Glue a pentagonal pyramid onto every face of a dodecahedron

Glue a pentagonal pyramid onto every face of a dodecahedron. Let\'s call the resulting solid S. How many vertices, edges, and faces does S have? Verify Euler\'s identity for S. Why is S not a perfect solid, even though every face is a triangle?

Solution

Faces

12

Edges

30 (6 + 24)

Vertices

20 (8 + 12)

Euler\'s identity => F + V E = 2

So 12+20-30=2 hence proved

it is impossible to make for four (or more) pentagonal faces to meet at a vertex, because they subtend more than 360 degrees. For hexagonal faces, three hexagons meeting at a point create another \"infinite solid\". It\'s also stated that no higher-order polygon can yield a solid, so the five solids already mentioned - tetrahedron, hexahedron, octahedron, icosahedron, and dodecahedron - are the only regular polyhedrons. It is not only proved that these solids exist, and that they are the only perfectly symmetrical solids.

Faces	12
Edges	30 (6 + 24)
Vertices	20 (8 + 12)