hexapent from Antiprism

Thinking Globally in 2016

Since around the late 1970s, when Synergetics was first published, a small cabal, not unlike the one surrounding General Semantics (Alfred Korzybski) has continued to mix geodesic domes with spherical geometry and cybernetics, following the authors’ lead. Where did that go?

If you’re into gaming, you probably know the next version of Civilization is about to make its debut, this October. Cybernetics means games and game action, like playing chess against “The Turk” a first AI simulation, actually a little man.

The chess-playing Turk, outwardly purely mechanical, might be remembered as the Napoleon-era artifact which booted AI as a concept, given how many believed in this hoax. Nowadays we play (and/or consult) Watson and Wolfram Alpha, not little men behind the curtain.

Like many classic board games, Civilization tiles the game-playing surface with hexagons, which stretch away in a great plain, the so-called “terraform” or map. These tend to come in levels. War game simulations have been using this hexagonal motif for like forever.

As the growing cabal of Divided Spheres readers know (that’s Popko’s primer on this whole spherical geometry topic), you’ll need 12 pentagons to close this expanding plain into a ball, a real planet. The resulting pattern, the “hexapent” is what “thinking globally” now looks like.

The discovery of buckminsterfullerene in the 1980s, along with nanotubes and graphene, have revolutionized the graphic arts in favor of 60–120 degree angles. XY & XYZ grids (all 90 degrees), no longer connote “high technology.” You might study Twitter to observe these trends, or check out the Hexagonal Awareness Project, since concluded, on Tumblr.

Latitude and longitude still rely on a more XY-like grid of spherical trapezoids and as a data structure that’s fine. Juxtaposing the whole Earth with a soccer ball pattern (or higher frequency) adds a visualization layer independent of the underlying model. Either / or thinking need not apply.

So am I claiming the next release of Civilization features a hexapent design? No, I have no special insider knowledge on that score. My claim is only that I’ve been lurking on the Civilization player forums and know many gamers have been clamoring for that feature, tantamount to “going planetary,” adding realism. A next open source game engine might do the same.

My claim is “civilization itself” (not just the game) has crossed a threshold, to where geometric fluency around spheres, has doubled at least three times since the 1970s. We’re at least eight times smarter about the hexapent motif. Yes, that’s hard to verify, but with social media, we can measure trends.

Before I end my story with the hexapent (like a disco ball but made of hexagons and 12 pentagons), let me point out another direction the zeitgeist has gone since the late 1970s: Elastic Interval Geometry (EIG).

Most readers will not have heard of that, but Tensegrity, a close relative, they more likely will have. Mathematicians simplified “tensegrity” to its uber-basics for animation purposes, and we got such as Darwin @ Home by Gerald de Jong, and the interactive Java program Springie by Tim Tyler.

“Tensegrity,” some might remember, is what the sculptor and computer animator Kenneth Snelson bequeathed to us. The word itself appears in Synergetics multiple times, with Buckminster Fuller’s own take on the topic.

The story behind the tug-o-war between Kenneth and Bucky traces back to Black Mountain College in North Carolina in the 1950s. If you’re looking for a door into recent intellectual history, here’s a good one.

The new high rise tower, replacing the Twin Towers of the World Trade Center in New York City, was at one point going to feature a Tensegrity radio tower. Visit the Hirshhorn Museum on the Washington, DC mall for a famous example, Needle Tower.

Finally, let’s talk about “modules,” 3D printable tetrahedral slivers that may be resized and combined to deliver what Synergetics names a “concentric hierarchy” of nested polyhedrons. If we 3D-print 24 identical MITEs and assemble them, we get a cube, ergo MITEs are space-fillers, given cubes are.

A long-running criticism of Aristotle is that somewhere he claimed that tetrahedrons fill space without gaps. His apologists keep saying “he never said regular” in which case he’s actually correct. A mathematician named Duncan MacLaren Young Sommerville gave us more of that story, with Michael Goldberg providing an addendum. Fuller’s “RITE” chopped into four, is another Sommerville tetrahedron.

In sum, I’ve identified three lineages stemming from the 1970s debut of Synergetics in two volumes (Macmillan): a new “world game” aesthetic featuring hexapents; a new breed of software application for modeling “tensegrities”; and a handy application for 3D printers, if looking to share about spatial geometry (the Minimum Tetrahedron or MITE, is made of two “A modules” and one “B module” each 1/24th of unit volume, just like the “T module”).

Given it’s “back to school” season, our growing cabal of Synergetics scholars is eager to pass the torch to its own, plus share with tourists. As a member of said cabal, I’ve been focussing on the Raspberry Pi of late, as a platform for generating hexapents using Adrian Rossiter’s Antiprism software, in conjunction with the ray tracer POV-Ray, both free and open source. I tweet these out through 4DsolutionsPDX sometimes. Check it out!

n-frequency hexapent