A regular solid with hexagonal faces cannot exist because if it did, the sum of the angles of any three hexagonal corners that met would already equal 360°, so such an object would be planar.
General properties of the Platonic solidsLet V = number of vertices, F = number of faces, E = number of edges, M = number of faces meeting at a vertex, N = number of edges and vertices associated with a face, and A = number of angles. Using Euler's formula for polyhedra, F - E + V = 2, allows us to complete the table below.
Each of these solids possesses an inscribed and a circumscribed sphere, which has the same center O. Further, the mid-points of all the edges of a Platonic solid also lie on a sphere again with center O. If we construct the inscribed sphere of a Platonic solid and join neighboring points of contact of the sphere with the faces of the polyhedron, there results within the sphere another regular polyhedron, which has the same number of vertices as the original solid has faces, and the same number of edges as the original solid. The cube yields an octahedron, the icosahedron a dodecahedron, and the tetrahedron another tetrahedron.
Related category SOLIDS AND SURFACES
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