Numbers and Proportions

Geometric gemstones

Friday, 6 December 2024 by Bruce Rawles

Recently 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).

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

Sunday, 24 November 2024 by Bruce Rawles

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):

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:

Enjoy!

Fractal and Geometric Animations

Friday, 25 October 2024 by Bruce Rawles

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

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

A few German sacred geometry books available for shipping costs only

Sunday, 21 July 2024 by Bruce Rawles

Update 21 July 2024: We’ve been in Arizona for exactly a year today, and I still have a handful of German books to find homes for, so I’m re-dating this post in case anyone would like to purchase a book for only my shipping costs!

(Originally published 25Jun2023):

We’re making good progress toward moving to Arizona (mid-July 2023) so I’m anticipating having more time to do in-depth posts after our move. Meanwhile, I’d still like to lighten our load and not move these books yet another time… I’m offering (until mid-July, while supplies last) – limited supply; limited offer) asking only for my shipping costs (US only) or free if you happen to be ~~on the central Oregon coast and can pick them up here in Yachats before we move!~~ in the Tucson, AZ metro area and can pick them up.

German sacred geometry books are available for shipping only (for a limited time!)

I have a handful of both of my German-translated books:

Mandalas der Heiligen Geometrie (MDHG) das Universale Design der Natur (SGDS German edition)
Bewusst malen – Heilige Geometrie

(If you live in Germany, you’ll probably want to order the Bewusst malen – Heilige Geometrie book, still in print, directly from the publisher, Silberschnur Verlag, since the shipping from the USA to Germany is likely more than buying it locally.)

If you speak German or know someone who does and has the least interest in geometry, I am offering special FREE books (very limited supply) of both of these books if you cover my postage costs. If you live near ~~Yachats, Oregon,~~ Green Valley, Arizona, you can pick them up here and skip the postage expense. For details on these limited-time, limited-supply books, contact me.

Note that you don’t have to be able to read or understand German to enjoy either of these books, as they are both lavishly illustrated – FAR more imagery than text. Also, each copy of the out-of-print MDHG book has a unique set of die-cut fold-up models of the 5 Platonic Solids (included behind the back cover) that make a neat mobile if you add your own thread.

Please contact me ~~before mid-July 2023~~ if you are interested in this special offer!

Here are a few more interesting geometry-related links (videos) that have crossed my path recently; several short introductory videos that add some useful and interesting additions to commonly cited facts:

Golden Spiral Symbol – What Does It Mean?
What Makes the Golden Spiral Symbol Special?
What is Sacred Geometry? (evidently using an AI-generated voice/script?)
How Does the Golden Spiral Relate to Sacred Geometry?
What is the Meaning of the Golden Spiral?
What are Some Examples of the Golden Spiral?
What is the Fibonacci Sequence?
… and several others…

AI generated geometry – revisited a few months later

Sunday, 31 December 2023 by Bruce Rawles

Since 2023 was a breakthrough year for AI, it will be interesting to follow how Artificial Intelligence tools will help geometers with a wide variety of interests leverage this mushrooming technology. I made a few preliminary “mixed bag” tests earlier this year with Dall-E 2. Here are a few more recent AI-generated geometric examples by others.

I was prepared to search for additional examples when I “switched gears” and thought I’d ask my friend and fellow geometer Dave Van Dyke if we could collaborate on some experiments with Dall-E 3 (since he has been using it a lot more than I have lately with fun and impressive results) and here’s what he generated (in less than an hour on a screen-sharing Zoom call) as we thought of prompts together that might provide interesting results; thanks, Dave! Note that Dall-E 3 still doesn’t quite know exactly how to make a precise (regular polyhedron) dodecahedron and other artistic liberties are taken, but still the results seem far more impressive than my more primitive results a few months ago. Here are some of the text prompts used to generate 4 sets of images at a time, some with “more strict” and some with “more creative” parameters selected:

make an image of (a) sea of mirrored geodesic spheres
or
make an image of (a) sea of mirrored dodecahedra

make an image of a translucent dodecahedron filled with multi-colored toroidal rings

make an image of a translucent toroidal ring filled with multi-colored platonic solids

make an image of a large translucent dodecahedron filled with multi-colored jellyfish

make an image of (a) sea of mirrored geodesic spheres with a large multi-colored Sri Yantra in the sky instead of clouds (evidently the reference to the prior image wasn’t used and instead the image was drawn from ideas more akin to mythical/fantasy/religious iconography/ideas)