Convex uniform honeycomb

They can be considered the three-dimensional analogue to the uniform tilings of the plane.

The individual honeycombs are listed with names given to them by Norman Johnson. (Some of the terms used below are defined in )

Compact Euclidean uniform tessellations (by their infinite Coxeter group families)

In addition there is one special elongated form of the triangular prismatic honeycomb.

The total unique prismatic honeycombs above (excluding the cubic counted previously) are 10.

Combining these counts, 18 and 10 gives us the total 28 uniform honeycombs.

The gyroelongated triangular prismatic tiling has the same vertex figure as one of the plain prismatic tilings; the two may be derived from the gyrated and plain triangular prismatic tilings, respectively, by inserting layers of cubes.

There are only 3 unique honeycombs from the square tiling, but all 6 tiling truncations are listed below for completeness, and tiling images are shown by colors corresponding to each form.

From these 9 families, there are a total of 76 unique honeycombs generated:

There are also 23 paracompact Coxeter groups of rank 4. These families can produce uniform honeycombs with unbounded facets or vertex figure, including ideal vertices at infinity: