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[W] Wenninger, 1974, has 119 figures: 1–5 for the Platonic solids, 6–18 for the Archimedean solids, 19–66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67–119 for the nonconvex uniform polyhedra.
Order-5 5-cell honeycomb; 120-cell honeycomb; Order-5 tesseractic honeycomb; Order-4 120-cell honeycomb; Order-5 120-cell honeycomb; Order-4 24-cell honeycomb; Cubic honeycomb honeycomb; Small stellated 120-cell honeycomb; Pentagrammic-order 600-cell honeycomb; Order-5 icosahedral 120-cell honeycomb; Great 120-cell honeycomb
{3,4} Defect 120° {3,5} Defect 60° {3,6} Defect 0° {4,3} Defect 90° {4,4} Defect 0° {5,3} Defect 36° {6,3} Defect 0° A vertex needs at least 3 faces, and an angle defect. A 0° angle defect will fill the Euclidean plane with a regular tiling. By Descartes' theorem, the number of vertices is 720°/defect.
Vertex the (n−5)-face of the 5-polytope; Edge the (n−4)-face of the 5-polytope; Face the peak or (n−3)-face of the 5-polytope; Cell the ridge or (n−2)-face of the 5-polytope; Hypercell or Teron the facet or (n−1)-face of the 5-polytope
Alternatively, if you expand each of five cubes by moving the faces away from the origin the right amount and rotating each of the five 72° around so they are equidistant from each other, without changing the orientation or size of the faces, and patch the pentagonal and triangular holes in the result, you get a rhombicosidodecahedron ...
A regular polyhedron is identified by its Schläfli symbol of the form {n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids ), and four regular star polyhedra (the Kepler–Poinsot polyhedra ), making nine regular polyhedra in all.
Vertex configurations [4] Faces [5] Edges [5] Vertices [5] Point group [6] Truncated tetrahedron: 3.6.6: 4 triangles 4 hexagons: 18 12 T d: Cuboctahedron: 3.4.3.4: 8 triangles 6 squares: 24 12 O h: Truncated cube: 3.8.8: 8 triangles 6 octagons: 36 24 O h: Truncated octahedron: 4.6.6: 6 squares 8 hexagons 36 24 O h: Rhombicuboctahedron: 3.4.4.4 ...
For example, the icosahedron is {3,5+} 1,0, and pentakis dodecahedron, {3,5+} 1,1 is seen as a regular dodecahedron with pentagonal faces divided into 5 triangles. The primary face of the subdivision is called a principal polyhedral triangle (PPT) or the breakdown structure. Calculating a single PPT allows the entire figure to be created.