20b
Category 20b - Snubs
Above are sections of sadi - the only uniform convex 4-D snub, rasdi - its spiky conjugate, and rappisdi - the head of a 3 member regiment. Also shown are two views of the vertex figure of the snub polychoron "rapsady" - the only known uniform polychoron (outside of prisms) with snub polyhedra as cells.
This category includes the 6 known "normal" snubs, the first 5 have ionic icoic symmetry and have 96 vertices each, while the last one has hyic symmetry with 2400 vertices. The snubs are as follows:
1123. Sadi - snub dis24 - cells are 24 ikes, 24 tets, 96 more tets (as snub cells), verf is a tridiminished icosahedron, sadi is convex.
1124. Rasdi - retrosnub dis24 - cells are 24 gikes, 24 tets, 96 more tets (as snub cells), verf is "trireplinished great icosahedron" (has triangle base with 3 pentagrams attached together as slanted walls, forming 3 spikes at the top). This one has very sharp spikey edges. Conjugate of sadi.
1125. Rappisdi - retroantiprismatosnub dis24 - cells are 24 gikes, 24 ikes, and 96 octs as snub cells. Verf resembles rapsady's above, but where the sirsid verfs are swapped for gike verfs and the trapezoids become squares.
1126. Sirhapsippady - small retrohemiantiprismatosnub prismatodis24 - cells are 24 gikes, 24 tets, 96 trips, and 96 thahs (snub cells) - belongs to rappisdi regiment.
1127. Girhapsippady - great retrohemiantiprismatosnub prismatodis24 - cells are 24 ikes, 24 tets, 96 trips, and 96 thahs (snub cells, oriented different from above) - belongs to rappisdi regiment.
1128. Rapsady - retroantiprismatosnub dis120 - cells are 120 sirsids (yes thats the yog-sothoth - the small inverted retrosnub!), 120 sesides, and 1440 paps (5 AP - the snub cells). This polychoron is the only known non-prismattic polychoron with snub polyhedra as cells - the sirsids are out front touching each other by 10 vertices. I discovered this one back in 1993.
There are many isogonal snubs, but there are two many conditions necessary to make them uniform, trying to make one of the cell types uniform stretches the other cells so they lose their uniformity, this is why there aren't many normal 4-D snubs, however there are 4 major non-Wythoffian snub regiments: The sidtaps and gidtaps (category 27), the idcossids (cat. 28), and the dircospids (cat. 29).
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