This is the third and final polyhedron described by Coxeter in his 1937 paper - "Regular Skew Polyhedra in Three and Four Dimensions and their Topological Analogues", Proc. London Math. Soc. 43, ser 2, 33-62, 1937. The first two are modeled in thing:2884052 and thing:2899221.

This polyhedron is composed of hexagons; the vertices of 6 hexagons meet at a point to form a 3 dimensional pattern. It is convenient to consider it as the four hexagonal faces of a truncated tetrahedron with the trigonal faces absent. Here are two models which illustrate how this pattern occurs.

Six_hexagons.stl is composed of three truncated tetrahedra taken from the infinite array. It clearly shows a point where 6 hexagonal vertices meet. Eight_tts.stl is composed of 8 truncated tetrahedra chosen to be easily and quickly printed. Minimal models which illustrate the packing of the two earlier polyhedra have been added for comparison. They are six_squares.stl and four_hexagons.stl.

OpenSCAD Code is included.

MakerBot

MakerBot Replicator (5th Generation)

Yes

No

0.2 mm

Eight_tts.stl, six_hexagons.stl. and four_hexagons.stl have flat bottoms and need no support. A shorter raft to model distance would help eight_tts.stl adhere to the raft. Six_squares.stl needs some support.

This pattern is a member of space group 227 with four hexagons as the repeating unit. Four hexagons arranged like those in the truncated tetrahedron can serve as the unit. This unit is 1/4 of the cell in the International Tables.rnrnThe two earlier polyhedra are both members of space group 229. Six_squares.stl and four_hexagons.stl each contain two copies of the smallest repeating unit. These repeating units are 1/2 the size of the cell in the International Tables. They are parts of larger arrays chosen to show points where 6 square or 4 hexagonal vertices meet. See thing:2884052 and thing:2899221.