Jim's Geometry 2008
ICOSAHEDRA
Regular lattices exhibit surprising geometrical regularities which, if
exploited in the right way, lead to very efficient algorithms for smoothing
gridded data. Smoothing data with the right degree and orientation of
anisotropy is especially challenging. Our anisotropic algorithms rely of the
so-called "Triad", "Hexad" and "Decad" methods in two, three and four
dimensions respectively. Taking the four-dimensional case, a "decad" refers to
a set of 10 lattice generators (irreducible integer-component 4-vectors)
configured in a particular way that allows the desired smoothing to be
effected by sequential line-filters along the directions the generators
indicate. The geometrical figure formed as the convex polyhedron having the 10
generators, and their negatives, as vertices, is a complicated object. In its
most symmetrical manifestation, its "surface" (a 3-dimensional boundary)
consists of 10 regular tetrahedra interlaced with 20 square-sided triangular
prisms, and the whole thing exhibits the symmetry group, "S_5", of all
permutations of five items. Fortunately, there is a way of visualizing some of
the important relationships among the generators by reference to two copies of simpler, three-dimensional polyhedra. The example shows one possible lattice decad tabulated in two of the 120 equally valid ways in which ten distinct generators are arranged into two sets of five (pentads). The association of ten representative generators of each tabulation, together with their negatives,
with the faces of a corresponding pair of regular icosahedra are also shown, and the two icosahedral representations are actually complementary. (The two maps are highly schematic since it is only the combinatorial relationships are relevant.) "Pentads" of (5) generators surrounding each vertex in either
icosahedron sum to the null vector. Also, "triads" of (3) generators that
immediately surround any given face of either icosahedron sum to the null
vector. All triads are represented somewhere on either one icosahedron or the other. The 60 proper rotational symmetries of the icosahedron, together with the
interchange of the complementary representations, exactly account for the symmetry group of 120 rotations in 4D of our original decad, but they can now be seen in the form of the 3D icosahedral group rotations, plus the interchange.
The transition between one decad and another as the degree and orientation
of anisotropy is altered actually involves an additional intermediate configuration of 12 (not just 10) generators, which are this time associated with the diameters of the celebrated "24-cell". This is a regular polyhedral
"solid" in 4 dimensions possessing many remarkable properties (including self-duality in the Legendre-Fenchel sense). The transition from a decad to the 12-generator configuration involves the removal of one generator (and of its negative, of course) from the decad, together with its replacement with three
new generators. The triangular diagrams schematically show the halves of our
complementary pair of icosahedra now orientated to place, dead-center, the triangle (or its opposite) that corresponds to the generator to be replaced. The transition to the new 12-generator object is shown to follow the same transition rules from either representation-the end result is the same. These rules inherit the three-fold cyclic symmetry of the icosahedron about the "central" face. The three-fold symmetry is therefore preserved in this new object-it is manifested in the natural partition of the new set of generators into the three (linearly deformed) orthoplexes that collectively make up the vertices of the 24-cell.
The 12-generator objects have a very high order (1152) of symmetry. The 12 + and - pairs of generators in their three orthoplex partitions, can be mapped to the surfaces of two complementary "rhombic dodecaheda" (shown here in stereographic projection), but this now involves double covering-the + and the - version of each generator maps to the same rhombic face. Triads (three 4-vectors summing to the null vector) now correspond to all the sets of three faces meeting at
the 120-degree vertices, but a sign-change for the indicated generator is implied when passing across any edge that is shown dotted. The topology of this suggested "Riemann-surface" is therefore not a sphere, but rather a "double-torus" (or "two-handled torus").
The pair of conjugate double-tori are most symmetrically visualized as two alternative rhombic tilings of the fundamental octagonal region of the hyperbolic plane into which the double-torus is smoothly mapped. Here they are shown side by side in stereographic projections-the rhombuses are actually congruent and we blame the projection for the apparently different sizes and shapes. All 16 distinct "triads" of the 24-cell are now found (twice) to correspond to the instances of three rhombuses meeting at a 120-degree junction on one or other of these complementary representations. Opposite edges of the octagonal fundamental region are understood to be identified in the
orientation-preserving way in order to recover the intended double-torus topology. The symmetry group of the rhombic tiling in one torus has order
96, plus interchanging the two complementary representations gives 192, and finally, the further permutations of the orthoplex labels, "A", "B" and
"C", gives the complete group of possible symmetries, 1152, equivalent to all the rotations and reflections in 4D of the original regular 24-cell.
We have illustrated ways by which discrete groups of orthogonal rotations in 4D can be partially mapped to simpler groups of rotations in 3D, plus transformations not so obviously of a geometrical kind (interchanges of alternative representations, permutations of orthoplex labels). But full group of continuous 4D rotations (SO(4)) has a covering that factors into a pair of (complex) SU(2) groups, each of which is intimately associated with the real rotations in 3D. Our discrete examples might be considered to correspond to special discrete restrictions of the more general continuous group factorization.