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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.

Digital dabbling with Dall-E

Wednesday, 29 March 2023 by Bruce Rawles

After several years of mixed results with Siri (on my iPhone and MacBook Pro) and cylindrical kitchen add-on Alexa, often amusing and sometimes rather useful, I think my current favorite “Jetson” technology still might be our humble shorter cybernetic cylinder, “Rosie” the Roomba – our household vacuuming robot weekly whisking an amazing amount of fur from feline friends in our home.

Out of curiosity, I recently have done some very preliminary explorations with 2 OpenAI artificial intelligence apps; ChatGPT – rather impressive for certain well-defined tasks – and Dall-E 2. Since the latter seems appropriate for the visual nature of geometric imagery, I gave it a few prompts and got the results below within a span of about 45 minutes. Most of that time was spent thinking up a geometric ideas and then reviewing (clicking on the smaller thumbnails to see the zoomed-in) results; the actual time that the application took to generate the images with about 10-20 seconds for each batch of 4 rendered thumbnails! At its present maturity, Dall-E seems to have some limited usefulness if mathematical precision is less important than artistic conceptual prototypes. What do you think? Fun to play with, of course! The text I typed is followed by the output for each experiment:

experiment 1:
golden glowing dodecahedron hovering above a shimmering sea of 144 smaller iridescent silvery icosahedra

experiment 2:
a fleet of 12 toroidal UFOs arranged in a circle beaming light to the apex of the Great Pyramid of Giza, Egypt with a rainbow gradient sky

experiment 3:
Mandelbrot set made of cedar sprigs in the style of M.C. Escher

experiment 4:
(Anthony James mirrored icosahedron variations by Dall-E; I used this image as input)

experiment 5:
Impressionist painting in the style of Vincent Van Gogh of CERN hardware

experiment 6:
A photo of the Eiffel Tower scaled to fit in the middle of Stonehenge with glowing orbs atop each megalith in the style of Salvador Dali

… with 4 more variations requested:

experiment 7:
7 purple Flower of Life disks wrapped around a silver cone reflected in a parabolic mirror by Rene Magritte (and 2 more sets of 4 variations)

experiment 8:
a photorealistic image of Dorothy from the Wizard of Oz holding hands with Alice in Wonderland standing in front of the Great Pyramid at Giza, Egypt.
(If you look closely at the faces, they look rather wonky at best, but zoomed out not too bad… Of course the 2nd, 3rd and 4th images used the 2nd “Kephren” pyramid which still has the capstone casing and not the Great Pyramid, but this is a learning technology, right. I wonder if the next time someone asks for a similar image, Dall-E will incorporate my comments and stick to just the Great Pyramid?)

Updated Donation Button

Thursday, 23 March 2023 by Bruce Rawles

Thanks to all of you for your support! Recently we updated our PayPal Donate button since the prior one stopped working; in case you tried to make a donation, the new button at the very bottom of the sidebar on every page now should work properly. Here is the same button for your convenience:

Email List Update! Please re-subscribe … again!

Wednesday, 1 March 2023 by Bruce Rawles

Hi all:

I’m not sure if this email went out on March 1, 2023, so I’m trying again on March 2, 2023; my apologies if you get this twice.

IMPORTANT UPDATE March 1, 2023: We are doing some major cleanup on our email list. We tried another email service provider but realized that it wasn’t providing what we needed, so we’re asking you to please re-subscribe, even if you may have just done this a few days ago. Thanks for your continued support!

To make sure you keep getting our monthly bulletins PLEASE re-subscribe using the form here. THANKS!!!

After you have re-subscribed – thank you! – please let me know if there are interesting items you’ve found that we should feature on the website or general topics related to mystical and/or fascinating geometric subjects by way of our contact form. THANKS also for your decades of support! If you find our “labor of love” helpful and/or interesting, donations using the PayPal Donate button on every page are always appreciated, as are purchases of our books and affiliate products.

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!