Cube capacity

Regular poly­hedra are natu­rally related to each other. This is both the duality of regular poly­hedra and all possible ways to inscribe one poly­he­dron into another. Yet the apparent encounter with this inter­re­la­tion is some­times surprising.

The basis of the model, which amazes both chil­dren and adults, is a cube without a top edge, made of glass or acrylic. The set includes several poly­hedra, which are better cut from a light­weight mate­rial, such as foam. If the size of the cube is large, the poly­hedra should have holes for fingers.

The most amazing poly­he­dron which can be put into a cube is a "large" tetra­he­dron. At first glance there is no way it could fit inside, but that is if we do not know that a tetra­he­dron can be inscribed into a cube so that all the vertices of the tetra­he­dron coin­cide with the vertices of the cube. As a vari­a­tion one can take two "oppo­site" tetra­he­drons. Such a poly­he­dron — the union of these tetra­he­drons — was consid­ered by Johannes Kepler, and he named it "stella octan­gula".

The crossing of the cube and its dual octa­he­dron, enlarged so that their edges inter­sect, is a cubo-octa­he­dron. This semi­reg­ular poly­he­dron appears in both the Star Trek movies and the Elite video game. Being obtained as a trun­ca­tion of a cube, it can be placed inside the cube.

It is inter­esting and useful to see "live" the way another regular poly­he­dron, the icosa­he­dron, fits into the cube: an icosa­he­dron edge will lie "in the centre" of each face of the cube.

And how many chil­dren can fit into a cubic metre? If you conduct an exper­i­ment by making a safe cube with an edge of one meter and no top edge, chil­dren will have fun and you will be surprised with the result!