### Math Monday: Boxtahedra

by Glen Whitney for the National Museum of Mathematics

There just happen to be a whole bunch of different-colored but otherwise identical cubical boxes lying around the MoMath Mondays making lab today. These make excellent fodder for geometric journeying on a relaxed morning, so let’s see what we can make. A simple way to connect them is to stick a classic brass paper fastener out one corner of one box, and into a corner of a different box, and then splay the two sides of the fastener.

One of the simplest linked structures you can make is a triangle of three boxes. But don’t stop there; keep adding more boxes. Here are some hints to keep you going: at least for me, it’s a lot easier to punch the paper fastener through the box from the inside out (rather than outside in), so on the receiving corner of a connection, first punch a fastener through from the inside and then pull it out and insert the pointy end of the fastener from the other box from the outside through the slot you just made. Also, if some of the splayed ends of the fasteners start to pull out from the boxes, you can tape them in place inside the boxes; once you close the lids, nobody will ever know.

If you keep going making only triangular openings among the boxes, pretty soon your creation will close up into this lovely “cuboxtahedron.”

If, on the other hand, you alternate triangular and square openings, then you will end up with a “rhombicuboxtahedron,” like this.

But even better yet, thanks to the existence of the cantellated cubic honeycomb, you can combine your two creations, linking them with more boxes, like so.

To get this far, you will at times need to connect to the lid end of some of the boxes. The best way to do that is to splay the pointy end of the paper fastener that will be going into the lid, tape one of the two prongs to the inside of the box, close the lid tightly over the fastener, and then tape the lid closed. (Note that this procedure requires that you have all of the connections to the lid side of the box in place before you close up the lid.) Another great thing about this honeycomb is that it is super-rigid and sturdy, so you can keep building wide and tall and your structures will stand on their own.

Send photos of your boxtahedral creations to mondays@momath.org!