## Regular polytopes |

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a-form analogous axes belong bounding hyperplanes cells centre circum-radius compound congruent transformation convex coordinates corresponding cos2 Coxeter cross polytope cube cuboctahedron defined denote diagonals dihedral angle direct dodecahedron edges equal equation equatorial polygon equilateral Euler's Formula faces finite groups follows four dimensions four-dimensional fundamental region Gosset graph group of order half-turn hedron Hence honeycomb hyperplanes icosahedron infinite inscribed integers invariant inversion ir/p ir/q isomorphic lattice lines of symmetry measure polytope mid-points nodes obtain occur octahedra octahedron opposite vertices orthogonal orthoscheme P0 Px parallel pentagonal permutations perpendicular Petrie polygon plane Platonic solids poly tope polyhedra projection pyramids reciprocal reflections regular polyhedron regular polytope rhombic rhombs rotation group Rx R2 Schlafli symbol Schoute sections segments sides skew polygon space sphere star-polytopes stella octangula stellated subgroup symmetry group tetrahedron theorem triangles truncation vectors vertex figure w-dimensional zonohedron