Uniform Honeycombs


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Honeycombs are three dimensional tesselations. Being able to fit in three dimensional space, honeycombs do receive quite a lot of study, though not as much as tilings. These are similar to the uniform polychora, and their existence in three dimensions makes for a good introduction to the uniform polychora in a more visualizable space. The list of uniform tilings is thought to be complete by academic sources (but not proven), but the list of uniform honeycombs Does Not Exist in academic sources, so that gives you an idea of how little we know about completeness. As always, inspiration is taken greatly from Jonathan Bowers, especially his list of uniform polychora, but also massive credits to Eric Binnendyk, who created the original honeycomb pages. This page and many of the category pages are essentially just regurgitations of his site with added corrections, details, and new discoveries.


Uniform Honeycomb Categories

Category A: Polygon-Apeirogon Duoprisms - This is the infinite set of duoprisms between an apeirogon and any regular polygon (note however that the apeirogonal duoprism is degenerate). Each of them looks like a column of n-gonal prisms, with apeirogonal prisms on the sides.

Category B: Blendic Infinitudes - It has been shown that there are infinitely complicated blends of many octets and possibly even riches. These work by finding edge lengths within the vertices of octet or chon that correspond to multiple orientations of larger octets, like how edges of length 5 in the vertices of squat create 2 types of squats in different orientations, and then finding that some of them blend. This process can then be repeated on the new sub-octets, leading to the sort of infinite recursive blending that makes orbiform enthusiasts cry. I haven't independently verified this, but I trust _Geometer with my life.

Category 1: Primaries - The two members of the chon regiment with full (o4o3o4x) or half (o3o4x *b3o) symmetry, as well as members of the octet regiment with full (x3o4o *b4o) or half (x3o3o3o3*a) symmetry. Octet, the 3D demicubic, quarter-cubic, and cyclo-simplex honeycomb, makes up the vast majority of this category.

Category 2: Truncates - These are the truncates of the honeycombs from category 1 that have uniform vertex figures. Also included here is the bitruncate, batch.

Category 3: Rich Regiment - Rich is the rectified cubic honeycomb. Its regiment contains 49 typical members, whose vertex figures are facetings of square prisms.

Category 4: Sphenoverts - These are the honeycombs with wedge-shaped vertex figures and their regiments (under only symmetry as a sphenovert, they all have subsymmetrics elsewhere).

Category 5: Greater Truncates - These are the great rhombates and great prismates and their kin. The vertex figures are various types of irregular tetrahedra.

Category 6: Prismatorhombates - These are the prismatorhombates (runcicantellates) and others with similar vertex figures, and their regiments. Their vertex figures are trapezoid pyramids.

Category 7: Triangular Podiumverts - These are the honeycombs with triangular podia or antipodia as their vertex figures, and their regiments. Also includes the chon members that exist under this symmetry.

Category 8: Gacoca Regiment - The gacoca regiment has 66 members (under typical symmetries) whose vertex figures are facetings of a square (rectangular) frustum. This regiment has some atypical subsymmetrics and blends, with both types getting their own categories later.

Category 9: Skewverts - These are the honeycombs with skewed wedges as vertex figures, and their regiments. Also includes the few regiment members of sphenoverts taken under skewvert subsymmetry. Both of the skewvert regiments have blending shenanigans in their own unique ways.

Category 10: Stut Cadoca Regiment - This category contains the 79 typical members of the stut cadoca regiment, as well as 30 fissaries. There are subsymmetrics and blends we will be seeing later.

Category 11: Prisms - This category contains the 52 prisms of uniform tilings. The azip prism is excluded for being in category A.

Category 12: Sabdiptica Regiment - This is a kaleidoscopic non-Wythoffian regiment formed by blending skivcadach from category 9 with quiprich from category 6. Think of it as a euclidean version of sabbadipady, except the verf can be faceted under only bilateral symmetry. It contains 168 members and 77 fissaries (though two of these fissaries are counted earlier in C5).

Category 13: Gacoca Subsymmetrics - This category is formed by taking gacoca under a nonwythoffian subsymmetry that allows the sircoes and goccoes to use their inner stops and ops. It can also be thought of as blends of gacoca members with compounds of 2 skivpacoca members. There were 46 of these, but new research has found further uncounted subsymmetries, so this will increase.

Category 14: Stut Cadoca Subsymmetrics - This category is ,, formed by taking stut cadoca under a nonwythoffian subsymmetry that allows the sircoes and goccoes to use their inner stops and ops. It can also be thought of as blends of stut cadoca members with compounds of sossa prisms. There are a large number of these, probably.

Category 15: Wavicac Blends - This category consists of the members of the wavicac superregiment, which sees blends of wavicac members with sossa prismatic honeycombs and with cuhsquahs. Also thrown in here at the beginning are the non-superregimental but still nonwythoffian wavicac subsymmetrics, which aren't numerous enough to get a category. There are 8 non-superregimentals and 30 superregimentals here, and 22 superregimental fissaries.

Category 16: Gacoca Blends - This category consists of members of the gacoca superregiment, which sees blends of gacoca members with cuhsquahs. There are only 20 of these under typical symmetries, but there is an unknown but seemingly large number abusing the subsymmetrics.

Category 17: Stut Cadoca Blends - This category consists of members of the stut cadoca superregiment, which sees blends of the stuts with octet regiment members. The subsymmetry comes into play severely here. Probably a huge category.

Category 18: Sabdiptica Blends - This category consists of members of the sabdiptica superregiment, which sees blends of sabdiptica members with rich members and sha prisms. This category has not even begun counting, but it is presumed to be utterly massive, possibly thousands exist.

Category 19: Prismatic Honeycombs - This category contains the stacks of prisms of uniform tilings, as well as any regiment members they have. These are usually formed by blending prismatic stacks with various arrangements of CA and C11 items. Also included here is a vast array of odd facetings of chon.

Category 20: Miscellaneous Slabs - This category contains other uniform prismatoids that do not belong in previous categories. The triangular tiling antiprism is included here, as well as some blends it forms.

Category 21: Gyrates and Elongates - This category forms from alternating or twisting stacks consisting of C11 members and C20 members (and C19 based on azap). The gyrated, elongated, and gyroelongated octet are here, and are our last convex honeycombs.

Category 22: Odd Octets - This category consists of all octet regiment members with strange nonwythoffian symmetries. This includes octet's members as stacks of trataps, under demi-brick symmetry, and any other odd symmetries discovered.

Category 23: Miscellaneous - This category contains strange outliers and small nonwythoffian regiments. Included here is apsirsh, our favorite euclidean ike snub, as well as iquipadah-like objects in some of the c7 regiments, and a few blends involving sacpaca from the prismatorhombates.


Other Pages

Here are other people's pages on similar polytopes.

List of uniform honeycombs by Eric Binnendyk. Eric's pages were the main inspiration for mine, infact many of my pages here are just updated versions of Eric's.

Uniform Polychora and Other Four Dimensional Shapes by Jonathan Bowers

List of hyperbolic honeycombs by lllllllllwith10ls

List of uniform honeycombs on hi.gher.space, the thread where many early uniform honeycomb discoveries were announced.

Username5243's honeycombs spreadsheet - Username5243 helped coin a lot of the names for these honeycombs and set up the first categorization.

Site by ThePokemonkey123, based on the work of Eric Binnendyk
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