaches its finest resolution; here’s a fold-up pattern for the Rhombic Triacontahedron (a.k.a. Kepler’s Solid). Here’s the animation via GIPHY.

Each edge of the triangular faces of this lovely icosahedron is subdivided 4 times. Here’s the animation via GIPHY.

… and a few example animations found using fractal as a search term on Giphy:

Fractal gears via GIPHY

Fractal dogs/ears via GIPHY

Koch snowflake via GIPHY

Whirling squares (golden ratio spiral) animation
whirling-squares-golden-ratio (frame from animation)

Thanks to fellow geometer Dave Van Dyke for pointing these out to me.

2025 Martineau Solar System Calendars available now

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Once again this perennial calendar is available! These timeless images portray amazingly accurate geometric representations of the planetary orbits in our very own solar system – now available in printed calendar form with dates for 2025!!

Martineau Solar System Calendars

Martineau solar system calendar -12 months

These 12 images are highly accurate geometric models of our solar system; details are here. They were inspired by the remarkable work of John Martineau (publisher of Wooden Books), who wrote “A Little Book of Coincidence“. Tiled images of planetary surfaces and related images form the backgrounds. Images are two or more orbits defined by simple 2D (circles, triangles, squares, pentagons, hexagons, octagons) or 3D geometries (e.g. tetrahedron, icosahedron, dodecahedron) typically with 99.9% or greater accuracy. The work picks up where Johannes Kepler’s mystical exploration left off. A newly revised calendar (which is updated every year) with images from the Martineau Solar System is again available for 2025.

… Don’t need a sacred geometry calendar … or a sacred geometry book? You can always support GeometryCode.com via our newly updated Tip Jar.  Our virtual “tip jar” accepts a variety of methods of making financial donations; any greatly appreciated! You can use Zelle, Venmo, Wise, PayPal, and a check or money order in US dollars works as well.tip jar for GeometryCode.com

Fun, whimsical, playful geometric art

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Since summer is often associated with vacations, play time and recreation, this month’s short but sweet post – thanks to my wife Nancy – is inspired by a link she sent me in her frequent scouring of social media for all things artsy. She thoughtfully keeps an eye out for geometric items. This one reminds me that in addition to being fascinating and profound for the scientific mind, geometry can also be inspiring for fun, loose, right-brained, artistic, non-linear, colorful and entertaining explorations and pastimes. This example was found in samples from this larger collection: Print & Pattern: Geometric Kindle Edition by Bowie Style and Marie Perkins

art: Neo Geo by Anne Waters
Neo Geo by Anne Waters