09 Apr

Tessellating Truncated Octahedra

With all of the recent activity at Studio Infinity on geometric units that can be automatically cut and scored, it was natural for the S∞ G4G14 giveaway to be the 14-sided Truncated Octahedron, which tessellates to fill space. The additional challenge here as compared to some of the earlier interlocking structures was to ensure that the resulting polyhedral units would be able to connect arbitrarily face-to-face, so that it’s possible to explore the three-dimensional structures you can build with multiple truncated octahedra.

This was accomplished by two measures. First, by leaving the square faces empty (fortunately the truncated octahedron is still rigid with its square faces deleted) and putting connection tabs similar to ITSPHUN units on the corresponding edges. That step allows any square face to connect to any other square face, in any of the four possible orientations.

Second, I cut out a triangle (not a hexagon!) from the center of each hexagonal face, and added similar connection tabs to those. That allows any hexagonal face to connect with any other hexagonal face, but only in the three orientations that allow the space-filling tessellation to continue.

These ideas result in the following SVG template:

The black lines are cuts and the magenta dashed lines are scores (all for mountain folds). If you are cutting directly from this SVG, you can also cut just the dashes of the magenta lines on very low pressure/high speed to create scores that fold well.There are two units because it takes two to create a single truncated octahedron, and to illustrate how they interleave for cutting, leaving very little unused material. (The outermost protruding tabs are for connecting these two pieces into one polyhedral unit.) In case it’s helpful, here are a PDF and the DXF design file for this unit.

See the previous post for full assembly instructions. I’ll leave you with an image of the smallest loop one can make, with six completed units.

If you build an interesting structure with these units, please post a picture in the comments!

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