Look inside this box.

What do you see reflected in the side mirrors?

This reflective cylinder correctly displays the deformed flat image

The geometric motif in the background is reflected multiple times by the 4 lateral mirrors and is displayed, a huge spherical surface.  

What mathematics are there?

La caixa és un tros de la piràmide de base quadrada que tindríem si allarguéssim els costats laterals fins que es trobessin en un punt, aquesta figura s’anomena tronc de piràmide.

Una qüestió interessant sobre l’esfera de la qual en visualitzem una part, és que no es pot construir físicament. En efecte no existeix un poliedre amb totes les cares quadrilàters.

This impossibility can be deduced from euler's well-known formula between C faces, vertex V and A edges of convex polyhedras, which says:


If all C-sides are quadrilateral, there would be 4· C vertices of polygons, but since in the vertices coincide 4 polygons, the vertices V will be 4· C/4= C

And as for the edges there would be 4 sides for each side, that is, 4· C, since the edges are made up of 2 sides, the number of A-edges must be 4. C/2 =2C

Replacing these equivalences of vertices V and edges A in Euler's formula would remain

C+C=2C +2

Which is clearly wrong.