1041.11 Seven Axes of Truncated Tetrahedron

1041.11 The prime generation of the seven axes of symmetry are the seven unique perpendiculars to the faces of the seven possible truncations of the tetrahedron:

4 original faces

4 triangular truncated vertexes

6 quadrilateral truncated edges

14 faces of the truncated tetrahedron, which produce seven unique pairs of parallel faces whose axes, perpendicular to their respective centers of area, generate the seven axes of symmetry. (See Secs. 100.103-.05 and Fig. 1041.11.)

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Fig. 1041.11 Truncated Tetrahedron Generates Seven Prime Axes of Symmetry: Four original faces + four truncated vertexes + six truncated edges = 14 faces whose midpoints interconnect in seven axes.

1041.12 The seven unique axes of the three unique sets (4 + 4 + 6) producing the 14 planes of the truncated tetrahedron are also identifiable with:

—the 14 planes that bound and enclosingly separate all biological cells;

—the 14 facets interbonding all bubbles in the bubble complexes; and

—the 25 and 31 unique planes generated by the seven sets of foldable great circles, which are the only such foldably unbroken sets (i.e., the 3, 4, 6, and 12 sets of the vector equilibrium and the 6, 10, and 15 sets of the icosahedron).

1041.13 Various high frequencies of modular subdividings of the tetrahedron produce a variety of asymmetrical truncatabilities of the tetrahedron. The dynamics of symmetry may employ any seven sets of the 56 foldable-great-circle variations of planar orientation. Thus it follows that both the biological cell arrays and the bubble arrays display vast varieties of asymmetries in their 14 enclosing planes, so much so that this set of interidentifiability with the 14 topological characteristics of the tetrahedron, the prime structural system of Universe, has gone unnoticed until now. (See Sec. 1025.14)