# Joined Truncated Octahedrons

2017/08/05

Truncated octahedrons pack the space without holes. You can join them at their square or hexagonal faces consistent with this packing. Jared McComb named such pieces Polytrocs and asked me to generate and count them. After I had written a program to do the job, I found Marc Owen's site where the pieces of order four and less are shown and animated. Back in 1986 Matthew Richards and Marc Owen coined the name Splatts for the pieces and suggested some constructions with them. In 2013 Marc Owen counted the pieces up to order 12 (OEIS A038180). I tried to add some new constructions and wanted to create real pieces with my recently assembled 3D printer.

Getting computer solutions is no problem but putting real pieces of order 4 together is hard.
• Even if the layers are given it's difficult to spot the right piece in a set of 44 tetratrocs. So I numbered the pieces and stick little labels on them.
• The computer program doesn't check if one piece gets through another one or if pieces are wrapped around each other. With polytrocs of order 4 or less the first case can only happen for the square tetratroc. But the second case applies for more pairs of pieces.
• Overhanging pieces can slide down if not fixed by other parts. Sometimes this problem can be prevented by turning the whole construction upside down.
If a real construction failed I took another computer solution and had to restart.

Two different layer types are used. If the squares of the truncated octahedrons are at the bottom of a figure, we get a square grid for all layers and the projections of the units are octagons. If the edges of the pieces are at the bottom, the grid of the layers is hexagonal, so are the projections of the units.

For the following sets I found some constructions. Click the sets to see pictures and solutions.

Set Number of Truncated Octahedrons Number of Pieces Total Volume
Tritrocs 3 6 18
Polytrocs of Order 3 or Less 1..3 9 23
Tetratrocs 4 44 176
Polytrocs of Order 4 or Less 1..4 53 199

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