Rich is the rectified cubic honeycomb. Its vertex figure is a square prism, and thus all the regiment members will be facetings of this squip. Rich has four different wythoffian symmetries, which gives its members a lot of diversity. The first parts of this regiment act a bit like the ico regiment, except because rich is less symmetric than ico we get more members. The second parts of this regiment act a bit like the afdec regiment, but because rich is more symmetric than afdec we get less members.
These are the ones with full chonnic symmetry, simply as a rectified cubic honeycomb. The verfs here have full squip symmetry.
41. Rich - Rectified cubic honeycomb. Cells are octs and coes. Its verf is a square prism. Symbols are o4x3o4o, o4x3o *b3o, x3o3x *b4o, and x3o3x3o3*a. Rich can also be seen as an octet diminishing, it is also the rectification of octet as well. Essentially, the rich regiment entirely consumes all of our rectates.
42. Oha - Octahedral hemiapeiratic honeycomb. Cells are octs and thas. Its verf is a square hourglass. Symbols are o4o3x∞x3/2*b and x3o3/2x3o3/2*a∞*c, it is only wythoffian as a podiumvert, as tha is only wythoffian as a trapezivert. This was previously known as Doha.
43. Coha - Cuboctahedral hemiapeiratic honeycomb. Cells are coes and thas.
44. Hohath - Hemioctahedral hemitrihexagonal honeycomb. Cells are ohoes and thats.
45. Hohah - Hemioctahedral hemiapeiratic honeycomb. Cells are choes and hohas.
46. Ohoha - Octahedral hemioctahedral hemiapeiratic honeycomb. Cells are octs, ohoes, and hohas.
47. Ohohath - Octahedral hemioctahedral hemitrihexagonal honeycomb. Cells are octs, choes, and thats.
These have octettic symmetry, as a rectified octet. The verfs have brick symmetry. There is another different octettic type later, which is why these have to be clarified as the isotoxal ones. There's not really a sensible order to these, there's a lot of them and they all look kinda the same. Some names have been changed from Eric's site in predictable ways, hemihexahedral hemioctahedral has been changed from haho to haheho, and hemitrihexagonal hemisquare has been changed from hathash to hathahash,. Other changes will be specifically noted.
48. Ricuhsquah - Rectified cubihemisquare honeycomb. Cells are thahs, coes, and squats. As you may imagine, this is indeed the rectified cuhsquah. Compare to ratho.
49. Retehtrah - Rectified tetrahemitriangular honeycomb. Cells are octs, ohoes, and thats. Symbol is x3o3x3o3/2*a3*c. This is only wythoffian under cyclotettic symmetry, because oho is only isogonal under tettic. This should hopefully be the last time I have to say that. Known as ridatha eons ago.
50. Rohtrah - Rectified octahemitriangular honeycomb. Cells are coes, choes, and thats.
51. Codaha - Cuboctahedral dihemiapeiratic honeycomb. Cells are coes, thas, and shas. This was previously known as cuhada.
52. Ocodaha - Octahedral cuboctahedral dihemiapeiratic honeycomb. Cells are octs, coes, thas, and shas.
53. Hahehodaha - Hemihexahedral hemioctahedral dihemiapeiratic honeycomb. Cells are thahs, ohoes, shas, and hohas.
54. Hahcohodaha - Hemihexahedral cuboctahedral hemioctahedral dihemiapeiratic honeycomb. Cells are thahs, coes, choes, shas, and hohas.
55. Hahcohohathahash - Hemihexahedral cuboctahedral hemioctahedral hemitrihexagonal hemisquare honeycomb. Cells are thahs, coes, ohoes, thats, and squats.
56. Hahehohathahash - Hemihexahedral hemioctahedral hemitrihexagonal hemisquare honeycomb. Cells are thahs, choes, thats, and squats.
57. Hahdhodahthahash - Hemihexahedral dihemioctahedral dihemitrihexagonal hemisquare honeycomb. Cells are thahs, ohoes, choes, thats in two orbits, and squats.
58. Hahdha - Hemihexahedral dihemiapeiratic honeycomb. Cells are thahs, thas, and shas.
59. Ohodaha - Octahemioctahedral dihemiapeiratic honeycomb. Cells are ohoes, thas, and hohas.
60. Chodaha - Cubohemioctahedral dihemiapeiratic honeycomb. Cells are choes, shas, and hohas.
61. Hahd Codaha - Hemihexahedral discuboctahedral dihemiapeiratic honeycomb. Cells are thahs, coes in two orbits, thas, and shas. I added a space here, previously known has hahdcodaha, which was weird to pronounce as one word.
62. Oco Ohodaha - Octahedral cuboctahedral octahemioctahedral dihemiapeiratic honeycomb. Cells are octs, coes, ohoes, hohas, and shas. Previously known as ohocodaha.
63. Ocochodaha - Octahedral cuboctahedral cubohemioctahedral dihemiapeiratic honeycomb. Cells are octs, coes, choes, thas, and hohas. Previously known as ocohodaha.
64. Hahdichohathadha - Hemihexahedral discubohemioctahedral hemitrihexagonal dihemiapeiratic honeycomb. Cells are thahs, choes in two orbits, thats, hohas, and shas. Previously hahdhohathadha.
65. Hahdhohasadha - Hemihexahedral dihemioctahedral hemisquare dihemiapeiratic honeycomb. Cells are thahs, ohoes, choes, squats, and hohas in two orbits. Previously hahdhohashadha.
66. Hahdohohathadha - Hemihexahedral disoctahemioctahedral hemitrihexagonal dihemiapeiratic honeycomb. Cells are thahs, ohoes in two orbits, thats, shas, and hohas. Previously hahohohathadha.
67. Dahohathadha - Dihemioctahedral hemitrihexagonal dihemiapeiratic honeycomb. Cells are ohoes, choes, thats, hohas, and shas.
68. Hahehohasadha - Hemihexahedral hemioctahedral hemisquare dihemiapeiratic honeycomb. Cells are thahs, choes, squats, thas, and hohas. Previously hahohashadha.
69. Hahehohathadha - Hemihexahedral hemioctahedral hemitrihexagonal dihemiapeiratic honeycomb. Cells are thahs, ohoes, thats, thas, and shas.
70. Cohohathadha - Cuboctahedral hemioctahedral hemitrihexagonal dihemiapeiratic honeycomb. Cells are coes, ohoes, thats, thas, and shas.
71. Ohohathadha - Octahedral hemioctahedral hemitrihexagonal dihemiapeiratic honeycomb. Cells are octs, choes, thats, thas, and shas.
72. Hahcohasadha - Hemihexahedral cuboctahedral hemisquare dihemiapeiratic honeycomb. Cells are thahs, coes, squats, and thas in two orbits.
These also have octettic symmetry, but as a podiumvert, not as a rectate. These can be considered analogous to sto and gotto. The subsymmetry allows the toes, usually confined to a noble compound, to split into two types and enter the picture. Names with disoctahedral were previously just octahedral.
73. Hotoh - Hemioctahedral tomooctahedral honeycomb. Cells are choes and toes.
74. Toha - Tomooctahedral hemiapeiratic honeycomb. Cells are toes and hohas.
75. Cotohath - Cuboctahedral tomooctahedral hemitrihexagonal honeycomb. Cells are coes, toes, and thats.
76. Hotoha - Hemioctahedral tomooctahedral hemiapeiratic honeycomb. Cells are ohoes, toes, and thas.
77. Dohotoh - Disoctahedral hemioctahedral tomooctahedral honeycomb. Cells are two types of octs, ohoes, and toes.
78. Dotohath - Disoctahedral tomooctahedral hemitrihexagonal honeycomb. Cells are two types of octs, toes, and thats.
These have cyclotettic symmetry, as a rectangular frustumvert. Think afdec, except the coes are octs now, so everything just becomes overwhelmingly supersymmetrical as a square prism. This can be seen as a combination of both octettic splittings, as it allows toes to split while also allowing coes and much of the internals to split as well. The symmetry can be seen as a rectified cyclotettic octet. The names with "disapeiratic" or "dishemiapeiratic" have been changed to "dihemiapeiratic", "da" to "dha"
79. Tohohath - Tomooctahedral hemioctahedral hemitrihexagonal honeycomb. Cells are ohoes, toes, and thats.
80. Dahotodha - Dihemioctahedral tomooctahedral dihemiapeiratic honeycomb. Cells are ohoes, choes, toes, thas, and shas.
81. Hotodha - Hemioctahedral tomooctahedral dihemiapeiratic honeycomb. Cells are choes, toes, hohas, and shas. Previously tohoda.
82. Cohotodha - Cuboctahedral hemioctahedral tomooctahedral dihemiapeiratic honeycomb. Cells are coes, choes, toes, thas, and hohas.
83. Cotohathadha - Cuboctahedral tomooctahedral hemitrihexagonal dihemiapeiratic honeycomb. Cells are coes, toes, thats, shas, and hohas.
84. Docohotodha - Disoctahedral cuboctahedral hemioctahedral tomooctahedral dihemiapeiratic honeycomb. Cells are two types of octs, coes, ohoes, toes, hohas, and shas.
85. Docohotohath - Disoctahedral cuboctahedral hemioctahedral tomooctahedral hemitrihexagonal honeycomb. Cells are two types of octs, coes, choes, toes, and thats.
86. Docotohathadha - Disoctahedral cuboctahedral tomooctahedral hemitrihexagonal dihemiapeiratic honeycomb. Cells are two types of octs, coes, toes, thats, hohas, and shas.
87. Dodhotodha - Disoctahedral dishemioctahedral tomooctahedral dihemiapeiratic honeycomb. Cells are two types of octs, ohoes, choes, toes, thas, and shas.
88. Dohotodha - Disoctahedral hemioctahedral tomooctahedral dihemiapeiratic honeycomb. Cells are two types of octs, ohoes, toes, thas, and hohas.
89. Dohotohathadha - Disoctahedral hemioctahedral tomooctahedral hemitrihexagonal dihemiapeiratic honeycomb. Cells are two types of octs, choes, toes, thats, thas, and shas.
C. Fitituoh - Fissal tetral tomooctahedral honeycomb. Cells are toes. This is a compound of 4 batches. Noble! This is one fourth of fituoh, you can get fituoh by placing one of these in each component of focoh.
The following honeycombs are self conjugate: all (trivially).
The following form conjugate pairs: none.
Category 2 --- Back to Honeycombs Home --- Category 4
Site by Benjamin Klein, based on the work of Eric Binnendyk
contact me via emailbenklein at gmail dot com or on the Polytope Discord
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