Polyhedra 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.