Monday, March 31, 2014

E Module Mensuration

Quoting from Dave:

T = 1/24 = .0416666
E = (√2/8)(ø^-3) = .0417313

So, the T & E modules are close in volume, and the difference was what Fuller expounded on in Synergetics 2.

If the Rhombic Triacontahedron is 120T modules, it has a volume of 5 tetra volumes. Alas, the "radius" was not exactly 1, but .999483.

So, the Rhombic Triacontahedron's volume was really 5.007758 or 120 E modules, when the radius is exactly 1. Kirby figured that out, how to get the radius for the exact volume 5 Rhombic Triacontahedron, which is .999483 or (2/3)^(1/3)((ø^1)/√2)).

The radius in question for the Rhombic Triacontahedron is from the origin to the center of the rhombus. The radius to the long leg to the short leg is ø^1:ø^0:ø^-1 or 1.618034:1.000000:.618034.

This was an epiphany for me since the E module derived from the Rhombic Triacontahedron had a radius of 1, so the legs were .618034 and .381966.

Increasing the edges by ø^1 we have a long diagonal on the Rhombic Triacontahedron that is the same as the icosahedron's edge of 2, which fully inscribes within the Super Rhombic Triacontahedron.

By increasing the radius by ø^1 for the 5.007758 E module Rhombic Triacontahedron, we get the Super Rhombic Triacontahedron.

Thusly, an E module is 120th of the 5+ volume Rhombic Triacontahedron or (√2/8)(ø^-3), and the next larger sized E module derived from the Super Rhombic Triacontahedron is ø^3 larger or √2/8 expressed as E3.

Quoting from me:

The blue icosa is the standard 18.51 of Synergetics, as is the yellow cubocta ("VE") of volume 20. The other shapes are all non-standard in having that 1.851 edge, 1/10th of the volume number, but here an edge.  The green cubocta, has those smaller edges, which is in turn the interval for the whole 4F tetrahedron.

Which is why the yellow cubocta sticks out. What defines the "non-standards" is the tetrahedron to which our standard icosahedron is flush.  The small green cubocta has a volume between 15 and 16 whereas the yellow one has volume 20 as you know.

The icosahedron + its dual = rhombic triacontahedron of whatever size.  In the jargon Koski and I have been using, the "super RT" would be the standard (18.51) icosahedron + its dual, combining to form this rather large combo.

It's when you scale down that super-RT by 1/phi that you get the RT mother-of-Emods i.e. 120th of such is what is named an "E module" in Synergetics.  David measures in those, and phi-up, phi-down versions of those, in terms of place value (base).  The video expresses volumes in "super-RT sized Es" plus standard Es, one could say.
Then in Synergetics we have the "T module", a fine distinction, in that the T's volume is exactly that of the A's and B's, whereas the E's is only really close (the exact ratio being a focus in Synergetics 2 another number with phi in it, though Fuller avoided greek letter stuff).