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sacred geometry interviews

4-Dimensional (and beyond) Polytopes – Revisited

Monday, 22 January 2024 by Bruce Rawles

Flatland - A Romance Of Many Dimensions by Edwin A. Abbott - pencil prankIn a recent metaphysical Zoom meeting, a friend mentioned a story about 2-dimensional creatures which I recognized as the classic book: “Flatland: A Romance of Many Dimensions by Edwin A. Abbott” first published in 1884 – note the very archaic labels on the cover illustration. (We’ve made several other references to flatland before on the GeometryCode website.) We got into a fun and mind-stretching conversation about how we believe we’re 3-dimensional creatures almost completely oblivious about higher dimensions and probably just as mindless about 1-dimensional and 2-dimensional realms. And of course, what would a 0-dimensional – non-dual beyond spacetime? – world be? No self-respecting 3-D creature wants to go there!

My favorite analogy is imagining the plight of someone trapped in 2-D (like Abbott’s Flatland) and having a “paranormal” experience when someone in 3-D (what Abbott calls “Spaceland”) decides to play a prank and push a 3-D pencil through some unfortunate 2-D denizen’s world – Let’s call her Trudy. At first, Trudy sees (hallucinates?) a black dot growing steadily larger (the lead of the pencil) which turns brown as it grows (the wood of the pencil) then morphs into a yellow hexagon which Trudy can only detect because of the 6 edges. The apparition remains a hexagon for a while until it abruptly changes into a slightly larger metallic circle, then a slightly smaller flexible pink circle (the eraser), and then, just as mysteriously completely vanishes as the 3-D pencil leaves the 2-D plane of Trudy’s paper-thin world.  What recourse does she have now but to phone the Flatland equivalent of National Enquirer to report an alien encounter and/or phone her therapist?

So what if we go the other direction and explore spatial dimensions of 4 and beyond? A couple of decades ago I had the good fortune to live not far from fellow geometer Russell Towle who lived in Dutch Flat, California. Russell spent several hours with me showing his brilliant work on his Mac along with lots of other amazing things, such as zonohedra and similar work from other math-savvy colleagues who had explored these realms. In 2013, I made a short post “In memory of Russell Towle” when I learned of his passing.

Not long after this, Russell gave me this link to one of Mark Newbold‘s pages about Russell Towle’s 4D Star Polytope Animations and I shared it on my Resources page – scroll down to Links to sites about polyhedra and higher dimensional polytopes. When I went to do a screen share of Towle’s animations for my Zoom colleagues, I discovered that they were made with a version of QuickTime that is no longer supported, so I quickly converted them to mp4 files and uploaded them to the GeometryCode YouTube channel (“shorts”) category here. Here is an excerpt from Mark’s pages about Russell’s work:

“These may be the first animations ever made of the solid sections of four-dimensional star polytopes. To get a better idea of just what these “polytopes” are, one should read H.S.M. Coxeter‘s “Regular Polytopes” (Coxeter01). Briefly, plane polygons are two-dimensional polytopes, and polyhedra, three-dimensional polytopes. Where polygons are bounded by line segments, and polyhedra by polygons, a 4-polytope is bounded by polyhedra.
Just as we may have any number of planes in three dimensions, in 4-space we may have any number of 3-spaces. Two 3-spaces might be a millionth of an inch apart and yet have no common point (thus the popular idea of parallel universes). It follows that, given a fixed direction in the 4-space, we can take solid sections of objects in the 4-space, perpendicular to that direction.

If you find these concepts difficult, you are not alone. Even when a person is blessed with some extraordinary faculty for visualizing objects in higher space–as was Alicia Boole Stott, a century ago–it is a matter of years, and considerable patience, before much progress is made in the subject.

In these animations, a 3-space is passed from one vertex of each star polytope, to the opposite vertex, and sections taken at small intervals. The star polytopes were constructed, and the sections found, using Mathematica 4.0. The sections were rendered in POV-Ray (a freeware ray-tracer).”

I had a copy of Regular Polytopes by Coxeter for several years, and I think I understood a small amount of it (on probably a very superficial level), but most of it was beyond my comprehension, yet fascinating!

Towle was also able to plumb the mental realms of 4+ dimensions further than I will likely ever venture and made some amazing computer animations of morphing polyhedra that represent projections of a 4-D polytope (a.k.a. polychoron) onto 3-D polyhedra, further projected down onto 2-D by representing 3-D polyhedra as 2-D animation frames. These amazing short animations may be metaphorically somewhat akin to consecutive 3-dimensional “slices” through a 4-dimensional shape, with each frame of the video being the next adjacent slice. Enjoy!

  • Russell Towle’s 4D Star Polytope Animation {52,3,5}vert
    1 frame (a 3D cross-sectional "slice" through) a 4D Star Polytope animation by Russell Towle: 52-3-5V
    1 frame (a 3D cross-sectional “slice” through) a 4D Star Polytope animation by Russell Towle: 52-3-5V


  • Russell Towle’s 4D Star Polytope Animation {3,3,52}vert
    1 frame (3D "slice") of a 4D Star Polytope animation by Russell Towle: 3-3-52V
    1 frame (3D “slice”) of a 4D Star Polytope animation by Russell Towle: 3-3-52V


  • Russell Towle’s 4D Star Polytope Animation {5,3,52}vert
    1 frame (a 3D cross-sectional "slice" through) a 4D Star Polytope animation by Russell Towle: 5-3-52V
    1 frame (a 3D cross-sectional “slice” through) a 4D Star Polytope animation by Russell Towle: 5-3-52V


  • Russell Towle’s 4D Star Polytope Animation {5,52,3}vert
    1 frame (a 3D cross-sectional "slice" through) a 4D Star Polytope animation by Russell Towle: 5-52-3V
    1 frame (a 3D cross-sectional “slice” through) a 4D Star Polytope animation by Russell Towle: 5-52-3V


  • Russell Towle’s 4D Star Polytope Animation {5,52,5}vert
    1 frame (a 3D cross-sectional "slice" through) a 4D Star Polytope animation by Russell Towle: 5-52-5V
    1 frame (a 3D cross-sectional “slice” through) a 4D Star Polytope animation by Russell Towle: 5-52-5V


  • Russell Towle’s 4D Star Polytope Animation {52,3,3}vert
1 frame (a 3D cross-sectional "slice" through) a 4D Star Polytope animation by Russell Towle: 52-3-3V
1 frame (a 3D cross-sectional “slice” through) a 4D Star Polytope animation by Russell Towle: 52-3-3V

Filed Under: 2D Geometries, 3D Geometries, 4D Geometries, Archimedean Solids, Platonic Solids, polyhedra, sacred geometry animation, sacred geometry art, sacred geometry books, sacred geometry interviews, sacred geometry videos

GeometryCode.com Annual Survey (December 2023)

Friday, 1 December 2023 by Bruce Rawles

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!

- Step 1 of 2
Hardly (or not at all)A littleSomewhatA lotVERY much
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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Learn about Sacred Geometry Design Sourcebook – Universal Dimensional Patterns by Bruce Rawles
Learn about The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation by Bruce Rawles
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Filed Under: 2D Geometries, 3D Geometries, 4D Geometries, applications, Archimedean Solids, audio, coloring books for adults, Fibonacci Numbers, Fractal Geometry, golden ratio, Hermetic Laws, modern physics, Numbers and Proportions, Platonic Solids, polyhedra, sacred geometry animation, sacred geometry apps, sacred geometry architecture, sacred geometry art, sacred geometry books, sacred geometry calendars, sacred geometry coloring books, sacred geometry interviews, sacred geometry jewelry, sacred geometry physics, sacred geometry toys, sacred geometry videos

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

Tuesday, 28 June 2022 by Bruce Rawles

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:

  • A 3D Printed Model of the Substructure of the Universe (the photo shows obvious icosahedral and dodecahedral symmetries)
  • The Golden Ratio and Us … and … Why is 1.61803399 considered a golden ratio?
  • … both the links above found from the Quantum Gravity Research site (which has a lovely polyhedra animation on the home page) and links to this nicely done video visually sharing the idea that the “20-group encodes 4-dimensional space” (from which the featured image for this post was derived.)
  • … and more in-depth physics here (The Einstein-Podolsky-Rosen Argument in Quantum Theory)

mirror-faced icosahedra descend on a city

 

 

Filed Under: 2D Geometries, 3D Geometries, 4D Geometries, modern physics, sacred geometry interviews, sacred geometry physics

Spirit of Science, Sacred Geometry and more – A conversation with Danushia Kaczmarek and Bruce Rawles

Friday, 2 March 2018 by Bruce Rawles

Danushia Kaczmarek interviews Bruce Rawles
Every moment is an opportunity to see the intrinsic sameness of everyone and everything – we’re all kindred spirits. – We explore, as Danushia says, seeing beyond the veil … and having fun with it! In this video conversation, Danushia Kaczmarek inteviews Bruce Rawles about “The Spirit of Science ~ Sacred Geometry ~ The Pyramids of Egypt” in her interview series about going from head to heart. We cover a wide range of subjects including Fibonacci numbers, spirals, the golden ratio, Platonic Solids, sterograms, biofeedback, catelysts, ACIM, the double helix of DNA, modern physics ideas such as quantum entanglement, the wave/particle (double slit) phenomenon, expectations in the mind and the undoing of the Newtonian world view, flat approximations for living on a spheroid, and much more. (Quick correction; in the conversation I mentioned Mauna Kea, but it’s actually Mauna Loa, Hawai’i that is at the 19.5° north latitude.)

(This interview took place Feb. 9, 2018)

Filed Under: 2D Geometries, 3D Geometries, Fibonacci Numbers, Hermetic Laws, modern physics, Numbers and Proportions, Platonic Solids, polyhedra, sacred geometry books, sacred geometry interviews, sacred geometry news, sacred geometry videos

Practice Makes Perfect in Gentle Mindfulness and Inclusive Awakening – a conversation with Marc Courtenay

Thursday, 1 February 2018 by Bruce Rawles

Updated 1Feb2018: Here is a link to the replay of the audio recorded this morning. – If you view this page after early February 2018, just scroll down until you find the conversation. Enjoy! :-)

Marc Courtenay

Tomorrow  Feb. 1, 2018 at 8:30 AM (Pacific Time) the topic of conversation (on “Groundhog Day eve”, so to speak) will be “Practice Makes Perfect in Gentle Mindfulness and Inclusive Awakening” when I will again be the guest on Marc Courtenay’s radio show “Finding Happiness and Good Health in Today’s Word” on (KCUP-1230 AM and BOSS 100.7 FM Radio in Newport, Oregon) and streaming on the web. You can listen live or via the archives of Marc’s shows.

TALK radio- 1230AM - Newport, OregonBOSS-FM

Filed Under: audio, sacred geometry interviews

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