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

Monday, 27 January 2025 by Bruce Rawles

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):

Balcony. July 1945, Lithograph (a coastal condo scene where the center “bulges” out in fisheye lens fashion)
Three Spheres I. September 1945, Wood Engraving. (I had a vibrant fluorescent “black light” poster of these stacked spheres on my wall in high school)
Reptiles. March 1943, Lithograph (including a 3D dodecahedron and a 2D tiling (tesselation) pattern of tiled reptile shapes)
Up and Down. July 1947, Lithograph (staircase/balcony composition combining bird’s eye and worm’s eye views of the same scene blended into one)
House of Stairs. November 1951, Lithograph (an impossible realm of stairs crawling with multi-legged creatures that can crawl or spiral roll)
Relativity, July 1953, Woodcut (another impossible realm of staircases)
Sun and Moon. April, 1948, Woodcut, printed from four blocks (Solar/lunar motifs overlap with space-filling birds)
Double Planetoid. December 1949, Wood Engraving printed from four blocks (star tetrahedron made from an “organic” tetrahedron and a “man-made” tetrahedral world that interpenetrate but do not appear to touch)
Order and Chaos II. August 1955, Lithograph (featuring a small stellated dodecahedron with pentagrams on the pentagonal bases)
Three Worlds, December 1955, Lithograph (leafs float on the surface of a fish pond reflecting bare winter tree branches which also seem like roots); this image was used as the original album cover for the Beaver & Krause vinyl album “In A Wild Sanctuary” which you can hear here, combined with their more recent album, Gandharva (celestial musician). Fascinated by the idea of musically interesting harmonics and proportions, I made (using a FORTRAN program) a spreadsheet using the CDC3400 our high school had access to once a week showing resonant frequencies and harmonics of the Grand Gallery of the Great Pyramid at Giza, Egypt (see other internal references to the amazing geometry of this pyramid) and other accessible chambers. On two occasions, I visited Beaver’s Los Angeles recording studio in the early 1970s and planned to join him to record electronic (Moog analog synthesier) music in the Great Pyramid at Giza, Egypt in fall 1973, but this trip never materialized, as the Middle East political situation then was hardly conducive to travel. In November 1992 (19 years later) I was able to explore the interior of the Great Pyramid and record some flute music by Gregg Braden (his website) with my wife Nancy and 18 others on a tour led by Braden.
Print Gallery, May 1956, Lithograph (a recursive image of an art gallery where the interior columns become exterior posts of an overhanging roofline); this is the image that reminded me of non-Euclidean Geometry and revisiting Escher’s art for this post.

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.

For further exploration, check out:

the official M. C. Escher website
the M. C. Escher Wikipedia page
Wolfram MathWorld references to Escher, such as Penrose Stairway, Freemish Crate, Impossible Figure, etc.
Lessons in Duality and Symmetry from M.C. Escher
The Influence of the Perspectives of M.C. Escher … just for starters!

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 Escher-inspired 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.

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