**Elementary Honeycombs**

It has long been recognised that space can be filled with uniform polyhedra. A 'uniform honeycomb' is a figure defined as a tiling of 3space where "all of whose cells are uniform polyhedra and whose vertices are transitive on its symmetry group, and which covers three-space exactly once" [1]. There are 28 known figures, these are listed by Banko Gr°nbaum in "Uniform tilings of 3-space" [2]. Images and models of all 28 uniform honeycombs are presented by Alex Doskey on his page at http://www.doskey.com/polyhedra/UniformHoneycombs.html. [3]

Inspired by Alex's work, I have begun to explore a similar set of honeycombs, which I term 'elementary honeycombs'. In place of the requirement that all cells are uniform polyhedra, I require each cell to be a regularly faced, elementary polyhedron: one which "cannot be separated by a plane into two smaller regularly-faced polyhedra" [4]. Candidates for such polyhedra include certain platonic and archimedean polyhedra and some of the Johnson Solids [5]. The restrictions that vertices are transitive on the symmetry group, and single coverage of three-space are unchanged.

Of the 28 uniform honeycombs, eleven are formed by taking a regular tessellation of the plane and replacing the polygons with prisms. These are then stacked to fill space. A further example is formed by gyrating alternative rows of triangular prisms, this is shown on Alex Doskey's page as number 23. Two more examples are formed by alternating rows of cubes and triangular prisms, these are shown on Alex Doskey's page as numbers 24 and 25. The two honeycombs differ as the alternate rows of triangular prisms can be set either parallel or perpendicular to each other.

In addition to the fourteen 'regular elementary honeycombs', I have discovered a further 13 examples. These are described in some detail below.

My notation for the elementary honeycombs is taken from the cells surrounding a vertex. The notation xA1-yA2 means that there are x examples of polyhedron A1 and y examples of polyhedron A2 around each vertex. The notation for individual cells is as follows:

**Pn** - n-gonal prism, including P3=triangular
prism and P4=cube.

**Qn** - n-gonal cupola, Q3=triangular cupola*,
Q4=square cupola.

**Sn** - n-gonal anti-prism, S3=octahedron*
(*=not elementary)

**T** - tetrahedron

**Yn** - n-gonal pyramid, Y4=square pyramid.

Where a non-regular polyhedron is used in the
formation of a honeycomb, a restriction has to apply to preserve vertex
transitivity. If the polyhedron has type A vertices and type B vertices in
the ratio a:b, then the number of type A vertices and the number of type B
vertices at any one vertex of the honeycomb must also be in the ratio a:b.
This means that the number of such polyhedra at each vertex of the honeycomb
must be a multiple of (a+b). For example, the square
pyramid has one apex of the form (3^{4}) and four base vertices of
the form (4°3^{2}). Any honeycomb containing square
pyramids must have a multiple of 5 at each vertex, of which 1/5 must have
apex vertices at the honeycomb vertex in question. The notation for such
honeycombs (such as 10Y4-8T) is therefore sufficient to define the number of
each type of cell vertex at the honeycomb vertex.

The following elementary honeycombs are presented in sequence

**Page 1:
Honeycombs derived from the octahedral/tetrahedral honeycomb part 1**

10Y4-8T-n

5Y4-4T-4P4

5Y4-4T-6P3-sq-ortho

5Y4-4T-6P3-sq-gyro

**Page
2: ****Honeycombs
derived from the octahedral/tetrahedral honeycomb part 2**

10Y4-8T-ortho

10Y4-8T-meta

10Y4-8T-para

**Page 3: ****Honeycombs
derived from the octahedral/tetrahedral honeycomb part 3**

5Y4-4T-6P3-tri-n

5Y4-4T-6P3-tri-ortho

5Y4-4T-6P3-tri-meta

5Y4-4T-6P3-tri-para

**Page 4: Honeycombs
derived from the runcinated alternated cubic honeycomb**

3Q4-T-2P8-P4

6Q4-2T

**Page
5: Honeycombs derived from the rectified cubic honeycomb (not elementary)**

6Q3-2S3-gyro

6Q3-2S3-ortho

3Q3-S3-2P6-2P3-gyro

3Q3-S3-2P6-2P3-ortho

Dr Richard Klitzing has kindly provided incidence
matrices for all of the above honeycombs. These can be found here,
or by clicking on the text '**Incidence Matrix**' by each honeycomb.

__References__

[1] Olshevsky G. (as quoted by Alex Doskey) - see [3].

[2] Gr°nbaum B. (1994) Uniform tilings of 3-space," Geombinatorics 4, 49-56

[3] Doskey A. [Online], available from http://www.doskey.com/polyhedra/UniformHoneycombs.html

[4] Cromwell P. (1997) Polyhedra, Cambridge University Press

[5] Johnson N.W. (1966) Convex Solids with Regular Faces, Canadian Journal of Mathematics, 18, 169-200