Acronym ...
Name n-fold dissected tetrahedral polytwister,
n-fold dissected tetter,
tetraswirlic 4n-choron
Circumradius ...
Especially ico (n=6)  
Confer
general polytopal classes:
isogonal   noble  

The Hopf fibration of the tetrahedron 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 triangular 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(3, 3, 2) = 6 in here.

For that specific value this swirlchoron then becomes ico. Then the edge length ratio can be evaluated as y : x = 1, i.e. ico will be uniform and esp. the traps happen to become achiral and regular.


Incidence matrix

4n |  2   6 |   9  3 |  6
---+--------+--------+---
 2 | 4n   * |   3  0 |  3  y
 2 |  * 12n |   2  1 |  3  x
---+--------+--------+---
 3 |  1   2 | 12n  * |  2
 3 |  0   3 |   * 4n |  2
---+--------+--------+---
 6 |  3   9 |   6  2 | 4n  chiral trap variant

© 2004-2021
top of page