Then there exists a convex polyhedron (unique up to translation) having facet unit normals u 1, u 2. u k be unit vectors that span the space, and a 1, a 2. Without giving an explicit definition it's easy to recognize a convex polyhedron: all its "real diagonals" (segments joining two vertices which don't belong to the same face) are inside. Remark about the sum of all the face angles: (Σα ij)/2π = a-f = s-2 The defect theorem (the sum of the defects at all vertices of a polyhedron is equal to 4π) is closely related to Euler's formula it holds for any convex polyhedron (and more generally for any polyhedron homeomorphic to a sphere). The solid angle's defect δ of a polyhedron's vertex is the difference between 2π and the sum of face angles at that vertex: δ=2π-Σa i radians (π radians = 180°). Nice polyhedra can be built in many different ways: glued wood, welded copper wire, origami. Generalization to non convex polyhedra: f + v = e + 2 - 2t where t is the number of tunnels of the polyhedron (its genus). Only five (semi-)regular polyhedra "fill the space": the cube, the triangular prism, the hexagonal prism, lord Kelvin's polyhedron (truncated octahedron) and the rhombic dodecahedron (dual of the cuboctahedron). The two snubs (and theirs duals) have no plane of symmetry they exist in two mirror images forms. We notice the poverty of the tetrahedron family: the tetrahedron truncated using the midpoints of the edges is the octahedron, the snub tetrahedron is the icosahedron, but the truncation using the thirds of the edges gives the truncated tetrahedron. The great rhombicuboctahedron is also the truncated cuboctahedron likewise the great rhombicosidodecahedron is the truncated icosidodecahedron.Īmong the Platonic and Archimedean solids the cuboctahedron is the only polyhedron inscribed in a sphere with radius equal to its edge (as the hexagon is the only regular polygon inscribed in a circle with radius equal to its side). Likewise the 30 square faces of the "rhombicosidodecahedra" belong to the faces of a rhombic triacontahedron (dual of the icosidodecahedron). In "rhombicuboctahedron" the prefix "rhombi" points out that 12 square faces belong to the faces of a rhombic dodecahedron (dual of the cuboctahedron). Pentakis-dodecahedron (isosceles triangle) Triakis-dodecahedron (isosceles triangle) Here are the characteristics of the semi-regular polyhedra of the first kind (for their duals we have just to exchange v and f): One can prove this relation with the help of Schlegel diagrams.Ĭonvex polyhedra have many other numerical properties among which e+6 ≤ 3f ≤ 2e and e+6 ≤ 3v ≤ 2e Other polyhedra satisfy the Euler formula: polyhedra without "holes", and more generally those which are topologically equivalent to a connected network (without intersections) drawn on a sphere. Actually, the Euler-Poincare characteristic f+v-e of a convex polyhedra is 2 (Descartes-Euler theorem).ĭiscussions between a professor and his students about the hypothesis of Euler's conjecture allows Imre Lakatos to give us an interesting analysis about the mathematical knowledge in Proofs and Refutations (Cambridge University Press, 1976). Euler's formula Euler's polyhedral formula: f + s = a + 2Īll convex polyhedra verify this relation between the numbers of faces, of vertices and of edges.