The net

What is the net of a poly­he­dron? It is simply a sheet of card­board whose folding gives the poly­he­dron. – May you answer. This is true, but there’s more. The concept of the net of a poly­he­dron contains some­thing more than just a sheet of card­board.

Which poly­he­dron can be achieved by folding the familiar Latin cross? The cube, of course. For this we should colour the edges, as our magic brush did (the edges of the same colour are glued to each other in the poly­he­dron).

However, it would be better to colour with different colours not the edges, but each pair of points. This should corre­spond to giving, as it is said in math­e­matics, the condi­tions of edges’ gluing.

After that the condi­tions of edges’ gluing are given, the edges located inside the sheet of card­board are uniquely defined, according to a theorem by A.D. Alek­san­drov.

So from the Latin cross, you can get a cube.

But it happens that if the condi­tions are given other­wise, you can get some­thing but a cube!

Our magic brush has coloured the edges here’s how. A final stroke of his and we already know how to define the edges within the piece of card­board. Then we will construct a poly­he­dron, following the condi­tions of gluing just designed: we get a pyramid!

Not long ago it was shown that giving different condi­tions of gluing the edges of the Latin cross, you can get 5 different types of convex poly­hedra.

So, as we have seen, the concept of net of a poly­he­dron dos not consist just of a sheet of card­board, but also of the gluing condi­tions of its edges. If these condi­tions are not defined, then from the same piece of card­board you may get different convex poly­hedra.