Category 4: Ico Regiment

There are 14 uniform polychora in the ico regiment, where ico itself has already been counted in category 1. Ico is the icosatetrachoron (24-cell). Only two members have icoic symmetry, ico and ihi - where ihi has an ionic icoic symmetry. Six members have tessic symmetry, while 6 have demitessic symmetry. Ico is symboled as oo'ox (icoic), oxo'o (tessic), or o9o (demitessic). There is also a compound in this regiment, which I call "Gico" short for great icosatetrachoron. Gico is the compound of 3 tesseracts. All members have 24 vertices. One interesting note, so far only ico, ratho, and gico (as 3 tesseracts) show up as terons (4-D facets) in uniform polytera, other than in prisms. All of these are self-conjugates. The tame ones are oh, shahoh, huhoh, ghahoh, ratho, and odho - all others are wild. Along with ico, oh and shahoh are orientable, all of the others are non-orientable.

Icoic Symmetry

This polychoron, like ico, has icoic symmetry (in this case only ionic icoic symmetry). It's verf along with ico's is shown above.

60. Ihi - (I hi) 24hemi24. Its cells are 24 thahs and 12 coes. Its verf is an ionic faceting of the cube. It has 480 pieces with three types, LOC = 20.

Tessic Symmetry

These polychora have tessic symmetry.

61. Oh - (O) 8hemi16. Its cells are 8 octs and 8 ohoes. Its verf is a square hourglass. It has 72 pieces with two types, LOC = 5. Oh is a dangly looking 4-D version of cho.

62. Shahoh - (SHAH ho) small 16hemi16. It has 16 octs and 8 ohoes as cells, and 8 pseudo-octs. Its verf is an inverted squip (squip meaning square prism). It has 80 pieces in two types, LOC = 6.

63. Huhoh - (HOO ho) 16hemi16. Its cells are 16 thahs and 8 choes while its verf is a spiky faceting of the cube. It has 256 pieces in two types, LOC = 16.

64. Hohoh - (HO ho) 16-8hemi16. It has 8 octs, 16 thahs, and 8 choes. Its verf is a square "bowtie-prism". It has 264 pieces in three types, LOC = 17.

65. Ohuhoh - (o HOO ho) 8-16hemi16. The cells are 8 octs, 16 thahs, and 8 coes. Its verf is a sort of square bowtie retroprism. It has 328 pieces in four types, LOC = 21.

66. Ghahoh - (GAH ho) great 16hemi16. Its cells are 16 thahs and 8 coes, it also has 8 pseudo-octs. Its verf is a pseudo-square bowtie-prism. It has 320 pieces in three types, LOC = 20.

Demitessic Symmetry

These have demitessic symmetry.

67. Ratho - (RA tho) rectified tho. Its cells are 8 octs, 8 thahs, and 4 coes. Its verf is a bowtie prism. It has 72 pieces in three types, LOC = 14.

68. Odho - (ODD ho) 8dishemi8. Its cells are 8 thahs, 4 ohoes, and 4 choes. Its verf is a bowtie hourglass. It has 160 pieces with three types, LOC = 20. It is no doubt the oddest looking polychoron in this regiment.

69. Hodho - (HODD ho) 16-8dishemi8. Its cells are 16 octs, 8 thahs, 4 ohoes, and 4 choes. Its verf is a cube with 2 bowtie-shaped windows. It has 176 pieces in five types, LOC = 24.

70. Hoh Honho - (ho HON ho) 16-8hemi16-8intercepted hemi8. It has 24 thahs (one group of 8 disalligned), 8 choes, and 4 coes. Its verf is a blend of hohoh's and odho's verf. It has 416 pieces in seven types, LOC = 52 - this is the most complex one of this category.

71. Dod Honho - (dod HON ho) dis8dishemi8intercepted hemi8. Its cells are 16 thahs (2 groups of 8), 4 ohoes, 4 choes, and 4 coes. Its verf is a bowtie prism with bowtie windows where the squares should be. It has 224 pieces with five types, LOC = 40.

72. Doh Honho - (doe HON ho) dis8hemi16intercepted hemi8. Its cells are 8 octs, 8 thahs, 4 coes, and 8 ohoes (2 groups of 4), it also has 8 pseudo-octs. Its verf is a cube faceting with 2 squares, 2 bowties, and 2 pseudo-squares where the squares would normally be. It has 232 pieces in six types, LOC = 30.

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