by wendy » Thu Sep 08, 2011 7:12 am
There are indeed star-tesselations. Densities are shown in square brackets after the polytope, as [d5].
In hyperbolic space, there is always {p,p/2} and {p/2,p}. When P is even, this becomes a compound of three parts. When p is odd, it is a star-tesselation on the lines of {5/2,5} and {5,5/2}. In higher dimensions, there are star-tiles like {5/2,5,3,3} [d4], {5,5/2,5,3,} [d6] and their duals. There are star tilings in H3, but none are regular. {5,5/2,5,3:} is a subgroup of order 4 of {5,3,4}, yields stars.
When one takes {6/2} and the stella octangula as stars on the line of {5/2}, one gets
{6/2,6} and {6,6/2} of order 3. To this, there is the 'asterix-star', formed by edges crossing in an asterix. Thence one has {6 * 3}, of d4, and {6/2*6} {6*6/2} of d12. This completes the regular stars of {6,3}
The simplest asterix-star is {4/2}, or +. These occur in the stellation of every {p,4} to give {p+p} [d2]. The stella octangula is an example of this, with p=3. But it occurs with all p, p=4 in euclidean space, and p>4 in hyperbolic. One finds also an {8/4} type edge in {8++8} [d6].
The stella octangula occurs as faces of stella tegmata, {3+3,4}, and as vertex figures of {4,3+3}. These both tile 4-space, as {3+3,4,3} [d4], {4,3+3,4} [d6], and {3,4,3+3} [d4], and again in H5, as {3+3,4,3,3} [d5], {4,3+3,4,3},[d10] {3,4,3+3,4} [d10], and {3,3,4,3+3}[d5].
In H3, one finds also {6/2,6,3} [d4], {6,6/2,6} [d6], {3,6,6/2} [d4], {4+4,4} and {4,4+4} both [d3]. The tilings {3+3,4,3+3}, {6/2,6,6/2} and {4+4+4} are all infinitely dense: the first two imply {3,3+3} and {3,6/2} which are also infinitely dense.
Over about 5 dimensions, stellated forms tend not to occur greatly.