the time star consists of five regular tetrahedra, configured so that
their vertexes are located at the centers of the twenty triangular
faces of the icosahedron. there are two ways to ways to create this
configuration, say lefthanded and righthanded.
TimeStar (JPEG 15k)
the symmetries contained in the tetrahedron are best revealed by
first bisecting the edges to create a central octahedron, and then
once more to create a cuboctahedron (vector equilibrium) inside
the octahedron. the central point of the VE is the central point
of the original tetrahedron, and in the time star configuration
these five centers coincide.
Bisection (JPEG 13k)
four of the VE's great circles correspond with the four planes
of the tetrahedron's faces VE (JPEG 16k),
and when the five sets of these
great circles are superimposed in the time star configuration
the result is not twenty great circles, but only ten, due to
pairing. the VE's themselves don't pair, since the vertexes
don't correspond, but only the great circles.
PAIRING (JPEG 14k)
the ten resulting great circles are exactly those ten circles
of the icosahedron which are generated by rotations around the
centers of opposite triangular faces. the surface image produced
by these ten circles is that of twelve star-shaped forms, connected
at their outer points. the radii of the original five VEs extend
out from the time star's center to the points of each star-shape's
the tetrahedron/octahedron combination contains all of the symmetries
of the isotropic vector matrix (sphere-packed space), reflecting
both threefold and fourfold symmetries. the icosahedron, on the other
hand, reflects fivefold symmetry. by placing five tetrahedra in
a time star configuration, we create a lowest-common-multiple of
all of these, thereby establishing a set of symmetries which embraces
every one of the prime geometrical forms.
if a coordinate system is to be found which is capable of accommodating
all of nature's symmetries, it must be based on a set of five
interpenetrating isotropic vector matrices oriented according to
the time star's tetrahedra.
it would seem sensible to conclude that, in contrast to "open space",
a close packing of multitudes of atoms in what we commonly refer to
as a solid, has the effect of "squeezing out" the fivefold symmetry
by forcing the timestar's tetrahedra to align. a particle essentially
alone in open space would "see" the full time star symmetry, but when
brought into close proximity with many others, the fivefoldedness is
suppressed and the IVM symmetries rule.
in a "centerless" configuration such as a bubble, the symmetry from
a macro point of view is clearly that of the time star, but viewing
the different vertex components of a bubble would reveal that there
were twelve vertexes experiencing time star symmetry while the rest
experienced a minor suppression of their fivefold symmetry due to
their being surrounded by other vertexes. in a dynamic system, this
suppression of fivefold symmetry would be easily 'transmitted' to
neighboring vertexes, causing the 'oddball' vertexes to pass their
fivefold symmetry to neighbors, once again being overpowered by
their environment and shifting back to fivefold suppression.
this view, holding the time star as rest-phase and IVM as a special
case where the time star's five IVM's coincide nearly perfectly,
will be the basis of my further geometrical exploration. i welcome