**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
triacontahedron’s 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 triacontahedron’s 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 triacontahedron’s 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.

RETURN

**
**** ****TABLE
OF CONTENTS **

HOME
PAGE