Sacred Geometry Media Resource Blog
Amazing that it’s been almost two decades since we photographed these unusual (to say the least) hieroglyphs in a lintel high above the floor of the ‘new’ temple (the one built over the top of the layer of the Osireion temple which features the famous “Flower of Life” and Seed of Life” designs) in Egypt. The lyrics to the song “When will we ever learn?” (Where have all the flowers gone) come to mind. Shall we close the patent office… again? :-)
Musing about how the interpretations of our physical senses are often misled by optical (and other) illusions, I was reflecting on being inspired decades ago by the classic M. C. Escher images in the book (The Graphic Work of M.C. Escher Introduced and Explained By the Artist Paperback – January 1, 1973
by M. C. Escher). The cover of this Escher book features an image of a small-stellated dodecahedron with cutouts in half of the lower portion of each triangular face of each stellating pentagonal pyramid permitting polychromatic reptile heads and limbs to protrude (Gravity, June 1952. Lithograph and Watercolour.) Many of Escher’s illustrations demonstrate these sensory deceptions – which can be beneficially used to inspire a willingness to see things from multiple (open-minded) perspectives – such as images in his Impossible Constructions and Mathematical galleries and other classic images employing space-filling tessellation patterns and numerous instances of multiple simultaneous perspectives. In addition to an early introduction to Platonic Solids, Archimedean Solids, stellations, and more in the polyhedral realms, Escher’s pioneering visionary art seemed to suggest to me the value of exploring perspectives beyond what our unreliable senses (and our misinterpreting minds) report and stoked a curiosity to explore realms beyond ordinary viewpoints. My other primary blog is devoted to these metaphysical explorations.
Examples which show the distortions of our sensory misinterpretations include these images (I haven’t copied the images due to copyright requirements, so please tap the links on this post to check out these example Escher gems (many more in these extensive archives: Selected Works by M.C. Escher):
Here is a close-up of Escher-inspired tiles that adorn the front yards of at least two local homes in Green Valley, AZ; photographed on nearby neighborhood walks during the past 18 months.
For further exploration, check out:
The end of the last article above reminded me that there’s a scene in the movie “Inception” (a fave flick) that had a physical implementation of one of these impossible (never-ending) staircases. featured in popular Escher image:
Escher’s art also explored Non-Euclidean Geometries, which will be featuring in next month’s post. The art below reflects an example of an Escher-inspired Non-Euclidean geometry.
In case you still need a 2025 calendar, there are 11 months left in the year as of the date of this post, and 2025 calendars are still available – as well as my over-quarter-century-old timeless classic reference book, Sacred Geometry Design Sourcebook – Universal Dimensional Patterns, heavily inspired by Escher, and other geometric luminaries.
This evening we finished watching the excellent 2-part documentary “Leonardo da Vinci” by Ken Burns (and others) which – like everything Ken touches – was wonderful. No surprise, and, of course, highly recommended. Please support PBS and your favorite public/community/independent media, too!
Initially, I thought I would use my default browser (Brave) to find some still images from the documentary, and using this search term: “sacred geometry in Ken Burns Leonardo documentary” got this interesting – not bad! – AI-generated response:
In the Ken Burns documentary about Leonardo da Vinci, sacred geometry is explored as a key aspect of da Vinci’s work and thought process. Da Vinci’s use of geometric shapes and proportions in his art and designs is highlighted as a way to understand the underlying structure of the universe.
- Geometric Lines and Proportions: The documentary shows how da Vinci used geometric lines and proportions in his paintings, such as the Mona Lisa and The Last Supper, to create a sense of balance and harmony. These geometric elements are also seen in his designs for machines and buildings.
- The Vitruvian Man: Da Vinci’s famous drawing, the Vitruvian Man, is featured in the documentary as an example of his use of sacred geometry. The drawing depicts a nude male figure inscribed within a circle and square, demonstrating the mathematical proportions of the human body.
- Nature and Geometry: The documentary explores how da Vinci saw geometry as a way to understand the natural world. He believed that the patterns and shapes found in nature, such as the spiral of a shell or the branching of trees, were evidence of a deeper geometric order.
- Da Vinci’s Notebooks: The documentary features da Vinci’s notebooks, which contain his thoughts on geometry, proportion, and the natural world. These notebooks show how da Vinci saw geometry as a way to understand the underlying structure of the universe.
Overall, the documentary presents sacred geometry as a central aspect of da Vinci’s work and thought process, and shows how it influenced his art, designs, and understanding of the natural world.
Check it out. The topic of sacred geometry – although not referred to by that name – makes a cameo appearance in this 2-part series among Leonardo’s wide range of mathematical and scientific interests which, in turn, are just a small portion of the depth and breadth of the realms explored by this amazing historical figure. I appreciated how this “Renaissance” icon appeared to embrace anything and everything as a classroom for for generalizing a deeper understanding, as Egyptologist John Anthony West told us was the goal in ancient Egypt, rather than encouraging specialization.
(The image at the top of this post is a screen snap from the documentary. There are way too many amazing illustrations, so this is just one of the numerous featured in this program.)
In addition to the Vitruvian Man, this PBS program also showed a generous assortment of da Vinci’s geometric and geometrically-oriented mechanical drawings of both natural and man-made objects.
When I was putting together my first book, I was inspired by the geometry of one of da Vinci’s sketches of an “ideal church” and made this (Claris CAD) image from that concept (page 191 from Sacred Geometry Design Sourcebook):
Some of the less obvious geometries in the “Vitruvian Man” (a.k.a. “Measure of Man”) by da Vinci – in addition to the more widely known phi proportion: toes-to-navel and navel to top of head – are revealed on page 190 of the SGDS book:
Enjoy!
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?)
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:
The illustrations below are color versions of similar illustrations from the book on pages 30 and 31:
Here are the chapter titles to give one a sense of the depth and breadth that this little gem covers:
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!