Blog

Fractal and Geometric Animations

Friday, 25 October 2024 by Bruce Rawles

Here are some interesting geometric animations… Lots more to explore!

A few example animations found using sacred geometry as a search term on Giphy:

Fractal (Octahedra) Polyhedra with Sierpiński triangle faces via GIPHY

A Flower of Life variation with animated rainbow colors via GIPHY

Fractal (Dodecahedra) Polyhedra with pentagon faces via GIPHY

A couple of example animations found using polyhedra as a search term on Giphy:

A Rhombic Triacontahedron appears when the animated mesh reaches its finest resolution; here’s a fold-up pattern for the Rhombic Triacontahedron (a.k.a. Kepler’s Solid). Here’s the animation via GIPHY.

Each edge of the triangular faces of this lovely icosahedron is subdivided 4 times. Here’s the animation via GIPHY.

… and a few example animations found using fractal as a search term on Giphy:

Fractal gears via GIPHY

Fractal dogs/ears via GIPHY

Koch snowflake via GIPHY

Whirling squares (golden ratio spiral) animation

Thanks to fellow geometer Dave Van Dyke for pointing these out to me.

2025 Martineau Solar System Calendars available now

Sunday, 15 September 2024 by Bruce Rawles

Once again this perennial calendar is available! These timeless images portray amazingly accurate geometric representations of the planetary orbits in our very own solar system – now available in printed calendar form with dates for 2025!!

Martineau Solar System Calendars

These 12 images are highly accurate geometric models of our solar system; details are here. They were inspired by the remarkable work of John Martineau (publisher of Wooden Books), who wrote “A Little Book of Coincidence“. Tiled images of planetary surfaces and related images form the backgrounds. Images are two or more orbits defined by simple 2D (circles, triangles, squares, pentagons, hexagons, octagons) or 3D geometries (e.g. tetrahedron, icosahedron, dodecahedron) typically with 99.9% or greater accuracy. The work picks up where Johannes Kepler’s mystical exploration left off. A newly revised calendar (which is updated every year) with images from the Martineau Solar System is again available for 2025.

… Don’t need a sacred geometry calendar … or a sacred geometry book? You can always support GeometryCode.com via our newly updated Tip Jar. Our virtual “tip jar” accepts a variety of methods of making financial donations; any greatly appreciated! You can use Zelle, Venmo, Wise, PayPal, and a check or money order in US dollars works as well.

Fun, whimsical, playful geometric art

Thursday, 29 August 2024 by Bruce Rawles

Since summer is often associated with vacations, play time and recreation, this month’s short but sweet post – thanks to my wife Nancy – is inspired by a link she sent me in her frequent scouring of social media for all things artsy. She thoughtfully keeps an eye out for geometric items. This one reminds me that in addition to being fascinating and profound for the scientific mind, geometry can also be inspiring for fun, loose, right-brained, artistic, non-linear, colorful and entertaining explorations and pastimes. This example was found in samples from this larger collection: Print & Pattern: Geometric Kindle Edition by Bowie Style and Marie Perkins

art: Neo Geo by Anne Waters — Neo Geo by Anne Waters

Plato’s Spirit, Carl Sagan’s Blue Dot, Contact, symbols and sources

Saturday, 27 July 2024 by Bruce Rawles

"Pale Blue Dot" Voyager 1 photo of earth (NASA/JPL-Caltech) — “Pale Blue Dot” Voyager 1 photo of Earth (NASA/JPL-Caltech)

“The Pale Blue Dot is a photograph of Earth taken Feb. 14, 1990, by NASA’s Voyager 1 at a distance of 3.7 billion miles (6 billion kilometers) from the Sun.”

*A year or so ago, a friend reminded me of this inspiring prose “The Pale Blue Dot” by Carl Sagan (copied below) inspired by the NASA image above… and the ever-increasing importance of finding and supporting the interests we ALL share on this tiny planetary sphere we call home:

“Look again at that dot. That’s here. That’s home. That’s us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every “superstar,” every “supreme leader,” every saint and sinner in the history of our species lived there-on a mote of dust suspended in a sunbeam.

The Earth is a very small stage in a vast cosmic arena. Think of the endless cruelties visited by the inhabitants of one corner of this pixel on the scarcely distinguishable inhabitants of some other corner, how frequent their misunderstandings, how eager they are to kill one another, how fervent their hatreds. Think of the rivers of blood spilled by all those generals and emperors so that, in glory and triumph, they could become the momentary masters of a fraction of a dot.

Our posturings, our imagined self-importance, the delusion that we have some privileged position in the Universe, are challenged by this point of pale light. Our planet is a lonely speck in the great enveloping cosmic dark. In our obscurity, in all this vastness, there is no hint that help will come from elsewhere to save us from ourselves.

The Earth is the only world known so far to harbor life. There is nowhere else, at least in the near future, to which our species could migrate. Visit, yes. Settle, not yet. Like it or not, for the moment the Earth is where we make our stand.

It has been said that astronomy is a humbling and character-building experience. There is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly with one another, and to preserve and cherish the pale blue dot, the only home we’ve ever known.”

― Carl Sagan, Pale Blue Dot: A Vision of the Human Future in Space

Recalling a cameo role played by a dodecahedron – a symbol for “Ether” (or Spirit) according to Plato – in a major motion picture, I recently re-watched (and enjoyed from a very different perspective) a movie I hadn’t seen in decades, Contact, written by Carl Sagan and Ann Druyan, starring Jodie Foster and Matthew McConaughey. The two stars seemed to search for meaning via science and religion respectively. It would appear that neither can find what they are searching for in isolation, just as we miss the “big picture” when we exclude other grander perspectives, putting our tiny lives in context. The very touching scene (toward the end of the movie when her dodecahedral “craft” fulfills its mission of transcending our understanding of space and time) where Foster’s character is reunited with her father in perhaps “another dimension” suggests that only our limiting perspectives and beliefs needlessly constrain us.

I recall attending a lecture in San Francisco, California in the 1970s by Dr. Ramamurti S. Mishra who compared science to a blind man and religion to a lame man; they are far more effective when they teamed up by the lame man metaphorically riding on the blind man’s shoulders. We’re all – it would seem – like individuals trying to make sense of our isolated perspectives like the proverbial “Blind men and an elephant.”

The movie also featured cameo appearances (in terms of locations) by the (now defunct) Arecibo Observatory Radio Telescope – a sizable geometric structure! – and the Very Large Array (VLA) Radio Telescope (a two-hour drive from Albuquerque, 50 miles west of Socorro, New Mexico which my wife & I visited a couple of decades ago) consisting of movable large radio telescope dishes on railroad tracks, allowing the virtual aperture to be adjusted (or something like that; my recall of the science is a bit rusty.) Both of these facilities employ lots of geometric engineering!

What prompted watching the Contact movie again (via a YouTube sidebar suggestion) was this brilliant video clip by Sagan (from his Cosmos series) that related 2D to 3D to 4th-dimensional polytopes – “flatland” squares to cubes t0 HyperCube tesseracts – and inspired my pencil metaphor decades later.

Plato (who studied and reported Socrates) suggested that symbols are not the source they represent. So if geometric symbols (such as spheres, dodecahedra, and the like) merely represent the eternal ideas that outlast and transcend the specific ephemeral instances of the forms that point to the concepts, principles, laws, and ideas they represent, it would behoove us to keep looking for the reality behind and beyond the forms that are limited by the particular spaces and times they appear to inhabit. Mind envelops matter; the Source contains – but is not limited by – symbols, which are mere shadows of what they represent, as Plato’s Allegory of the Cave would suggest. The content of the mental universe is more profound than any examples in form could possibly convey.

If you scroll down on any page of this website, you will notice a symbolic hand with a forefinger pointing up; this image was cropped from Plato’s representation in “The School of Athens.” Read more about this in a post from earlier this year.

A few German sacred geometry books available for shipping costs only

Sunday, 21 July 2024 by Bruce Rawles

Update 21 July 2024: We’ve been in Arizona for exactly a year today, and I still have a handful of German books to find homes for, so I’m re-dating this post in case anyone would like to purchase a book for only my shipping costs!

(Originally published 25Jun2023):

We’re making good progress toward moving to Arizona (mid-July 2023) so I’m anticipating having more time to do in-depth posts after our move. Meanwhile, I’d still like to lighten our load and not move these books yet another time… I’m offering (until mid-July, while supplies last) – limited supply; limited offer) asking only for my shipping costs (US only) or free if you happen to be ~~on the central Oregon coast and can pick them up here in Yachats before we move!~~ in the Tucson, AZ metro area and can pick them up.

German sacred geometry books are available for shipping only (for a limited time!)

I have a handful of both of my German-translated books:

Mandalas der Heiligen Geometrie (MDHG) das Universale Design der Natur (SGDS German edition)
Bewusst malen – Heilige Geometrie

(If you live in Germany, you’ll probably want to order the Bewusst malen – Heilige Geometrie book, still in print, directly from the publisher, Silberschnur Verlag, since the shipping from the USA to Germany is likely more than buying it locally.)

If you speak German or know someone who does and has the least interest in geometry, I am offering special FREE books (very limited supply) of both of these books if you cover my postage costs. If you live near ~~Yachats, Oregon,~~ Green Valley, Arizona, you can pick them up here and skip the postage expense. For details on these limited-time, limited-supply books, contact me.

Note that you don’t have to be able to read or understand German to enjoy either of these books, as they are both lavishly illustrated – FAR more imagery than text. Also, each copy of the out-of-print MDHG book has a unique set of die-cut fold-up models of the 5 Platonic Solids (included behind the back cover) that make a neat mobile if you add your own thread.

Please contact me ~~before mid-July 2023~~ if you are interested in this special offer!

Here are a few more interesting geometry-related links (videos) that have crossed my path recently; several short introductory videos that add some useful and interesting additions to commonly cited facts:

Golden Spiral Symbol – What Does It Mean?
What Makes the Golden Spiral Symbol Special?
What is Sacred Geometry? (evidently using an AI-generated voice/script?)
How Does the Golden Spiral Relate to Sacred Geometry?
What is the Meaning of the Golden Spiral?
What are Some Examples of the Golden Spiral?
What is the Fibonacci Sequence?
… and several others…