Acronym | ... |
Name |
n-fold dissected great-dodecahedral polytwister, n-fold dissected gaditer, great-dodecaswirlic 12n-choron |
Circumradius | ... |
Especially | sisp (n=10) |
Confer |
The Hopf fibration of the great dodecahedron maps its vertices to according great circles, its edges into twisted (i.e. non-flat but smoothly curved) faces (then looking like a Möbius strip), and the faces get mapped into twisters, which are solid rings bounded by those twisted faces and having thereby throughout the polygonal cross-section of the pre-image, i.e. are pentagonal here. Further each twister then gets dissected into n identical chiral antiprisms. This isochoric construction moreover happens to come out to be isogonal as well, so in total provides a noble polychoron.
This polychoron will have 2 types of edges, one describes the right-up lacing edges of the antiprisms (y), while all its remaining edges belong to the other type (x), simply because the neighbouring twister attaches its cross-secting base polygons next to the left-up lacings of the former. That is, the whole polychoron happens to be chiral in general.
In fact, right this connectedness of the mutually swirling individual twisters does further restrict that n after all. This thus brings back into play the former vertex figure of the starting polyhedron – in addition to the so far only considered faces thereof (the cross-sections of the twisters, i.e. the bases of the antiprisms). Because there also is a full inversion symmetry of the outcome of that fibration, we thus finally have to consideder n = LCM(p, q, 2) for a starting polyhedron {p, q}, i.e. n = LCM(5, 5/2, 2) = 10 in here.
For that specific value this swirlchoron then becomes sisp. Then the edge length ratio can be evaluated as y : x = 1, i.e. sisp will be uniform and esp. the paps happen to become achiral.
12n | 2 10 | 15 5 | 10 ----+---------+---------+---- 2 | 12n * | 5 0 | 5 y, edge figure: {5/2} 2 | * 60n | 2 1 | 3 x ----+---------+---------+---- 3 | 1 2 | 60n * | 2 5 | 0 5 | * 12n | 2 {5} ----+---------+---------+---- 10 | 5 15 | 10 2 | 12n chiral pap variant
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