If you're up on Bucky Fuller's Synergetics, as distinct from Haken's, and you could be up on both (or neither) then you'll likely know of BEAST, if not by that acronym. BASKET = BEAST + K.
I'm talking about the A & B modules, a pair, one easily distorted into the other, or "morphed", both wedges, tetrahedrons, slices of space.
Then come the T & E modules, exactly the same shape as one another (tetrahedrons) but one radiating outward enough further from a common center to leave a gap, like a biosphere of a planet, between the two rhombic triacontahedrons in question, the one of volume 5 (120 T), and the one of volume 5 plus (120 E).
And finally: the S, wedged between the octahedron of volume 4 (D edge length; D for diameter) and the eight faces-inscribing 20-faced icosahedron, or a “dozeneighteyes” in Struppi's dozenal namespace. 24 S slices define the difference, a dozen left and a dozen right.
All these A, B + E, T + S modules are handed (L and R), come as inside-outs of one another (same diff).
Enter the K, and hence BASKET.
The K has the same T & E shape, so KET or TEK is a logical triple, as is BAT or TAB (all 1/24). The K is 1/120th of a rhombic triacontahedron of volume not 5, not 5+, but 7.5 i.e. the RT made of Ts, scaled up 3/2 times, volume-wise. K volume = (1/24)(3/2) = 1/16, or half the MITE volume of 1/8 (MITE = B+ A+ A- = A+ A- B-), though not shaped that way.
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| RD 6 yellow; RT 7.5 red; Octa 4 green; Cube 3 blue |
The K is allowed to resonate with DK, or David Koski, in helping us remember the timeline and the fact that the RT of volume 7.5 did not occur in concentric hierarchy renderings until later.
The 7.5 volume RT shares vertices with the volume 3 cube and by extension with tips of rhombic dodecahedron short diagonals. That's the RD of volume 6, made of As and Bs.
In addition to the 7.5 RT, Koski and Kirby (KU, myself) talked a lot about the 21.21 RT (15√2), the one of volume 20 times Syn3 (Syn3 = 2nd root of 9/8). I often will say "2nd root" instead of "square root" given Synergetics addresses this very prejudice. Python lets me customize the namespace.
That's four RTs of interest, in order of increasing size: 120 Ts (5), 120 Es (5+), 120 Ks (7.5), SuperRT (21.21...).
Here are the volumes we're talking about again, this time arbitrarily extended to 50 decimals:
