2D 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.

Spirals, Ferns, and Universal Unfolding Motifs

Thursday, 1 December 2022 by Bruce Rawles

While I try not to obsess about geometric forms, I do observe when certain motifs appear more frequently in various locales. When we visited San Miguel de Allende, Mexico earlier this year, I noticed an abundance of small stellated dodecahedra (and similar shapes) particularly evident in metal lamps, yet also in other art forms as well; each pentagonal pyramidal stellation arm encodes the golden ratio. …

Having just returned from a second visit to New Zealand, it seemed that one of the most ubiquitous geometric shapes we saw on this trip was the spiral, which appears in Maori jewelry, carving, tattoos, art, and culture, connecting the natural world with the art of the indigenous people of Aotearoa. Below I made a mirrored tiling pattern from an image of an unfurling silver fern frond which exhibits an excellent example of fractal geometry.

Wikipedia says “The koru is a spiral shape based on the appearance of a new unfurling silver fern frond. It is an integral symbol in Māori art, carving, and tattooing, where it symbolizes new life, growth, strength, and peace.”

Reflecting intuitively on our spiral “journey” through life like the homeward trajectory Dorothy embarks upon in the Wizard of Oz when she follows the yellow brick road, we find context in the cyclical, yet ever-evolving, ever-unfolding awareness with each revolution of the spiral. In three dimensions, we have the double helix of the DNA molecule in everything we typically call life in this world, embodying the golden ratio in each pentagonal molecule appendaged to the helix and also in the (Fibonacci proportion) 34/21 Angstrom ratio of each complete turn of the helix. Every year, each season has familiar patterns and signs, yet there is a much larger progression we discover when we “zoom out” a bit from our myopic, provincial perspectives. The Chambered Nautilus shell has a pleasing uniformity as one spins the shell around, since each accretion uses the same angular template as the preceding one, and builds upon the strength of sameness throughout.

Here are a few more spirals from our most recent NZ trip; enjoy!

New Zealand: Napier museum; Maori wood carvings with numerous spiral forms

Auckland Airport ceiling lamp fixture — Auckland NZ Airport ceiling lamp fixture

Auckland NZ Airport ceilng art: spiral fish school

New Plymouth, NZ art gallery staircase; spiral flourishes

Napier NZ boardwalk art with overlapping complementary spirals

The Geometry Code (Kindle edition) turns 10 today!

Wednesday, 28 September 2022 by Bruce Rawles

On this day in 2012 – a few months after introducing the print version, I made the Kindle version of my second book available: The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation. I’ve appreciated all the support and encouragement during the past decade for this work, intended to be a bridge between two primary interests: the principle of interconnectedness reflected in geometric archetypes … and pure non-dual metaphysics.

The cover art and the opening graphics for the last 7 chapters in the book are symbolic interpretations of the 7 Hermetic Laws – covered in detail on this website.

However, I just realized that I never gave very many details on the first geometric image used in the book, so here are a few. The image below depicts a geometer’s compass spanning an infinity symbol – to represent ideas that are “all-encompassing” – and the Great Pyramid of Giza, Egypt (the last remaining of the seven wonders of the ancient world) with the Great Sphinx in the foreground. There is abundant fascinating geometry in both the exterior and interior of the Great Pyramid including the phi and pi proportions, squaring the circle, many terrestrial proportions, etc. which are covered in various talks and essays on this website spanning several decades.

The Pyramid’s foundation idealizes the square, connecting the compass (vertical axis) and square (horizontal plane of materiality.) The four sides of the pyramid and the overall shape of the compass are triangular, signifying the reconciliation of polarities, meeting in the capstone atop the pyramid and the apex of the compass where both vision and inspired guidance are possible through the exalted perspective that sees all as equal.

Here is the foreward to the book by Gary Renard:

I met Bruce Rawles in Medford, Oregon in 2008 and again in 2009 and 2010 in Ashland, Oregon when he and Ed Karlovich hosted and promoted seminars I did there. During one of the car rides to the airport, he mentioned a book that he was working on that sounded unique and original. I was curious as to how he would incorporate the diverse disciplines of sacred geometry (which he has studied extensively), Hermetic philosophy with its unchanging laws of nature, and the non-dual metaphysics that is the essence of my life work, best known to many by the spiritual masterpiece, A Course In Miracles (ACIM).

He was quick to point out that it’s easy to get distracted or diverted by external indicators like geometric symbols and get bogged down in specifics or interpretations that lead away from wholeness and toward the problematic belief in the concept we call separation or ego. At best, symbols can be mirrors or reflections within a dualistic world that attest to principles guiding our minds back to an awareness that has no opposite, an experience of love that transcends anything that can be conveyed by words or symbols of any kind. These symbols, seen from that view that embraces oneness, reflect a healed mind, but are never the cause of it.

He also recognized that the very ‘universal’ laws that can seem to keep us stuck in the dream (to our detriment) can also be reinterpreted for everyone’s benefit by the inner teacher of kindness that ACIM refers to as Holy Spirit and my teachers abbreviate metaphorically as ‘J’. We can learn to trust this inner teacher and use every bit of feedback as cues or reminders to practice, practice, practice the spiritual art of quantum forgiveness, which sees our ‘problems’ as causeless, and therefore completely benign and forgivable.

Having acknowledged these two caveats, he has done a remarkable job of sharing how – by developing trust in the inner Inclusive Guide we all have in common – we can include the symbolic contents of our unconscious mind in very familiar forms, such as geometric ones, as feedback devices or memory jog aids to deepen our trust in our inner teacher. For example, Bruce points out how egos use circles dualistically to exclude and banish ‘others’ to the outer regions (and the equally unhelpful ‘special love’ for those ‘within the circle’), but can just as easily be reinterpreted to be infinitely large, including All as One.

Similarly, with his explanations of seven laws of nature, it becomes obvious that these unchanging rules (starting with the fact that the universe exists only in mind) can be directed by either the 100% sane thought system or the 100% insane thought system that we all share. The choice is up to us, moment by moment. Without the specific people and events of our lives – including the exemplified laws and forms we encounter – our unconscious guilt would never be revealed, released, and healed.

Countless other examples demonstrate the importance of forgiving the dreamer of separation; this work is a fun and useful addition to the list of ACIM studies books. Bruce and I share the kinship of all (not just those one would call students) learning to practice forgiveness, gently awakening our dreaming minds to Love that is beyond any words or symbols. Enjoy, and awaken.

– Gary R. Renard, the best selling author of The Disappearance of the Universe

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.