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Numbers and Proportions

Monthly miscellany for mathematical musing

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Bubble colors (close up, hue shifted)

golden spirals from pentagram fractal by Geotryptamine

Similar golden ratio, pentagram/pentagon, spiral constructions – with numerous variations! – are on pages 146-156 (among others) of Sacred Geometry Design Sourcebook.

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

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book cover: Proportion in Art and Architecture by Michael Schneider
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!

Mirrored geometric solid cameo on new whodunit

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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.)

Glass Onion: Mirrored dodecahedron in center

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:

Anthony James mirrored icosahedron

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!

Spirals, Ferns, and Universal Unfolding Motifs

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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.

mirrored tiling pattern: unfurling silver fern frond

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.”

Wizard of Oz geometry: Dorothy on yellow brick road spiral

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
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
Auckland NZ Airport ceilng art: spiral fish school
Spiral fern head; Kitekite Falls, NZ
Spiral fern head; Kitekite Falls, NZ
Spiral yard art; Hamilton, NZ
Spiral yard art; Hamilton, NZ
New Plymouth, NZ art gallery staircase; spiral flourishes
New Plymouth, NZ art gallery staircase; spiral flourishes
Napier NZ boardwalk art with overlapping complementary spirals
Napier NZ boardwalk art with overlapping complementary spirals

Geometric Inspiration and Education

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geometric art, color-filled intersections of simple 2D shapes shapesReflecting on the growth – explosion, really – of information available to us online compared to what we had access to just a few decades ago, I recalled that in the early 1970s when I was an undergraduate electrical engineering student at UC Davis, California, I used the card catalog in a long row of tidy wooden cabinets at the Shields Library – a building which covers an entire block of the campus – to look up everything I could find (at that time) on the Golden Ratio, Golden Number, Golden Section, Divine Proportion, Phi (ø) or Tau (τ) Proportion, Sacred Cut, Extreme and Mean Ratio, etc. and other synonyms for the ubiquitous (1 + √5)/2 ≈ 1.6180339887… value. This fascination was my response to a gift of two books – both with references to the golden ratio – from my mother who indirectly introduced me to this enthralling subject which later was popularized with the term “sacred geometry”:

I found a few dozen books that led to other books and on and on into the mystical realms of the intersections of mathematics, ornamentation, art, music, architecture, history, and above all – mind. … But after checking out those books, I felt I had somewhat exhausted the more obvious references to this topic in this library. … Not anymore!

Fast forward a half-century; now googling Sacred Geometry, as of today (21 April 2022) we get a staggering 45 million results for this niche topic (including my evergreen minimalist GeometryCode Introduction to the subject now way past overdue for a revisit and update) and 481 million for the Golden Ratio!

The same curiosity still remains in my mind: What can be counted on that does NOT change? The laws of mathematics in general and geometry, in particular, seem relatively stable, fortunately, at least within the context of the Cartesian space-time paradigm. When I was in high school, I read Education and Ecstasy by George Leonard and I realized that to really dive deep and learn immersively, one needed to be motivated, inspired or at least have their curiosity seriously piqued or their assumptions about life challenged. Exploring some of the ideas that these readings opened up for me was enough!

Now there are a multitude of online offerings that can motivate young and old alike to explore esoteric topics like interconnectedness (e.g. quantum entanglement), geometric symbolism and synchronicities, and so much more. Another vital element that has a multiplying effect is FUN!

Here are a few quickly-found resources for educators and explorers in this realm (including teachers, home-schoolers) and philosophers of all stripes:

Enjoy! :-)