3D Geometries

In light of the current (highly appropriate) global practice of social distancing to minimize the spread of a global pandemic, the ideas and issues of space, isolation, and closeness are now in sharper focus than ever before for billions of humans on this planet. One social extreme generally considered severe punishment is “solitary confinement” where prisoners are cut off from almost all communication with others. Human examples of the other extreme might be a crowded party or bus where many of us would gasp for “breathing room” or perhaps a “dogpile” or gymnastic feat of some sort involving cramming bodies together – phone booths and VW bugs come to mind so I guess I’m dating myself with those examples. :-) Interesting how just a few short weeks can shift worldwide perception about situations involving physical proximity that we perhaps welcomed before on occasion, but now avoid – both literally and figuratively – like the plague!

Geometric Metaphors: “as close as you can get”

In the realm of geometry, we have 2-dimensional tiling patterns or tesselations where space is filled without any gaps by simple geometric archetypes. In my first book, Sacred Geometry Design Sourcebook – Universal Dimensional Patterns (SGDS), pages 16 through 84 of the 256 pages are devoted to these grids of space-filling patterns: 30 pages of regular polygon tesselations, 18 pages of square, rectangular, and circular arc tilings, 10 pages of hexagon-based or dodecagon-based tilings, and 9 pages of pentagonal tesselations, just a tiny sampling of an almost endless assortment of permutations of patterns that can fill a 2-dimensional space and not leave any gaps. Here’s an example of regular polygonal tilings from page 27 of Sacred Geometry Design Sourcebook showing a few variations (indicated by thicker lines) of “unit cell” patterns that fill space in two dimensions:

The classic work of graphic artist M. C. Escher brought some unique and elegant examples of tilings and tesselations to the world a few decades ago. Here’s a video with a few examples and related software for making tesselation patterns in 2D. There are countless examples of both periodic (uniform) and aperiodic (non-uniform, such as Penrose) tilings in nature, art, and architecture.

What about 3D Space-Filling Patterns (Tesselations)?

One of the most familiar examples in three dimensions of a space-filling pattern is a cubic lattice. Here’s an (Escher) example of a 3-D grid (think steel girders on a high-rise building). Although not space-filling per se and not 100% density, sphere packing takes on various geometric patterns, including Hexagonal Close Packing, and Cubic Close Packing. Page 239 in SGDS has a chart of radius ratios and diagrams for Coordination Numbers of 3,4,6,8 and 12 of spherical packing, as typified in ionic chemical bonding on a molecular level. Here’s an excerpt from that page:

Here are some of the less familiar 3-dimensional space-filling patterns. and related honeycomb geometries. … One of the more exotic classes of 3-D space-filling forms is Rhombic Spirallohedra; here’s a related fold-up pattern. Russell Towle introduced me to these amazing space-filling shapes almost two decades ago.

So what if “close” isn’t close enough?

In our “mind’s eye” we can visualize intersecting shapes that don’t have physical counterparts in the everyday material world around us. One example might be two identical intersecting spheres where each sphere exactly touches the other at its center; we can use ray-tracing software with opacity less than 100% (partially translucent shapes) to model how these might look (see translucent spheres intersecting in Vesica Piscis cross-section rendering below)

The 2D “shadow” (or cross-sectional slice) of the imagined 3D shape would be the famous Vesica Piscis shape. This might be analogous to intersecting (colliding) spiral galaxies where their forms interpenetrate; it might be a bit chaotic in such a galactic struggle where each “PacMan” galaxy tries to absorb the other!

Check out Russell Towle’s amazing animations – each frame of these videos being a “3D slice” through a 4D polytope or 4-dimensional polyhedron… You can see how we only get a glimpse of what these 4D shapes must be like; a reminder that our senses are limited, distorted and fallible, yet can remind us to beyond them if we’re looking for the “bigger picture.”

We’re still dealing with separate identities, separate shapes, though. The great mystical traditions, along with modern physics and non-dual metaphysics suggest that – if time and space (since both are inextricably linked in the fabric of spacetime) are an illusion – or at least a woefully incomplete and inaccurate view of reality, then even identical forms inhabiting the exact same space and time in a completely congruent fashion would still be “separate” if their identities could be distinguished from one another somehow. Recent breakthroughs in quantum computing that further validate the theory of quantum entanglement – where entangled particles are synchronized (e.g. their spin axis) instantaneously, regardless of the magnitude of the space between them – give impetus to exploring this idea.

Beyond Close

A few years ago, when my wife was about to leave that morning for an extended planned work situation in another state, while she was asleep next to me in bed, I realized that I was missing her already! How could that be possible? That “Aha!” moment further propelled my renewed exploration into the mystical study and practice of non-dual metaphysics, as I realized that she – and ultimately everyone! – lives in my mind.

Let’s do what can with zoom, Skype, web, email and whatever other technologies make sense to stay connected and support each other, especially when we appear to be more physically isolated than ever while considering that – just maybe – the imagined distance between us might just be subject to reinterpretation. This mindfulness provides all the more reason for us to be kind, compassionate and generously gracious in our assessment of each other and ourselves – as ourselves. What if the mystical Oneness alluded to down through the millennia in poetry, song, and prose  – that we usually are too busy to consider – holds the effective key to not only pragmatic concerns like economic and medical infrastructure – but also to sustainable peace of mind as well? A forgiving and inclusive perspective on life would then make space (or lack thereof) a less important variable. We’re all in the same boat, fighting the same difficult battle – the silly, erroneous belief in separate interests. Sacred geometry and being mindful of clues reminding us of our interconnectedness in the world can be guideposts for the mind’s journey back to the Oneness we never left.