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Digital dabbling with Dall-E

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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?)

 

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!

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

Are Physics, Geometry and Science pointing to (Non-Dual) Metaphysics?

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I just finished listening to an extended (almost 3 and half hour) fascinating podcast interview of cognitive scientist and author Donald Hoffman by Lex Fridman speaking on “Reality is an Illusion – How Evolution Hid the Truth” – (Here is a YouTube clip, evidently excerpted from the longer podcast.) If nerdy math and physics is over your head (as much of this was for me) you might want to skip to the last half hour or so where Hoffman delves into a more philosophical discussion with Fridman where more arcane subjects like Amplituhedron, Permutation matrices, Birkhoff polytopes and cosmological polytopes (new or minimally explored concepts to me) give way to a conversation that seems to lead one toward non-dual metaphysics (with which I’m much more familiar.) Here’s a related TEDtalk by Hoffman: Do we see reality as it is? I don’t pretend to understand much of the in-depth new math and physics that Hoffman talks about but I particularly appreciate his willingness to consider the idea that matter derives from consciousness and not necessarily the other way around! Here’s a quote that hopefully, I haven’t taken out of context: “The notion of self can be an artifact of projection…” (at about 2:16:39 into the podcast).

While doing some very preliminary perusals of some of the links above, I found a few interesting geometric links, including:

mirror-faced icosahedra descend on a city

 

 

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! :-)