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 Anti-Prisms

• 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 anti-prisms. 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 anti-prisms.

Geodesic Spheres and Geodesic Hemispheres

• 1-Frequency Tetrahedral Geodesic Sphere
• 1-Frequency Hexahedral Geodesic Sphere
• 1-Frequency Octahedral Geodesic Sphere
• 1-Frequency Dodecahedral Geodesic Sphere
• 1-Frequency Icosahedral Geodesic Sphere
• 2-Frequency Tetrahedral Geodesic Sphere
• 2-Frequency Hexahedral Geodesic Sphere
• 2-Frequency Octahedral Geodesic Sphere
• 2-Frequency Dodecahedral Geodesic Sphere
• 2-Frequency Icosahedral Geodesic Sphere
• 3-Frequency Tetrahedral Geodesic Sphere
• 3-Frequency Octahedral Geodesic Sphere
• 3-Frequency Icosahedral Geodesic Sphere
• 2-Frequency Icosahedral Geodesic Hemisphere
• 3-Frequency Octahedral Geodesic Hemisphere
• 1-Frequency Hexahedral Geodesic Hemisphere
• 2-Frequency Octahedral Geodesic Hemisphere
• 2-Frequency Hexahedral Geodesic Hemisphere
• 2-Frequency Tetrahedral Geodesic Hemisphere
• 1-Frequency Snub Icosidodecahedral Geodesic Sphere
• 2-Frequency Snub Icosidodecahedral Geodesic Sphere
• 1-Frequency Truncated Tetrahedral Geodesic Sphere
• 2-Frequency Truncated Tetrahedral Geodesic Sphere
• 1-Frequency Snub Cuboctahedral Geodesic Sphere
• 2-Frequency Snub Cuboctahedral Geodesic Sphere
• 4-Frequency Tetrahedral Geodesic Sphere
• 4-Frequency Octahedral Geodesic Sphere
• 4-Frequency Icosahedral Geodesic Sphere
• 4-Frequency Tetrahedral Geodesic Hemisphere
• 4-Frequency Octahedral Geodesic Hemisphere
• 4-Frequency Icosahedral Geodesic Hemisphere
• 6-Frequency Tetrahedral Geodesic Sphere
• 6-Frequency Octahedral Geodesic Sphere
• 6-Frequency Icosahedral Geodesic Sphere
• 6-Frequency Tetrahedral Geodesic Hemisphere
• 6-Frequency Octahedral Geodesic Hemisphere
• 6-Frequency 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 non-triangular face is triangulated by connecting its vertices to a new vertex placed at the center of the face. You will then have a 1-frequency 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. 2-frequency geodesics bisect each edge of their associated 1-frequency geodesics by using 4 triangles for each mesh; 3-frequency geodesics trisect each edge and add one vertex to the center of each face of their associated 1-frequency geodesic by using 9 triangles for each mesh.