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Encyclopedia of Polyhedra By George W. Hart … and other geometric gems

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Apples and Oranges by George W. Hart https://www.georgehart.com/Applorng.html
Apples and Oranges by George W. Hart
https://www.georgehart.com/Applorng.html

It’s time to revisit the work of other polyhedra enthusiasts. Of particular note (having crossed my digital path again recently) is George W. Hart who has shared some excellent resources along these lines (faces, vertices, etc.) for decades. Here’s one gold mine to explore: Virtual Polyhedra: The Encyclopedia of Polyhedra By George W. Hart. Here is part of the table of contents and highly recommended if you want to explore the genres and categories of polyhedra; lots of great imagery, rotatable 3D models (rotate them with your mouse), facts and details:

Perhaps the most well-known and ubiquitous polyhedral shapes have fold-up models (and related patterns) for the first 3 categories above in Sacred Geometry Design Sourcebook (SGDS pages 196-220) and also on this website for the 5 Platonic Solids and 13 Archimedean Solids as well as Illustrations of Platonic And Archimedean Solids, Math Tables for Platonic And Archimedean Solids (and much more about polyhedra on this website.

compound of five cubes by George W. Hart
compound of five cubes by George W. Hart

If you exhaust this generous resource (or somehow overlook a few gems) here are more of his contributions – the videos are particularly helpful if you want to replicate his constructions:

Stained Glass Ball By George W. Hart https://www.georgehart.com/stained_glass.html
Stained Glass Ball By George W. Hart
https://www.georgehart.com/stained_glass.html

GeometryCode.com Annual Survey (December 2023)

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Thanks to all of you for your continued and ongoing support of this labor of love website!

It has been quite a while since we’ve had a reader survey or poll! Here are the results of the last one from a couple of decades ago:

 

Here’s an opportunity to share your geometric interests, passions, curiosities, favorite topics, and whatever you think will be helpful, useful, and fun!

We’ll replicate this survey each year, provide results annually, and hopefully guide our content to where you find existing and unexplored subjects!

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sacred, mystical or symbolic geometry
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books on sacred geometry & related topics
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geometric art, graphics, jewelry, decor
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geometric apparel, printed items, wares
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geometric tools, toys, products
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geometric software, applications, links
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geometric videos and animation
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geometry interviews, conversations
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geometry in the natural world, biology, mineralogy
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geometry in man-made world, architecture, technology
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interconnectedness & related topics
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proportions, numbers (e.g. golden ratio, Fibonacci Series)
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fractals, Mandelbrot Set, etc.
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tilings, tesselations, space-filling geometries
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hyperdimensional geometries
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physics, quantum theory, metaphysics
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macrocosmic geometry (e.g. planets, galaxies)
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microcosmic geometry (e.g. crystals, molecules)
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primal 2D geometric forms (e.g. polygons, curved shapes)
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primal 3D geometric forms (e.g. polyhedrons: Platonic, Archimedean Solids, etc.)
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geometry and sound, music, acoustics
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geometry videos, tutorials
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geometric construction techniques and tips
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other geometric topics (please specify)
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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?)

 

Sacred Geometry for healing

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Sometimes I get questions via email that seem appropriate for generalizing and sharing… Here’s one in that category. Thanks to all of you for over a decade for sharing your insights and musings!

Bruce,
Do you or anyone you know, have books on sacred geometry to heal physical and emotional issues…. any sacred geometrical patterns for detoxification????
I’d love to know!

Great question, thanks! (I’m taking the liberty of making this an anonymous blog post…)

I’m leaning more and more toward the idea that – here comes the ‘heresy’ for
geometry fans!  the particular form isn’t nearly as important as the feeling and intention of the person…

Having said that, I’d recommend using patterns and forms that have the greatest
resonance/meaning/kind & benevolent associations as possible… Aside from that,
I’ll keep my ears and eyes open… My interest lies mostly in the area of Universal
principles rather than emphasizing physical effects… It seems to me that if we
restore our awareness to wholeness (no separation), then whatever healing
can occur in the material world will be optimized.

I hope that’s helpful!

Cheers and blessings!

Intention and Symbols

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Today’s post is an edited response to an email I just received, asking about the pentagram (or pentacle) and its cultural associations.

Hello again. It’s been a few years since I last emailed you and this time I have to ask you for some advice. … I have been interested in sacred geometry since I visited your previous site. I think it was called ‘Intent’. … I felt a strong correlation between the traditions of …, sacred geometry, alchemy, and mystic symbolism.

I am currently designing a range of posters around the four seasons which is loosely based on I think it’s the ‘Ryder’ tarot deck. I am drawn to using some of the symbolism but the one that I am having a bit of difficulty deciding on, is the use of the Pentacle or Pentagram. I think it is also referred to the ‘morning star’.

The rub for me is not so much the symbols themselves but their association with Illuminati or satanic symbolism that is used as subliminals in some advertising. I might be barking up the wrong tree but I feel the miss use of these elements might be clouding the higher message that these symbolic elements and resultant graphic language hold for humanity. This is the message that I understand in your work. I really want to help people realise a higher potential and level of conciousness in themselves but the last thing I wish to do is propagate any luciferic conciousness.

I guess my confusion stems from not knowing enough, and maybe I shouldn’t be meddling in things I don’t understand. Could you please steer me in the right direction with regard to this issue.
Sincerely (name omitted)

To answer your question about pentagrams, or pentacles, I always like to “back up” to a more fundamental question. What is the thought system, intention, perspective, or attitude of the question? If it is an inclusive, all-encompassing, transpersonal one, then the issues around any malevolence disappear, regardless of the symbol or form, because the content will be one of kindness and integrity. If the thought system is divisive, petty, personal, and based on the fantasy of separation that inevitably occurs when we take sensory data as evidence of isolation or differences, then we will see whatever symbol appears to us from an egoic orientation.

In the specific case of pentagrams, or pentacles, synchronistically I’ve been reviewing the movie “The DaVinci Code” this week and the “Langdon” character played by Tom Hanks in the movie describes (I’m paraphrasing here) how the pentagram has been used by pagan cultures as a symbol for the divine feminine principle. I would suggest that any shape with 5-sided (pentagonal) symmetry intrinsically has metaphoric connections to the principle of interconnectedness since the pentagram has the golden ratio (approximately 1.618) encoded in every proportion. There are numerous references on my GeometryCode.com website (particularly the tutorial page) about how this ratio represents the idea that there is no separation, with examples from the cross-section of every DNA molecule (and also the shape of several of the molecular chains that are attached to each turn of its helical design, to spiral galaxies, Chambered Nautilus shells and countless other examples in nature, art and architecture, including the Great Pyramid at Giza, Egypt.

Like anything else, motivation is the key. We can use a hammer to build or destroy a house, and our thoughts and the intentions they serve are no different in this regard. Historically, there have been associations with the pentagram and both benevolent and malevolent intentions. But the same can be said about any other shape, symbol, or design. The swastika has been used by Nazi Germany as an emblem of world conquest, yet also by native Americans to represent healing.

The vesica piscis has been used by a major financial institution as their commercial (credit card) logo, on the cover of the Chalice Well in Glastonbury, Somerset, England, and countless examples of medieval religious art to represent Christ.

(I also found this interesting geometric connection – Villarceau circles – with the torus, while exploring this idea this morning):

One might suggest that the PRINCIPLE of Christ (that there IS no separation) is what is being depicted in this symbol. Returning to the original premise of this writing, perhaps one can – and I make every effort to do so! – recall that ANY symbol can be used in the service of truth (beyond our finite perceptions limited by space, time, matter, and duality) or enslaved by the realm of apparent separation and its attendant motivations.

I hope this is helpful!