Poly includes all of the following polyhedra:
Platonic Solids
 Tetrahedron
 Cube
 Octahedron
 Dodecahedron
 Icosahedron
Each platonic polyhedron is constructed using (multiple copies of) a single regular polygon; the same number of polygonal faces is
used around each vertex. A polygon is regular if all of its edges have the same length and all of its interior angles are equal. Both the equilateral triangle
and square are regular polygons.
Archimedean Solids
 Truncated Tetrahedron
 Cuboctahedron
 Truncated Cube
 Truncated Octahedron
 Rhombicuboctahedron
 Great Rhombicuboctahedron
 Snub Cube
 Icosidodecahedron
 Truncated Dodecahedron
 Truncated Icosahedron
 Rhombicosidodecahedron
 Great Rhombicosidodecahedron
 Snub Dodecahedron
The Archimedean solids were defined historically by Archimedes, although we have lost his writings. All of the Archimedean solids
are uniform polyhedra with regular faces. A polyhedron with regular polygonal faces is uniform if there are symmetry operations that take one vertex through
all of the other vertices and no other points in space. For example, the cube has rotation by 90° around an axis and reflection through a plane perpendicular
to that axis as its symmetry operations.
A common heuristic for the Archimedean solids is that the arrangement of faces surrounding each vertex must be the
same for all vertices. Although all of the Archimedean solids have this property, so does the elongated square gyrobicupola (Johnson solid #37), which is not
an Archimedean solid.
Prisms and AntiPrisms
 Triangular Prism
 Pentagonal Prism
 Hexagonal Prism
 Octagonal Prism
 Decagonal Prism
 Square Antiprism
 Pentagonal Antiprism
 Hexagonal Antiprism
 Octagonal Antiprism
 Decagonal Antiprism
After the Platonic and Archimedean solids, the only remaining convex uniform polyhedra with regular faces are prisms and
antiprisms. This was shown by Johannes Kepler, who also gave the names commonly used for the Archimedean solids.
Johnson Solids
 Square Pyramid (J1)
 Pentagonal Pyramid (J2)
 Triangular Cupola (J3)
 Square Cupola (J4)
 Pentagonal Cupola (J5)
 Pentagonal Rotunda (J6)
 Elongated Triangular Pyramid (J7)
 Elongated Square Pyramid (J8)
 Elongated Pentagonal Pyramid (J9)
 Gyroelongated Square Pyramid (J10)
 Gyroelongated Pentagonal Pyramid (J11)
 Triangular Dipyramid (J12)
 Pentagonal Dipyramid (J13)
 Elongated Triangular Dipyramid (J14)
 Elongated Square Dipyramid (J15)
 Elongated Pentagonal Dipyramid (J16)
 Gyroelongated Square Dipyramid (J17)
 Elongated Triangular Cupola (J18)
 Elongated Square Cupola (J19)
 Elongated Pentagonal Cupola (J20)
 Elongated Pentagonal Rotunda (J21)
 Gyroelongated Triangular Cupola (J22)
 Gyroelongated Square Cupola (J23)
 Gyroelongated Pentagonal Cupola (J24)
 Gyroelongated Pentagonal Rotunda (J25)
 Gyrobifastigium (J26)
 Triangular Orthobicupola (J27)
 Square Orthobicupola (J28)
 Square Gyrobicupola (J29)
 Pentagonal Orthobicupola (J30)
 Pentagonal Gyrobicupola (J31)
 Pentagonal Orthocupolarotunda (J32)
 Pentagonal Gyrocupolarotunda (J33)
 Pentagonal Orthobirotunda (J34)
 Elongated Triangular Orthobicupola (J35)
 Elongated Triangular Gyrobicupola (J36)
 Elongated Square Gyrobicupola (J37)
 Elongated Pentagonal Orthobicupola (J38)
 Elongated Pentagonal Gyrobicupola (J39)
 Elongated Pentagonal Orthocupolarotunda (J40)
 Elongated Pentagonal Gyrocupolarotunda (J41)
 Elongated Pentagonal Orthobirotunda (J42)
 Elongated Pentagonal Gyrobirotunda (J43)
 Gyroelongated Triangular Bicupola (J44)
 Gyroelongated Square Bicupola (J45)
 Gyroelongated Pentagonal Bicupola (J46)
 Gyroelongated Pentagonal Cupolarotunda (J47)
 Gyroelongated Pentagonal Birotunda (J48)
 Augmented Triangular Prism (J49)
 Biaugmented Triangular Prism (J50)
 Triaugmented Triangular Prism (J51)
 Augmented Pentagonal Prism (J52)
 Biaugmented Pentagonal Prism (J53)
 Augmented Hexagonal Prism (J54)
 Parabiaugmented Hexagonal Prism (J55)
 Metabiaugmented Hexagonal Prism (J56)
 Triaugmented Hexagonal Prism (J57)
 Augmented Dodecahedron (J58)
 Parabiaugmented Dodecahedron (J59)
 Metabiaugmented Dodecahedron (J60)
 Triaugmented Dodecahedron (J61)
 Metabidiminished Icosahedron (J62)
 Tridiminished Icosahedron (J63)
 Augmented Tridiminished Icosahedron (J64)
 Augmented Truncated Tetrahedron (J65)
 Augmented Truncated Cube (J66)
 Biaugmented Truncated Cube (J67)
 Augmented Truncated Dodecahedron (J68)
 Parabiaugmented Truncated Dodecahedron (J69)
 Metabiaugmented Truncated Dodecahedron (J70)
 Triaugmented Truncated Dodecahedron (J71)
 Gyrate Rhombicosidodecahedron (J72)
 Parabigyrate Rhombicosidodecahedron (J73)
 Metabigyrate Rhombicosidodecahedron (J74)
 Trigyrate Rhombicosidodecahedron (J75)
 Diminished Rhombicosidodecahedron (J76)
 Paragyrate Diminished Rhombicosidodecahedron (J77)
 Metagyrate Diminished Rhombicosidodecahedron (J78)
 Bigyrate Diminished Rhombicosidodecahedron (J79)
 Parabidiminished Rhombicosidodecahedron (J80)
 Metabidiminished Rhombicosidodecahedron (J81)
 Gyrate Bidiminished Rhombicosidodecahedron (J82)
 Tridiminished Rhombicosidodecahedron (J83)
 Snub Disphenoid (J84)
 Snub Square Antiprism (J85)
 Sphenocorona (J86)
 Augmented Sphenocorona (J87)
 Sphenomegacorona (J88)
 Hebesphenomegacorona (J89)
 Disphenocingulum (J90)
 Bilunabirotunda (J91)
 Triangular Hebesphenorotunda (J92)
After taking into account the preceeding three categories, there are only a finite number of convex polyhedra with regular faces
left. The enumeration of these polyhedra was performed by Norman W. Johnson.
Catalan Solids
 Triakis Tetrahedron
 Rhombic Dodecahedron
 Triakis Octahedron
 Tetrakis Hexahedron
 Deltoidal Icositetrahedron
 Disdyakis Dodecahedron
 Pentagonal Icositetrahedron
 Rhombic Triacontahedron
 Triakis Icosahedron
 Pentakis Dodecahedron
 Deltoidal Hexecontahedron
 Disdyakis Triacontahedron
 Pentagonal Hexecontahedron
The Catalan solids are duals of Archimedean solids. A dual of a polyhedron is constructed by replacing each face with a vertex, and
each vertex with a face. For example, the dual of the icosahedron is the dodecahedron; the dual of the dodecahedron is the icosahedron.
Dipyramids and Deltohedra
 Triangular Dipyramid
 Pentagonal Dipyramid
 Hexagonal Dipyramid
 Octagonal Dipyramid
 Decagonal Dipyramid
 Square Deltohedron
 Pentagonal Deltohedron
 Hexagonal Deltohedron
 Octagonal Deltohedron
 Decagonal Deltohedron
Dipyramids are duals of prisms; deltohedra are duals of antiprisms.
Geodesic Spheres and Geodesic Hemispheres
 1Frequency Tetrahedral Geodesic Sphere
 1Frequency Hexahedral Geodesic Sphere
 1Frequency Octahedral Geodesic Sphere
 1Frequency Dodecahedral Geodesic Sphere
 1Frequency Icosahedral Geodesic Sphere
 2Frequency Tetrahedral Geodesic Sphere
 2Frequency Hexahedral Geodesic Sphere
 2Frequency Octahedral Geodesic Sphere
 2Frequency Dodecahedral Geodesic Sphere
 2Frequency Icosahedral Geodesic Sphere
 3Frequency Tetrahedral Geodesic Sphere
 3Frequency Octahedral Geodesic Sphere
 3Frequency Icosahedral Geodesic Sphere
 2Frequency Icosahedral Geodesic Hemisphere
 3Frequency Octahedral Geodesic Hemisphere
 1Frequency Hexahedral Geodesic Hemisphere
 2Frequency Octahedral Geodesic Hemisphere
 2Frequency Hexahedral Geodesic Hemisphere
 2Frequency Tetrahedral Geodesic Hemisphere
 1Frequency Snub Icosidodecahedral Geodesic Sphere
 2Frequency Snub Icosidodecahedral Geodesic Sphere
 1Frequency Truncated Tetrahedral Geodesic Sphere
 2Frequency Truncated Tetrahedral Geodesic Sphere
 1Frequency Snub Cuboctahedral Geodesic Sphere
 2Frequency Snub Cuboctahedral Geodesic Sphere
 4Frequency Tetrahedral Geodesic Sphere
 4Frequency Octahedral Geodesic Sphere
 4Frequency Icosahedral Geodesic Sphere
 4Frequency Tetrahedral Geodesic Hemisphere
 4Frequency Octahedral Geodesic Hemisphere
 4Frequency Icosahedral Geodesic Hemisphere
 6Frequency Tetrahedral Geodesic Sphere
 6Frequency Octahedral Geodesic Sphere
 6Frequency Icosahedral Geodesic Sphere
 6Frequency Tetrahedral Geodesic Hemisphere
 6Frequency Octahedral Geodesic Hemisphere
 6Frequency Icosahedral Geodesic Hemisphere
The first geodesic dome was designed by Walter Bauersfeld
and was built in 1922. Several decades later, Buckminster Fuller
popularized and extended the ideas behind geodesic constructions.
To construct a geodesic sphere, you first choose a convex polyhedron
which will serve as a framework for the construction.
The next step is to ensure all of the polyhedron’s faces are triangular:
each nontriangular face is triangulated by connecting its vertices
to a new vertex placed at the center of the face.
You will then have a 1frequency geodesic sphere if all of the vertices
are made to be equidistant from the center of the polyhedron by
moving them directly away from or towards the center.
Higher frequency geodesic spheres may be constructed by replacing
each face with a regular triangular mesh and then ensuring all of
the new vertices are equidistant from the center.
2frequency geodesics bisect each edge of their associated 1frequency geodesics by using 4 triangles for each mesh;
3frequency geodesics trisect each edge and add one vertex to the center of each face of their associated 1frequency
geodesic by using 9 triangles for each mesh.
