Uniform tilings . . . Uniform tetracombs

This is an attempt to list all known uniform Euclidean 3-dimensional honeycombs. The style of this list and the names were inspired by Jonathan Bowers' list of uniform polychora, which you should really check out.

For the purposes of this webpage, a **honeycomb** is a tessellation of polyhedra (cells) in flat 3-dimensional space. The polyhedra can overlap, and their faces can intersect, as long as every cell touches exactly one other cell at each of its faces. Cells can be infinite tilings of the plane as well as finite polyhedra, and faces can be apeirogons (an endless row of line segments) as well as polygons.

Formally, a honeycomb is an abstract polytope of rank 4 embedded in 3-space. In particular, this means that:

- It is strictly connected (it is not a compound, and neither are elements, element-figures, or element-figures of elements)
- It is dyadic (all ridge figures, ridge figures of a cell, ridge figures of a face, and edges have two vertices)

In order to be a proper honeycomb, I also require that:

- Two distinct elements cannot share the same set of ridges (a property sometimes called coincidicity)
- Two edges cannot have the same endpoints (this prevents digons from occurring).

Each vertex is identified with a point in Euclidean 3-space, satisfying the following conditions:

- Each face is planar.
- It is monal, meaning no two vertices overlap (this rules out fissary cases)

In order for a honeycomb to be **uniform** (by my definition), it must have the additional properties:

- It is vertex transitive
- Every cell is a uniform polyhedron or uniform tiling of Euclidean 2-space. In particular, every element of rank
*n*can be either inscribed in an (*n*- 1)-sphere or lies in flat (*n*- 1)-space.

Finally, in order for a honeycomb to not be placed in Category B and excluded in the main count, it must not be a **subdivision**: it cannot have two adjacent cells that are circumscribable in the same sphere or embeddable in the same plane. Equivalently, the vertex figure cannot have two coplanar faces meeting at an edge.

**Category A: Polygon-apeirogon duoprisms** - This is the infinite set of duoprisms between an apeirogon and any regular polygon (not counting an apeirogon). Each of them looks like a column of n-gonal prisms, with apeirogonal prisms on the sides.

**Category B: Subdivisions** - These are the honeycombs with multiple apeiratic cells in the same plane. Their vertex figures have coplanar faces. For each honeycomb in this category, there is an analogous honeycomb somewhere else where each stack of apeiratic cells is replaced by a single apeiratic cell.

**Category 1: Primaries** - The two members of the chon regiment with full (o4o3o4x) or half (o3o4x *b3o) symmetry, as well as the fully symmetric members of the octet regiment.

**Category 2: Truncates** - These are the truncates of the honeycombs from category 1 that have uniform vertex figures.

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

**Category 4: Sphenoverts** - These are the honeycombs with wedge-shaped vertex figures and their regiments.

**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.

**Category 8: Gacoca regiment** - The gacoca regiment has 66 members whose vertex figures are facetings of a square podium (frustum).

**Category 9: Skewverts** - These are the honeycombs with skewed wedges as vertex figures, and their regiments.

**Category 10: Ditetrahedronaries** - These are the members of the octet regiment with cyclotettic symmetry. They have tetrahedrally symmetric vertex figures.

**Category 11: Stut Cadoca regiment** - This category contains the 79 members of the stut cadoca regiment.

**Category 12: Prismatic honeycombs and their regiments**

**Category 13: Prisms**

**Category 14: Other slabs**

**Category 15: Gyrates and elongates**

Here is a spreadsheet of all known uniform honeycombs. Some of the Bowers-style acronyms are official, but most are unofficial.

Here are other people's pages on similar polytopes.

**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 most uniform honeycomb discoveries were announced ("polychoronlover" is me)

**Username5243's honeycombs spreadsheet** - Username5243 helped coin most of the names for these honeycombs and set up the categorization.

*The vertex figure images were created using Miratope, a polytope rendering software by the Miratope team.*