golden ratio

Monthly miscellany for mathematical musing

Sunday, 30 April 2023 by Bruce Rawles

Thanks to fellow geometer, James Barrett, for this interesting update about tiling patterns: Newly discovered ‘einstein’ shape can do something no other tile can do.
A portion of this Coast-to-Coast program is on crop circles, which have exhibited some amazing geometries for decades.
The Quantum Revolution (12 episodes): “Gaia is proud to present a comprehensive series from Nassim Haramein who offers a quantum revolution to the way we think about the universe, quantum mechanics, consciousness, and our spiritual connection.” Geometry and physics are inexorably linked! I attended one of Nassim’s fascinating all-day workshops a couple decades ago, give or take.
2022’s Biggest Breakthroughs in Math; one example: like 3D creatures able to discern situations that Flatlanders can fathom, sometimes we just need to add another dimension – in this case to solve a bubble problem.
Sacred Geometry circle art and some helpful how-to “Geometryptamine” construction videos by Zak Korvin.

$golden spirals from pentagram fractal by Geotryptamine$

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)

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!

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!