This category consists of the honeycombs whose vertex figures are either triangular podiums or antipodiums. Well, loosely. Chon is a full octahedron, and cytatoh is a triangular antiprism, rather than an antipodium. But you get the idea. The first two regiments here, chon and cytatoh, get regiment members stolen from them due to supersymmetrical shenanigans. The sratoh and getdoca regiments get to live in peace though.
These are members of the chon regiment, under symmetry as x4o3o4x, which makes it a podivert. All hypercubic honeycombs will be allowed to become antipodites at some point in their career. The vertex figures of these are facetings of oct with triangular antiprismatic or pyramidal symmetry. Dacha and dapha have batchic symmetry, napca is only chonnic, as x4o3o4y.
154. Dacha - Dicubatihemiapeiratic honeycomb. Cells are cubes and shas. Symbol is o3o4x∞x4/3*b. The cubes here are only the bases of the antipodium.
155. Dapha - Disprismatohemiapeiratic honeycomb. Cells are cubes and shas. The cubes here are only the lacing triangles of the antipodium, as square prisms.
156. Napca - Spinoprismatocubatiapeiratic honeycomb. Cells are cubes of two types and shas. The cubes are half of the bases and half of the laces.
This is the only regiment with cyclotettic symmetry that doesn't immediately collapse to some much higher symmetry. Nonetheless, some of them, cytatoh itself, datha, and idatha, have a doubled cyclotettic symmetry (not octettic), which I guess I should call "cytatic symmetry" or something. This symmetry is even more unimportanter compared to batchic. We won't see anything else cytatic until the odd octets, and then after that never again. The other two things here have singular, normal cyclotettic symmetry. Cytatoh itself is convex! This regiment's verfs are facetings of a triangular antiprism, sometimes under only pyramidal symmetry. It's like spid.
157. Cytatoh - Cyclotruncated tetrahedral-octahedral honeycomb, Cells are tets and tuts. Symbol is o3o3x3x3*a. This is often called the bitruncated tetrahedral-octahedral honeycomb, or batatoh, even though this is not really true at all. It's also often called the "quarter-cubic honeycomb" which makes even less sense. It's cytatoh now.
158. Datha - Ditetrahemiapeiratic honeycomb. Cells are tets and thas. Symbol is o3o3x∞x3/2*b. Yes, the verf of datha is just two unit edged tetrahedra kissing. In fact, datteha is four unit-edged tetrahedra kissing. These tetrahedra are getting 𝓯𝓻𝓮𝓪𝓴𝔂.
159. Idatha - Inverted ditetrahemiapeiratic honeycomb. Cells are tuts and thas.
160. Tota - Tetratetraapeiratic honeycomb. Cells are tets, tuts, and thats.
161. Nateta - Spinotetratetraapeiratic honeycomb. Cells are tets, tuts, and hohas.
These have octettic symmetry. Their vertex figures are facetings of a triangular podium. There are subsymmetrics here, mostly scaliform, but one uniform we'll see in the miscs. Sratoh is convex!
162. Sratoh - Small rhombated tetrahedral-octahedral honeycomb. Cells are tets, cubes, and sircoes. Symbol is x3o4x *b3o.
163. Stedoca - Small tetradicubatiapeiratic honeycomb. Cells are tets, cubes, tics, and tosquats. Symbol is o3o3x4x4/3*a.
164. Canach - Cubatiapeirointercepted cubatic honeycomb. Cells are tics, tosquats, and sircoes.
165. Scoftaca - Small cubatifaceted tetracubatiapeiratic honeycomb. Cells are tets, tosquats, and soccoes.
166. Stafcadoch - Small tetrafaceted cubatidicubatic honeycomb. Cells are cubes, tics, and soccoes.
167. Santadoch - Small spinotetradicubatic honeycomb. Cells are tets, tics, and srohs.
168. Sandoca - Small spinodicubatiapeiratic honeycomb. Cells are cubes, tosquats, and srohs.
These have octettic symmetry. Their verfs are facetings of a triangular antipodium, These are very thin, due to containing quitsquats, but not so thin i'd consider them most spikiest contenders. There are subsymmetrics here, mostly scaliform, but one uniform we'll see in the miscs.
169. Getdoca - Great tetradicubatiapeiratic honeycomb. Cells are tets, cubes, quiths, and quitsquats. Symbol is x3o3o4x4/3*a.
170. Qratoh - Quasirhombated tetrahedral-octahedral honeycomb. Cells are tets, cubes, and quercoes. Symbol is x3o4/3x *b3o.
171. Dacca - Dicubatiapeiratic honeycomb. Cells are quiths, quitsquats, and quercoes.
172. Gacoftaca - Great cubatifaceted tetracubatiapeiratic honeycomb. Cells are tets, quitsquats, and goccoes.
173. Gotafcadoch - Great tetrafaceted cubatidicubatic honeycomb. Cells are cubes, quiths, and goccoes.
174. Gantadoch - Great spinotetradicubatic honeycomb. Cells are tets, quiths, and grohs.
175. Gandoca - Great spinodicubatiapeiratic honeycomb. Cells are cubes, quitsquats, and grohs.
The following honeycombs are self conjugate: dacha, dapha, napca, cytatoh, dapha, idatha, tota, nateta (all trivially).
The following form conjugate pairs: stedoca-getdoca, sratoh-qratoh, canach-dacca, scoftaca-gacoftaca, stafcadoch-gotafcadoch, santadoch-gantadoch, sandoca-gandoca.
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