First in three dimensions. Here's a (not very good l-) picture of a cube:
A cube has six faces (one on top, one at the bottom and four round the sides); eight vertices (i.e. corners - four at the top and four at the bottom); and twelve edges (four at the top, four at the bottom, and four vertical ones up the sides). If we add up the number of faces and vertices we get 6+8=14, which is two more than the number of edges.
Here's a tetrahedron instead:
Here there are four faces (the bottom face and the three sides), four vertices (one at the top, three at the bottom), and six edges (three round the bottom and three going up to the vertex at the top). Adding the faces and vertices we get 4+4=8, which is again 2 more than the number of edges.
It turns out that this relation faces + vertices = edges + 2 holds for all polyhedra.
Finally, if we just look at a square on a sheet of paper:
There are clearly four edges and four vertices (corners) but we only get the two faces we need for F+V=E+2 if we count the outside of the square as one face. For a square this isn't particularly interesting but it's more useful for more complicated diagrams.
(If you have two totally unconnected squares then it doesn't hold either, but I think we can sweep that under the carpet for the purposes of deconstructing silly quizes.)
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Date: 2003-02-15 07:58 am (UTC)First in three dimensions. Here's a (not very good l-) picture of a cube:
A cube has six faces (one on top, one at the bottom and four round the sides); eight vertices (i.e. corners - four at the top and four at the bottom); and twelve edges (four at the top, four at the bottom, and four vertical ones up the sides). If we add up the number of faces and vertices we get 6+8=14, which is two more than the number of edges.
Here's a tetrahedron instead:
Here there are four faces (the bottom face and the three sides), four vertices (one at the top, three at the bottom), and six edges (three round the bottom and three going up to the vertex at the top). Adding the faces and vertices we get 4+4=8, which is again 2 more than the number of edges.
It turns out that this relation faces + vertices = edges + 2 holds for all polyhedra.
Finally, if we just look at a square on a sheet of paper:
There are clearly four edges and four vertices (corners) but we only get the two faces we need for F+V=E+2 if we count the outside of the square as one face. For a square this isn't particularly interesting but it's more useful for more complicated diagrams.
(If you have two totally unconnected squares then it doesn't hold either, but I think we can sweep that under the carpet for the purposes of deconstructing silly quizes.)