937.20 Doubleness of Octahedron

937.21 The octahedron always exhibits the quality of doubleness. When the octahedron first appears in the symmetrical contraction of the vector equilibrium jitterbug system, it appears with all of its vectors doubled (see Fig. 460.08D). It also takes two sets of three great circles each to fold the octahedron. You might think you could do it with one set of three great circles, but the foldability of the octahedron requires two sets of three great circles each. (See Secs. 835 and 836.) There are always six great circles doubled up in the octahedron to reappear only as three. (See Fig. 937.20.)

937.22 And we also recall that the octahedron appears as the prime number 2 in the geometrical hierarchy, while its volume is 4 when the tetrahedron is taken as volumetric units (see Table 223.64).

The tetrahedron’s prime number identity is 1

The octahedron’s prime number identity is 2

Both cubes and rhombic dodecahedra are 3

And icosahedra and vector equilibria are 5

They first occur volumetrically, respectively, as

1, 4, 3, 6, 18.51, and 20.