3D Geometries

Michael Schneider’s new book! Proportion in Art and Architecture (Wooden Books)

Thursday, 19 January 2023 by Bruce Rawles

book cover: Proportion in Art and Architecture by Michael Schneider

Long-time – as in decades-long – readers of GeometryCode.com and its predecessors (such as the bibliography in Sacred Geometry Design Sourcebook (SGDS)– Universal Dimensional Patterns published in 1997) will recall that I’ve often recommended – more than any other book – A Beginner’s Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science (released September 29, 1995) by geometer colleague Michael S. Schneider as an ideal companion to SGDS. In that lavishly illustrated journey of imagery and text through number and geometry, he explores – in over 350 pages – as the subtitle suggests: mathematical archetypes on a voyage from 1 to 10. It’s a great book!

Schneider also authored 6 geometric activity books spanning 730 pages plus an animated DVD: Constructing The Universe: A Journey From 1 to 12 – brilliant, very easy to follow, and clearly explained, beginning at the bare basics and culminating in compelling cosmologies. Each of the first 12 counting numbers receives a first-class tour through realms mundane to mystical, deepening our appreciation of numbers and their geometric interconnections from many disciplines, traditions, and perspectives. In addition, he also authored the fascinating DVD Fibonacci Garden: Mathematics in the Plant World.

Michael’s most recent work, another great book – Proportion in Art and Architecture – is among the 60 delightful Wooden Books titles by a variety of authors, published by John Martineau who will also be very familiar to readers of this website. This book, like the Beginner’s Guide to Constructing the Universe book, has wonderful explanations, illustrations, and analyses – and most of the analyses in it are original! The UK version is available now, but it’s inexpensive to ship to the US. The US version will be out in March 2023.

The image below is page 13 of this excellent new book in the chapter entitled: Frozen Music: the geometry of harmony:

Frozen Music - from Proportion in Art and Architecture page 13 by Michael Schneider — Frozen Music: the geometry of harmony – from Proportion in Art and Architecture page 13 by Michael Schneider

The illustrations below are color versions of similar illustrations from the book on pages 30 and 31:

pentagonal construction underpinnings: Greek, Egyptian, Mayan art - studies by Michael S. Schneider — pentagonal construction underpinnings: Greek, Egyptian, Mayan art – studies by Michael S. Schneider

Here are the chapter titles to give one a sense of the depth and breadth that this little gem covers:

Introduction
Macro-Meso-Microcosm
Therapeutic Arts
Mathematics in Design
Static and Dynamic
Proportion
Frozen Music
Harmonic Rectangles
Harmonious Proportions
Hotspots and Alignments
Regular Polygons
Ad Triangulatum
Triangular Proportions
Ad Quadratum
Square Proportions
Pentagonal Proportions
Hexagonal Proportions
Heptagonal Proportions
Octagonal Proportions
Ten and Twelve
Canonical Rectangles
Rabatment
The Root-Two Rectangle
Root-Two Proportions
The Root-Three Rectangle
The Root-Four Rectangle
The Root-Five Rectangle
The Golden Rectangle
Golden Proportions
Proportional Quotations

There’s even a cameo appearance of the Mona Lisa in the book which also had a cameo in the Glass Onion movie!

Check out Michael’s website: ConstructingTheUniverse.com

Aside: I interviewed Michael Schneider for local TV in Grass Valley when he first arrived in CA 25 years ago on FCAT (Foothills Community Access Television) which was great fun!

Cedar sprig symbolism: Freedom or fragmentation?

Monday, 16 January 2023 by Bruce Rawles

Button art: Yachats New Year's Day Peace Hike 2023 — Button art: Yachats New Year’s Day Peace Hike 2023 – art by Loren Dickinson

I started the new year by participating in the annual Peace Hike here in Yachats which I look forward to every year since moving to the Oregon coast 6 years ago – it’s always a great reminder to share in the willingness to look at past atrocities of non-connected thinking and use the contrast of that insanity to the peace we really want to motivate us to deepen an all-inclusive sense of community – and not make avoidable mistakes by forgetting the lessons we have been given. I feel blessed to live in a community where that egalitarian spirit is highly valued all the while realizing that this is the longing of every heart and mind around the world, however frequently ignored or seemingly deeply buried.

In addition to the hike from the Commons Building to the Amanda Statue (near the new bridge*) the yearly ritual involves giving all participants a little sprig of cedar leaves to put on the common fire to symbolically represent our renewed commitment to peace and anchor our intention. The very fractal geometry of cedar sprigs gives us an opportunity to look at our ongoing motivations for everything we think and do – we can either metaphorically follow the smallest branches back through larger and larger gatherings of self-similar shapes to find the source or root where we realize we share the same pattern, the same intrinsic design, the same essential identity. If we traverse the fractal in the opposite direction – from unity to individuality, we may lose sight of the unifying generalization by a preoccupation with individual, specific divergent paths. This is one of the countless examples of how a geometric symbol can be used to join and strengthen … or as a tool for divisive thinking. It all depends on how we look at life each moment, either from the freeing kindness of inclusion … or the cruel fragmentation of exclusion. This is explored in greater depth in The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation.

Here are a few more fractal images from prior posts on this website and other media I’ve made:

IMception: fractal flotilla? — source: https://img.einnews.com/large/196490/imception-02.png#4096×2304

fractal dodecahedron from sketchup.google.com

pentagram fractal (made with Adobe Illustrator, Bruce Rawles)

Mandelbrot Set - Deep Dive — Mandelbrot Set – Deep Dive

tetrahedral-fractals - low resolution image of page 226 from SGDS (Sacred Geometry Design Sourcebook - Universal Dimensional Patterns by Bruce Rawles) — Tetrahedral-Fractals – low-resolution image of page 226 from SGDS (Sacred Geometry Design Sourcebook – Universal Dimensional Patterns by Bruce Rawles)

toroidal fractal by Bruce Rawles (POV-Ray) — toroidal fractal by Bruce Rawles (made using POV-Ray software)

* Last year’s theme for the local Polly Plumb Productions banner project was “Paths to Yachats” to coincide with the completion of the new suspension footbridge. If you look for my (Bruce Rawles) banner art on this page, you’ll see my notes about my submission:
“Inspired by the symbolism of new higher, more sustainable footbridge connecting Yachats to Cape Perpetua, I combined the multi-ridge skyline of the cape as seen from Yachats Ocean Road with the general shape of the bridge from a photo I took shortly after the bridge was opened. Realizing that there are more than just human “paths” to our lovely community, I thought of adding an eagle or seagull or two but ended up just complementing the human traffic with the vast aquatic migration by means of a sea star and a whale fluke. I’m reminded that I must keep the bridges in my mind open to graciously receive all, regardless of form.”

Since I now have two artist depictions of a suspension bridge as visual bookends for this post, it reminds me of a geometric shape that occurs with a rope, chain, or other flexible material suspended between two fixed points, the catenary, which sounds like a cross between a cat and a canary – or perhaps the shape of the belly of a cat that ate a canary? The actual shape of suspension bridges with additional non-uniform weight is probably different, but it’s a good place to start for a first approximation.

Mirrored geometric solid cameo on new whodunit

Friday, 30 December 2022 by Bruce Rawles

Spoiler Alert: This post will reveal the Platonic Solid that stole the show (at least for me) on the latest flick by director Rian Johnson. If this isn’t an issue, read on! May I have the envelope, please…

I don’t often watch murder mysteries unless there’s a generous amount of tongue-in-cheek involved, such as my favorite classic of this genre, “Murder by Death” which I recommend if you haven’t seen it. The “22 Twain” punny address of the mansion at the beginning of that film had me hooked at the get-go.

Based on Johnson’s prior whodunit, we thought we’d give his new one a go, and enjoyed it, also. So… for silly fun, we recently watched (streamed at home) “Glass Onion: A Knives Out Mystery” which featured – in addition to the expected absurd assemblage of characters with a preposterous plot, and a fun puzzle box invitation featuring a “whirling squares” (Fibonacci) spiral, other familiar geometries plus assorted over-the-top glass sculptures in a billionaire island mansion with an infinity pool – a few witty pop-culture-for-nerds cameos like a Boston Dynamics “dog robot” porter and momentary dialogue about Google alerts – which I find helpful every week for feeding me news about sacred geometry and other metaphysical topics.

Regular readers will recall that this isn’t the first time I’ve featured the lovely geometric artwork of artist Anthony James … The (virtual envelope) contents of that award-winning geometric art (from my perspective) is … cue the drum roll … James’ Mirrored Icosahedron with the illuminated interior edges presenting the viewer with an exotic labyrinthine landscape of triangular tunnels. That’s it! At the focal point heart of the bulbish-shaped centerpiece glass onion sphere-like feature of the larger mansion sits a mirrored icosahedron which sure looks a lot like James’ handiwork! (If anyone knows any different, please let me know.)

The screen snap of a frame from the trailer doesn’t do it justice… Here’s a frame close-up from a video giving the “star” more exposure:

Speaking of mirrored and/or virtual polyhedra, see also: Sara Frucht’s Kaliedoscapes!

Alternate ending: If you were hoping for a dodecahedron, you might want to read this post on my other niche blog; the original model that appeared in the Lawrence Livermore Lab bulletin my Dad brought home early in the development of laser fusion showed a translucent plexiglass dodecahedron with 12 plexiglass cones centered on each pentagonal face all converging at a hydrogen pellet target.

Happy New Year all!

In memoriam: William Goldstein

Friday, 14 October 2022 by Bruce Rawles

I recently learned of the passing of William Goldstein (a.k.a. Will Gold), who left us on August 11, 2022. William was a fellow geometer, researcher, historian, mystic, and much more. It was only fairly recently (September 2021) that I discovered his work, our many common interests, and favorite books while exploring his extensive fascinating writings which are a great complement to my more mathematical leanings. I began a brief stint as his webmaster and I was hoping to be able to share many recorded conversations between William and myself with my GeometryCode readership, but that was not to be.

Rather than have this important body of work disappear, I offered his family the opportunity to keep the site going so that others might benefit from his insights, and helpful resources including an extensive library of unique and interesting books, most of which are still available. The newly relocated website (formerly Chartres-sacred-geometry.com) is now live at Chartres.GeometryCode.com. It is my hope that my readers will avail themselves of his thoughtful and insightful writing and excellent book (and other media) recommendations of particular value for geometry enthusiasts, particularly the in-depth cultural and historical underpinnings behind so many important discoveries and real human progress.

Here is the prior article I wrote about William’s site which I would recommend reading as well.

Thank you, William, and to your family as well!

2023 Martineau Solar System Calendars available now

Friday, 23 September 2022 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 2023!

Martineau Solar System Calendars

These 12 images are highly accurate geometric models of our solar system; details 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 2022.