sacred geometry news

more mirrored polyhedra

Thursday, 25 March 2021 by Bruce Rawles

Over the years, I’ve been fascinated with the exquisite possibilities of using mirrors for the interior faces of polyhedra (particularly Platonic Solids and Archimedean solids, and among those, particularly ones with dodecahedral and icosahedral symmetries.

One of my earliest fun discoveries was the Kaleidoscapes work of colleague Sara Frucht. She showed me some amazing reflected interior spaces made simply by using 3 triangular mirrors with a shared vertex, carefully designed to replicate the interior angles of a variety of polyhedra.

There are a number of other examples I found over the years by other geometric artists and this latest exhibit at Melissa Morgan Fine Art in Palm Desert, Calif. featuring the geometric artwork of artist Anthony James is quite stunning. I would check it out if I lived closer to southern California! :-) It appears that he’s using uniform edge lighting (probably LED that wasn’t available not too many years ago) and 2-way mirrors to make particularly lovely 3D models that can be viewed from many angles. Here are additional photos and a short video of this exhibit.

I’ve seen other versions (including a dodecahedron at the Exploratorium in San Francisco, California several decades ago) which used regular (100% reflective) mirrors for all but one face, with the one “viewing face” either open (affording putting one’s head inside the shape for an even more immersive experience) or with regular glass.

When the Olympics was held in Greece some years ago, I thought it would be a cool attraction (with homage to the home of both Plato and Archimedes for whom their respective polyhedra are credited) to have huge versions of the aforementioned polyhedra big enough to walk inside (perhaps with elevators bringing spectators in and out) and then an additional bonus would be making acoustic recordings inside each of these structures and having a multisensory experience enhanced by the unique acoustic resonances of each structure, much as Paul Horn did with his classic recordings of flute music in the Taj Mahal in India and the Great Pyramid in Egypt. I made a very short recording inside the latter structure in the fall of 1992.

Here’s a video about another seasoned artist (Frank Stella) who has incorporated – appropriately enough, given his last name! – stellated polyhedra in his work.

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

Monday, 28 September 2020 by Bruce Rawles

If 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.

In memoriam: Keith Critchlow

Sunday, 31 May 2020 by Bruce Rawles

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.

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:

Mandelbrot Set: Deep Dive

Wednesday, 19 February 2020 by Bruce Rawles

Most geometers are familiar with the Mandelbrot set, the pioneering fractal geometry archetype credited to polymath Benoit Mandelbrot in 1980. Fractal images now appear in countless contemporary geometric art forms including fractals in film and video, both static and animated. Landscapes are particularly suitable subjects; my animator buddy Dave Van Dyke makes use of them frequently.

The Wikipedia page for the Mandelbrot set has a generous assortment of interesting links, including the Buddhabrot, previously featured here in GeometryCode.com’s archives – and numerous presentation I’ve given on sacred geometry over the past 16 years – and intriguing shapes such as the Mandelbulb:

In case you’d like a more extended visual meditation on this fascinating geometry, here’s a 1-hour video offering a deep dive into the intricacies of this quintessential modern geometric classic.

For those with both mathematic and metaphysical interests, fractals seem like a perfect metaphor for symbolizing both the thought system of inclusion – ever joining and restoring the mind to the interconnectedness and ultimate Oneness of our essential, eternally shared nature … and exclusion – divisively fragmenting, bifurcating and alienating seemingly separate selves into warring factions of countless varieties … while the intrinsic Unity we all share patiently awaits our return to that unchanging awareness. I have much more to share about this on another blog I maintain, ACIMblog.com.

Fibonacci Association – treasure trove of Fibonacci Series … and related … info!

Sunday, 30 June 2019 by Bruce Rawles

The Fibonacci Quarterly – Official Publication of the Fibonacci Association (since 1963)

I recently “re-discovered” the Fibonacci Quarterly which is the Official Publication of the Fibonacci Association; they have been publishing fascinating information about the Fibonacci Series, the Golden Ratio and related mathematical topics … since 1963 – over half a century! Check it out!