BUCKYVERSE

988.100 Octa-icosa Matrix

988.110 The icosahedron positioned in the octahedron describes the S Quanta Modules. (See Fig. 988.100.) Other references to the S Quanta Modules may be found at Secs. 100.105, 100.322, Table 987.121, and 987.413.

988.111 As skewed off the octa-icosa ma trix, they are the volumetric counterpart of the A and B Quanta Modules as manifest in the nonnucleated icosahedron. They also correspond to the \sfrac{1}{120}th tetrahedron of which the triacontahedron is composed. For their foldable angles and edge-length ratios see Figs. 988.111A-B.

988.12 The icosahedron inscribed within the octahedron is shown at Fig. 988.12.

988.13 The edge lengths of the S Quanta Module are shown at Fig. 988.13A.

988.14 The angles and foldability of the S Quanta Module are shown at Fig. 988.13B. upon the planar VE they reveal the following rational fraction edge increments:

+:-----------------------+:-----------------------+:------------+:----------+
| | | | |
±-----------------------±-----------------------±------------±----------+
| B-31 _ 1 | B-C _ 1 | > B-32 _ 1 | B-33_ 3 |
| | | > | |
| D-D 10 | D-D 6 | > D-D 4 | D-D 10 |
±-----------------------±-----------------------±------------±----------+
| B-D _ 1 | B-E _ 1 | > B-62 _ 1 | B-67 _ 3 |
| | | > | |
| D-D 2 | D-D 6 | > D-D 4 | D-D 10 |
±-----------------------±-----------------------±------------±----------+
| B-73 _ 3 | B-H _ 1 | | |
| | | | |
| D-D 8 | D-D 2 | | |
±-----------------------±-----------------------±------------±----------+

Figure 40: 988.00 Polyhedral Evolution: S Quanta Module: Comparisons of skew polyhedra
Figure 40: 988.00 Polyhedral Evolution: S Quanta Module: Comparisons of skew polyhedra
Figure 41: Fig. 988.100 Octa-Icosa Matrix: Emergence of S Quanta Module: A. Vector equilibrium inscribed in four-frequency tetrahedral grid. B. Octahedron inscribed in four-frequency tetrahedral grid. C. Partial removal of grid reveals icosahedron inscribed within octahedron. D. Further subdivision defines modular spaces between octahedron and icosahedron. E. Exploded view of six pairs of asymmetric tetrahedra that make up the space intervening between octa and icosa. Each pair is further subdivided into 24 S Quanta Modules. Twenty-four S Quanta Modules are added to the icosahedron to produce the octahedron.
Figure 41: Fig. 988.100 Octa-Icosa Matrix: Emergence of S Quanta Module: A. Vector equilibrium inscribed in four-frequency tetrahedral grid. B. Octahedron inscribed in four-frequency tetrahedral grid. C. Partial removal of grid reveals icosahedron inscribed within octahedron. D. Further subdivision defines modular spaces between octahedron and icosahedron. E. Exploded view of six pairs of asymmetric tetrahedra that make up the space intervening between octa and icosa. Each pair is further subdivided into 24 S Quanta Modules. Twenty-four S Quanta Modules are added to the icosahedron to produce the octahedron.
Figure 42: Fig. 988.12 Icosahedron Inscribed Within Octahedron: The four-frequency tetrahedron inscribes an internal octahedron within which may be inscribed a skew icosahedron. Of the icosahedron’s 20 equiangular triangle faces, four are congruent with the plane of the tetra s faces (and with four external faces of the inscribed octahedron). Four of the icosahedron’s other faces are congruent with the remaining four internal faces of the icosahedron. Two-fifths of the icosa faces are congruent with the octa faces. It requires 24 S Quanta Modules to fill in the void between the octa and the icosa.
Figure 42: Fig. 988.12 Icosahedron Inscribed Within Octahedron: The four-frequency tetrahedron inscribes an internal octahedron within which may be inscribed a skew icosahedron. Of the icosahedron’s 20 equiangular triangle faces, four are congruent with the plane of the tetra s faces (and with four external faces of the inscribed octahedron). Four of the icosahedron’s other faces are congruent with the remaining four internal faces of the icosahedron. Two-fifths of the icosa faces are congruent with the octa faces. It requires 24 S Quanta Modules to fill in the void between the octa and the icosa.
Figure 43: Fig. 988.13A S Quanta Module Edge Lengths: This plane net for the S Quanta Module shows the edge lengths ratioed to the unit octa edge (octa edge = tetra edge.)
Figure 43: Fig. 988.13A S Quanta Module Edge Lengths: This plane net for the S Quanta Module shows the edge lengths ratioed to the unit octa edge (octa edge = tetra edge.)
Figure 44: 988.13B S Quanta Module Angles: This plane net shows the angles and foldability of the S Quanta Module.
Figure 44: 988.13B S Quanta Module Angles: This plane net shows the angles and foldability of the S Quanta Module.
Figure 45: Fig. 988.13C S Quanta Module in Context of Icosahedron and Octahedron
Figure 45: Fig. 988.13C S Quanta Module in Context of Icosahedron and Octahedron

988.20 Table: Volume-area Ratios of Modules and Polyhedra of the Primitive Hierarchy:

                     *Volume*   *Area*   *Volume /Area*   *ArealVolume*

A Module * *
P II
E
S
B
Tetrahedron
Icosahedron**
Cube
Octahedron
Rhombic dodecahedron
Icosahedron

* Volume and area of A Module considered as unity.
** Icosahedron inside octahedron.