Saturday, January 19, 2008

Koski on T-Modules

Dave Koski is doing some detailed chronology regarding his explorations of five-fold symmetric polyhedra, starting with his stumbling upon the golden mean, then the rhombic triacontahedron, its stellate and dimpled complements.

Dave writes (#37501, typo fixed):
The rhombic triacontahedron that is dimpled in at 8 locations can have 8 regular rh triac surround it by filling in at the dimples. The space left over can be filled in with more 8 dimpled rh triac. So the 8 dimpled rh triac and regular rh triac compliment each other in filling all space.

The rhombic triacontahedron that is dimpled in only four locations is an all space filler by itself because before there was an 8 dimple and a reg rh triac complementing each other. But with only four dimples, the 8 dimple concept is shared with reg rh triac so they all have four dimples and look the same. And that looks like closest packing to me, since they stack tetragonally. Too lazy to make image, sorry. I would love to give credit to whomever discovered this one. It was shown to me by man named David Noble (I think), out in LA, back in the eighties. I do not think I would have thought of it.
Dave's obsession with golden ratio geometry led him to study Fuller's Synergetics, which is where he and I started to overlap, in LA (me just passing through) and Santa Monica (his domicile at the time). We were both recipients of the Synergetics Explorer Award, presented by Fuller's grandson Jaime at BFI (this was before I became BFI's first web wrangler).

Back in the early 1990s, David, myself, Russ Chu, Robert Orenstein, and Bonnie Goldstein (later DeVarco) were trying to realize Hal Hildebrand's vision of a Smalltalk storage system for all of Bucky's many boxes of papers, collectively referred to as the Chronofile (plus he had a bound volume set by that name). Safe to say, with an 8 MB RAM motherboard (purchased by Russ), and no budget, we were ahead of our time.

Anyway, Koski dissected what Fuller named the T-module, 1/120th of a rhombic triacontahedron, into discrete numbers of smaller T-modules, edges scaled by 1/phi (some say tau or vice versa) i.e. 0.618... (volume scaled by a 3rd power of same) at each smaller size level -- except another shape tends to stand out in this project, what Dave termed a Remainder Tet, also recursively disassembling.

David is chronicling all this on Synergeo these days, complete with uploaded pictures made with vZome by Scott Vorthmann, a nifty free tool for Zome users.

T-mod, classic dissection
in vZome by D. Koski