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In accomplishing these transformations of the uniformly-perimetered
symmetrical shapes, it is also of significance that
the area of six equiangular, uniform-
edged triangles is reduced to four such triangles. Therefore,
it would take more
equiperimeter triangular tiles or squares to pave a
given large floor area than it would
using equiperimetered hexagons. We thus discover that
the hexagon becomes in fact the
densest-packed patterning of the circles; as did the
rhombic dodecahedron become the
minimal limit case of self-packing allspace-filling
in isometric domain form in the
synergetical from-whole-to-particular strategy of discovery;
while the rhombic
dodecahedron is the six-dimensional state of omni-densest-packed,
nuclear field domains;
as did the two-frequency cube become the maximum subfrequency
self-packing, allspace-
filling symmetrical domain, nuclear-uniqueness, expandability
and omni-
intertransformable, intersymmetrical, polyhedral evolvement
field; as did the limit-of-
nuclear-uniqueness, minimally at three-frequency complexity,
self-packing, allspace-filling,
semi-asymmetric octahedron of Critchlow; and maximally
by the three-frequency, four-
dimensional, self-packing, allspace-filling tetrakaidecahedron:
these two, together with the
cube and the rhombic dodecahedron constitute the only-four-is-the-limit-system
set of
self-packing, allspace-filling, symmetrical polyhedra.
These symmetrical realizations
approach a neatness of cosmic order.
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