THE GEOMETRY OF THE TIME STAR

by Gerald de Jong


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 central pentagon.

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 fellow explorers!


:: gerald_de_jong/rotterdam, http://www.xs4all.nl/~gdj
:: Binary Artifact: place 1 inside 0 as its axis of rotation.