polyhedra

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.

2025 Martineau Solar System Calendars available now

Wednesday, 18 September 2024 by Bruce Rawles

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

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 is greatly appreciated! You can use Zelle, Venmo, Wise, PayPal, or a check or money order in US dollars, which works as well.

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

Saturday, 27 July 2024 by Bruce Rawles

"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

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.