Acronym ... Name dodecaswirlic triacosihexacontachoron,thirtyfold dissected dodecahedral polytwister,thirtyfold dissected doter Circumradius ... Confer more general: n-doe-swirl   general polytopal classes: isogonal   noble Externallinks

The Hopf fibration of the 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 squars 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, 3, 2) = 30 in here.

For that specific value it results in this swirlchoron. Here the edge length ratio can be evaluated as y : x = sqrt[(15-sqrt[75+30 sqrt(5)])/10] = 0.554994.

Incidence matrix

600 |   2    6 |    9   3 |   6
----+----------+----------+----
2 | 600    * |    3   0 |   3  y
2 |   * 1800 |    2   1 |   3  x
----+----------+----------+----
3 |   1    2 | 1800   * |   2
5 |   0    5 |    * 360 |   2
----+----------+----------+----
10 |   5   15 |   10   2 | 360  chiral pap variant