APPENDIX B

The three polytope family of the truncated rhombic triacontahedron.

The solution set, 43,200(1), is a polytope that has 44,132 faces and 43,200 vertices (centers of optimal spherical packing of circles that are 1 degree in diameter).   Each of the truncated rhombic triacontahedronís thirty faces is optimal packed with 1440 circles per face.  The faces are spherical polygons, 1880 triangular, 42,240 rhombic, and 12 pentagonal, all of which have edges of 1 degree.

The solution set, 10,800(2), is a polytope that has 11,012 faces and 10,800 vertices (centers of optimal spherical packing of circles that are 2 degrees in diameter). The truncated rhombic triacontahedronís thirty faces are each optimal packed with 360 circles per face.  The faces are spherical polygons, 440 triangular, 10,560 rhombic, and 12 pentagonal, all of which have edges of 2 degrees.

The solution set, 2700(4), is a polytope that has 2,732 faces and 2,700 vertices (centers of optimal spherical packing of circles that are 4 degrees in diameter). The truncated rhombic triacontahedronís thirty faces are each optimal packed with 90 circles per face.  The faces are spherical polygons, 80 triangular, 2,640 rhombic, and 12 pentagonal, all of which have edges of 4 degrees.

The ten polytope family of the truncated rhombic dodecahedron.

The solution set, 4800(3), has the form of a truncated rhombic dodecahedron V 4.3 that has 12 rhombic faces optimal packed with 400 circles per face.  The polytope has 9134 faces and 4800 vertices (centers of optimal spherical packing of circles that are 3 degrees in diameter).  The polytope has 8672 triangular spherical polygons faces, and 462 square ones, all of which have edges of 3 degrees. 

The solution set 1728(5), has the form of a truncated rhombic dodecahedron that has 12 rhombic faces optimal packed with 144 circles per face.  The polytope has 3182 faces and 1728 vertices (centers of optimal spherical packing of circles that are 5 degrees in diameter).  The polytope has 2912 triangular spherical polygons faces, and 270 square ones, all of which have edges of 5 degrees

The solution set, 1200(6), has the form of a truncated rhombic dodecahedron that has 12 rhombic faces optimal packed with 100 circles per face.  The polytope has 2174 faces and 1200 vertices (centers of optimal spherical packing of circles that are 6 degrees in diameter).  The polytope has 1952 triangular spherical polygons faces, and 222 square ones, all of which have edges of 6 degrees.

The solution set, 432(10), has the form of a truncated rhombic dodecahedron that has 12 rhombic faces optimal packed with 36 circles per face.  The polytope has 734 faces and 432 vertices (centers of optimal spherical packing of circles that are 10 degrees in diameter).  The polytope has 608 triangular spherical polygons faces, and 126 square ones, all of which have edges of 10 degrees.

The solution set, 300(12), has the form of a truncated rhombic dodecahedron that has 12 rhombic faces optimal packed with 25 circles per face.  The polytope has 494 faces and 300 vertices (centers of optimal spherical packing of circles that are 12 degrees in diameter).  The polytope has 392 triangular spherical polygons faces, and 102 square ones, all of which have edges of 12 degrees.

The solution set, 192(15), has the form of a truncated rhombic dodecahedron that has 12 rhombic faces optimal packed with 16 circles per face.  The polytope has 302 faces and 192 vertices (centers of optimal spherical packing of circles that are 15 degrees in diameter).  The polytope has 224 triangular spherical polygons faces, and 78 square ones, all of which have edges of 15 degrees.

The solution set, 108(20), has the form of a truncated rhombic dodecahedron that has 12 rhombic faces optimal packed with 9 circles per face.  The polytope has 158 faces and 108 vertices (centers of optimal spherical packing of circles that are 20 degrees in diameter).  The polytope has 104 triangular spherical polygons faces, and 54 square ones, all of which have edges of 20 degrees.

The solution set, 48(30), has two possible forms, a truncated octahedron V 3.4 that has 8 triangular faces optimal packed with 6 circles per face or a truncated rhombic dodecahedron V 4.3 that has 12 rhombic faces optimal packed with 4 circles per face.   The first polytope has 62 faces, the second polytope has 50 faces and they both have 48 vertices (centers of optimal spherical packing of circles that are 30 degrees in diameter).  The first polytope's faces are 32 triangular spherical polygons, and 30 square ones.  The second polytope's faces are 8 triangular spherical polygons, 30 square ones, and 12 rhombic ones, all of which have edges of 30 degrees. 

The solution set, 12(60), is the field of the cuboctahedron or the faces of the tetrahedron with 3 optimal packed circles per face.  The cuboctahedron is a truncated tetrahedron V 4.3!  The polytope has 14 faces and 12 vertices (centers of optimal spherical packing of circles that are 60 degrees in diameter).  The faces are spherical polygons, 8 triangular, and 6 square, all of which have edges of 60 degrees.

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