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Symbols of Interconnectedness Everywhere

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As we close out a very surreal year, we wish you the peace that arises from the realization that we’re all interconnected and therefore kindness is appropriate every where and every when. Note the ring of Vesica Piscis shapes around the middle cylinder in this ceiling decoration in a hotel lobby. :-) The Vesica Piscis is the “lens” that is formed when two identical circles (symbolizing two identical seemingly separate selves) exactly touch overlapping so that one circle touches the “heart” or center of the other and vice versa. I imagine Plato (if he were around to comment) might suggest that even these helpful symbols are not the perfection – that transcends their apparent form – that the circles merely represent; the reality or essence in our minds is beyond any specific instance in space or time that the shapes, no matter how elegant or refined, could possibly convey. Like holograms, each of us contains the whole although not apparent to our senses. May 2021 be an opportunity for all of us to deepen our appreciation of the wholeness within every seeming part! :-)

Vesica Piscis ring around hotel ceiling fixture

Metatron Cube 3D Animation – Flower of Life – Platonic Solids – video

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Platonic Solid vertices from Metatron's Cube

Here’s a fun 3-D video animation showing how the vertices of the five Platonic Solids and their “shadow” projections from 3-dimensional space to the 2-dimensional plane align with intersections of lines in the classic “Seed of Life”, “Flower of Life”, “Fruit of Life” and “Metatron’s Cube” line art illustrations, progressing from Tetrahedron to Star Tetrahedron to Cube (Hexahedron) to Octahedron to Icosahedron to Dodecahedron.

If you’d like to explore the relationship of “Metatron’s Cube” and “Flower of Life” (progressing from “Seed of Life”) line art illustrations with the Platonic Solids (all illustrated in the video above) and much more, check out my first book, Sacred Geometry Design Sourcebook: Universal Dimensional Patterns!

The “Seed of Life”, “Flower of Life”, and “Fruit of Life” illustrations (with numerous variations) cover are on pages 85 through 93. Pages 94 and particularly  page 95 (a low-res image shown above) explore the Metatron’s Cube relationships in detail.

Sacred Geometry Design Sourcebook - cover

 

Penrose and other Tilings: periodic and aperiodic – video exploration

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Homage To Kepler's model of the solar system
Kepler’s Model of the Solar System (nested Platonic Solids) and other fun and somewhat boggling geometric explorations unfold in this visually engaging video presentation about periodic … and aperiodic pentagonal tilings discovered by Roger Penrose, quasicrystals, and – of course – cameo appearances by the Golden Ratio and the Fibonacci Series. Enjoy! :-)

Thanks to Shaun L. for alerting me to this gem of a video!

Kepler's solar system models - outer and inner

If you’d like to explore the various periodic tilings and/or Platonic Solids (both mentioned in the video above) and much more, check out my first book, Sacred Geometry Design Sourcebook: Universal Dimensional Patterns!

Sacred Geometry Design Sourcebook - cover

 

3-D Polyhedra Software: Modeling, Art, Animation, Apps and more

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Stella software: Rhombicosidodecahedron - example showing polyhedron, partially folded, and flattened netIf you want to experiment with 3D geometry software, check out Stella from Software3D; lots of possibilities here including the nets (a.k.a. fold-up patterns) for making polyhedra cut out from flat material, such as paper, cardboard or whatever. The animations showing the flattened 2-dimensional nets morphing into their completed 3-dimensional counterparts are particularly fun and useful since you can see intuitively how these polygons all work together to make a whole greater than the sum of its parts. :-)

Among the many other helpful and interesting products and items on this site is the “Worlds Kaleid!” live camera kaleidoscope app for Android.

Eiffel Tower through "Worlds Kaleid" app geometric morphing

 

 

In memoriam: Keith Critchlow

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Keith Critchlow
Keith Critchlow

I recently learned of the passing of Keith Critchlow through this obituary in The Guardian. One of my first introductions – in the early 1970s – to what has become known as sacred geometry was through the books of Keith Critchlow, notably Order In Space and later, Islamic Patterns: An Analytical and Cosmological Approach. The former book has small illustrations of the tesselations (2-D tiling patterns) comprised of regular polygons. I thought these classic regular tilings all merited a full page (along with numerous archetypical variations) – so these make up a considerable portion of my first book, including pages 16-42, plus other tiling patterns inspired by these on pages 43-95, many unique to SGDS.

Here’s an example (page 27) of one of the small sketches from Prof. Critchlow’s book that I expanded and gave an entire page, in order to facilitate photocopying, coloring – for adults or “kids” of any age, arts, crafts or what have you…

You’ll note that on these 2D tesselation pages I’ve also added “unit cell” variations that show some of the possible groupings of polygon components that can make “step and repeat” (via horizontal and/or vertical translation) patterns that tile the plane in two dimensions.

Sacred Geometry Design Sourcebook example: 4 Triangle, 2 Square Grid (page 27) copyright (c) 1997 Bruce Rawles

I’m not sure if Order In Space was my first exposure to the 5 Platonic Solids and 13 Archimedean Solids, but this may be likely, and pages 196-214 in Sacred Geometry Design Sourcebook – Universal Dimensional Patterns were certainly inspired – in part – by Critchlow’s illustrations. You can find other useful tables in the back of SGDS that were also helped by the example of Critchlow’s generous offerings of useful patterns, ideas, and interconnections.

Elsewhere in Order In Space, he shows a matrix grid chart of the various possible permutations of nesting the 5 Platonic Solids inside each other. Again, these lovely little sketches seemed to merit full-page treatment, so they contributed to pages 224-229 in SGDS.

When I was nearing completion of SGDS in 1997, I sent a review copy to Professor Critchlow, and he sent a very thoughtful reply which included this testimonial:

“A valuable source book of geometric patterning, very useful data, a reminder of the supra human source of such mathematical forms.”

He also provided some very appropriate advice (which certainly aligned with my own experience) in advance of the initial publication, so I added this underlined sentence to page 5 of SGDS paying homage to what must be many millennia of geometers using straightedge and compass for discovery:

“I strongly encourage experimentation and particularly duplication with your own hand.”

Thanks for many years of inspiration, Keith!

Lots more to explore!!! In the process of researching archives online as well as my own prior work to find 2D tilings and tessellations, I found this excellent Wikipedia page: List of regular polytopes and compounds, which expands this topic into many fascinating connected subjects. Here’s a particularly lovely example image from the Hyperbolic star-tilings section: