Polytwisters
In four dimensions, there's a new kind of geometric figure in town - I call them polytwisters. Polytwisters have some similarity to polytopes but with an interesting new twist, instead of having vertices, edges, faces, and cells - they have rings, full twist Mobius strips, and "twisters" that swirl around each other. I discovered the uniform polytwisters last year (2001). The cells (which I call twisters) of a polytwister are best described as follows: take a polygonal rod (somewhat bowed out) give it a full 360 degree twist, curve it around into 4 space to form a ring. The faces of this twister will be a full twist Mobius band, while the edges will be rings (circles). A regular polytwister is a polytwister with only one kind of regular twister (a twister based off of a regular polygon), and with all of its rings identical. A uniform polytwister is a polytwister with like rings, and with regular twisters (not necessarily alike). The polygonal rods are bowed out due to proper fitting and to keep the twisters flush (in the same way that the tube face of a cylinder is flush (not bowing inwards or outwards). Polytwisters have swirlprism symmetries. A regular convex polytwister such as the dodecatwister would make an unusual die, the dodecatwister has 12 identical pentagonal twisters, 30 full twist strips, and 20 rings all swirling around each other (notice the similarity to the dodecahedron). If it was rolled like a die, it would land on a side and could still roll while on that side, if tilted over to another side, it would roll off at an angle.
There are 18 regular polytwisters, 2 based off of each regular polyhedron:
tetratwister: convex, has 4 trigonal twisters, 6 full twist bands, and 4 rings
quasitetratwister: non-convex, has 4 trigonal twisters, 6 bands, and 4 rings (the quasi-cases have twisters orthogonal to the normal cases and they cut in deeper, both of the twisters use the same rings)
cubetwister: convex, has 6 square twisters, 12 bands, and 8 rings
quasicubetwister: non-convex, has 6 square twisters, 12 bands, ans 8 rings
octatwister: convex, has 8 trigonal twisters, 12 bands, and 6 rings
quasioctatwister: non-convex, has 8 trigonal twisters, 12 bands, and 6 rings
dodecatwister: convex, has 12 pentagonal twisters, 30 bands, and 20 rings
quasidodecatwister: non-convex, has 12 pentagonal twisters, 30 bands, and 20 rings
icosatwister: convex, has 20 trigonal twisters, 30 bands, and 12 rings
quasicosatwister: non-convex, has 20 trigonal twisters, 30 bands, and 12 rings
gadtwister: has 12 pentagonal twisters, 30 bands, and 12 rings (has star shaped "ring figure")
quasigadtwister: has 12 pentagonal twisters, 30 bands, and 12 rings (has star shaped "ring figure")
sissidtwister: has 12 star twisters, 30 bands, and 12 rings (pentagon ring figure)
quasisissidtwister: has 12 star twisters, 30 bands, and 12 rings (pentagon ring figure)
giketwister: has 20 trigonal twisters, 30 bands, and 12 rings (star ring figure)
quasigiketwister: has 20 trigonal twisters, 30 bands, and 12 rings (star ring figure)
gissidtwister: has 12 star twisters, 30 bands, and 20 rings (triangle ring figure) - the rings are quite peaked
quasigissidtwister: has 12 star twisters, 30 bands, and 20 rings (triangle ring figure)
There's also an infinite series of regular twisters as well, based off of polygons, the twisters in this case resemble ringed twisted rods with a () shaped section ( the () is sort of a bowed out line segment). Each polygon has one of these twisters since the quasi-case gives the same result. An n/d-gonic twister has n twisters, n bands, and 2 rings (which are orthogonal). Uniform polytwisters are based off of the uniform polyhedra, one such example is the sirsidtwister which is a very complicated polytwister based off of the small retrosnub icosicosidodecahedron (sirsid).
Polytwisters don't exist in 3 or less dimensions, or in any prime dimension other than as a prism. However non-prime dimensions seem promising in having polytwisters of some kind, this will require further research. I have took a glance at 6 dimensional cases, so far it appears that 6-D cases require stronger conditions, they would be based off of polytetra (5-D polytopes) but the polytetron itself may require swirl sub-symmetries for it to work. I was also hoping to find polychoron based "polyspheristers" which would have spheres swirling around each other, but so far there seems to be clashing going on. 6-D polytwisters has yet to be confirmed.
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