Category 4: Ico Regiment

ico ihi oh shahoh huhoh hohoh ohuhoh ghahoh ratho odho hodho hoh honho dod honho doh honho


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.
Ico Regiment

Icoic Symmetry

Verf of Ihi

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

Verf of Tessic Ico Troops

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

Verf of Demitessic Ico Troops

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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