List of Regular Shapes by Schläfli Symbol

The following is a list of shapes by their Schläfli symbol

0-Dimensional

• $()$ - Point

1-Dimensional

• $\{\}$ - Line segment

2-Dimensional

Spherical Tilings

• $\{1\}$ - Monogon
• $\{2\}$ - Digon
• $\{3\}$ - Triangle
• $\{4\}$ - Square
• $\{5\}$ - Pentagon
• $\{\frac{5}{2}\}$ - Pentagram
• $\{6\}$ - Hexagon
• $\{7\}$ - Heptagon
• $\{\frac{7}{2}\}$ - Heptagram
• $\{\frac{7}{3}\}$ - Great heptagram
• $\{8\}$ - Octagon
• $\{\frac{8}{3}\}$ - Octagram
• $\{9\}$ - Enneagon
• $\{\frac{9}{2}\}$ - Enneagram
• $\{\frac{9}{4}\}$ - Great enneagram
• $\{10\}$ - Decagon
• $\{\frac{10}{3}\}$ - Decagram

Euclidean Tilings

• $\{\aleph_0\}$ - Apeirogon

3-Dimensional

Monogonal Faces

• $\{1,2\}$ - Monogonal dihedron

Digonal Faces

Spherical Tilings

• $\{2,1\}$ - Monogonal hosohedron
• $\{2,2\}$ - Digonal hosohedron
• $\{2,3\}$ - Trigonal hosohedron
• $\{2,4\}$ - Square hosohedron
• $\{2,5\}$ - Pentagonal hosohedron
• $\{2,6\}$ - Hexagonal hosohedron
• $\{2,7\}$ - Heptagonal hosohedron
• $\{2,8\}$ - Octagonal hosohedron

Euclidean Tilings

• $\{ 2,\aleph_0 \}$ - Apeirogonal hosohedron

Triangular Faces

Spherical Tilings

• $\{3,2\}$ - Trigonal dihedron
• $\{3, 3\}$ - Tetrahedron
• $\{3, 4\}$ - Octahedron
• $\{3, 5\}$ - Icosahedron
• $\{ 3, \frac{5}{2} \}$ - Great icosahedron

Euclidean Tilings

• $\{3,6\}$ - Triangular tiling

Compact Hyperbolic Tilings

• $\{3,7\}$ - Order-7 triangular tiling
• $\{3,8\}$ - Order-8 triangular tiling

Paracompact Hyperbolic Tilings

• $\{3,\aleph_0\}$ - Infinite-order triangular tiling

Square Faces

Spherical Tilings

• $\{4,2\}$ - Square dihedron
• $\{4,3\}$ - Cube

Euclidean Tilings

• $\{4,4\}$ - Square tiling

Compact Hyperbolic Tilings

• $\{4,5\}$ - Order-5 square tiling
• $\{4,6\}$ - Order-6 square tiling

Paracompact Hyperbolic Tilings

• $\{4,\aleph_0\}$ - Infinite-order square tiling

Pentagonal Faces

Spherical Tilings

• $\{5,2\}$ - Pentagonal dihedron
• $\{5, 3\}$ - Dodecahedron
• $\{ 5, \frac{5}{2} \}$ - Great dodecahedron

Compact Hyperbolic Tilings

• $\{5,4\}$ - Order-4 pentagonal tiling
• $\{5,5\}$ - Order-5 pentagonal tiling

Paracompact Hyperbolic Tilings

• $\{5,\aleph_0\}$ - Infinite-order pentagonal tiling

Pentagrammic Faces

• $\{\frac {5}{2}, 3\}$ - Great stellated dodecahedron
• $\{\frac {5}{2}, 5\}$ - Small stellated dodecahedron

Hexagonal Faces

Spherical Tilings

• $\{6,2\}$ - Hexagonal dihedron

Euclidean Tilings

• $\{6,3\}$ - Hexagonal tiling

Compact Hyperbolic Tilings

• $\{6,4\}$ - Order-4 hexagonal tiling

Paracompact Hyperbolic Tilings

• $\{6,\aleph_0\}$ - Infinite-order hexagonal tiling

Heptagonal Faces

Spherical Tilings

• $\{7,2\}$ - Heptagonal dihedron

Compact Hyperbolic Tilings

• $\{7,3\}$ - Order-3 heptagonal tiling

Octagonal Faces

Spherical Tilings

• $\{8,2\}$ - Octagonal dihedron

Compact Hyperbolic Tilings

• $\{8,3\}$ - Order-3 octagonal tiling

Apeirogonal Faces

Euclidean Tilings

• $\{ \aleph_0, 2 \}$ - Apeirogonal dihedron

Paracompact Hyperbolic Tilings

• $\{\aleph_0,3\}$ - Order-3 apeirogonal tiling
• $\{\aleph_0,4\}$ - Order-4 apeirogonal tiling
• $\{\aleph_0,5\}$ - Order-5 apeirogonal tiling
• $\{\aleph_0,6\}$ - Order-6 apeirogonal tiling
• $\{\aleph_0,\aleph_0\}$ - Infinite-order apeirogonal tiling

4-Dimensional

Digonal Faces

Trigonal hosohedral cells

Spherical Tilings

• $\{2,3,2\}$ - Trigonal dihedral hosochoron
• $\{2,3,3\}$ - Tetrahedral hosochoron
• $\{2,3,4\}$ - Octahedral hosochoron
• $\{2,3,5\}$ - Icosahedral hosochoron

Euclidean Tilings

• $\{2,3,6\}$ - Triangular tiling hosochoron

Square hosohedral cells

Spherical Tilings

• $\{2,4,2\}$ - Square dihedral hosochoron
• $\{2,4,3\}$ - Cubic hosochoron

Euclidean Tilings

• $\{2,4,4\}$ - Square tiling hosochoron

Pentagonal hosohedral cells

• $\{2,5,2\}$ - Pentagonal dihedral hosochoron
• $\{2,5,3\}$ - Dodecahedral hosochoron

Hexagonal hosohedral cells

Euclidean Tilings

• $\{2,6,3\}$ - Hexagonal tiling hosochoron

Triangular Faces

Tetrahedral Cells

Spherical Tilings

• $\{3,3,2\}$ - Tetrahedral dichoron
• $\{3,3,3\}$ - Pentachoron
• $\{3, 3, 4\}$ - Hexadecachoron
• $\{3,3,5\}$ - Hexacosichoron
• $\{3, 3, \frac{5}{2}\}$ - Grand hexacosichoron

Paracompact Hyperbolic Tilings

• $\{3,3,6\}$ - Order-6 tetrahedral honeycomb

Octahedral Cells

Spherical Tilings

• $\{3,4,2\}$ - Octahedral dichoron
• $\{3,4,3\}$ - Icositetrachoron

Paracompact Hyperbolic Tilings

• $\{3,4,4\}$ - Order-4 octahedral honeycomb

Icosahedral Cells

Spherical Tilings

• $\{3,5,2\}$ - Icosahedral dichoron
• $\{3, 5, \frac {5}{2}\}$ - Faceted hexacosichoron

Compact Hyperbolic Tilings

• $\{3,5,3\}$ - Order-3 icosahedral honeycomb

Great Icosahedral Cells

• $\{3, \frac {5}{2}, 5\}$ - Great faceted hexacosichoron

Triangular Tiling Cells

Euclidean Tilings

• $\{3,6,2\}$ - Order-2 triangular tiling honeycomb

Paracompact Hyperbolic Tilings

• $\{3,6,3\}$ - Order-3 triangular tiling honeycomb

Square Faces

Cubic Cells

Spherical Tilings

• $\{4,3,2\}$ - Cubic dichoron
• $\{4, 3, 3\}$ - Tesseract

Euclidean Tilings

• $\{4,3,4\}$ - Cubic honeycomb

Compact Hyperbolic Tilings

• $\{4,3,5\}$ - Order-5 cubic honeycomb

Paracompact Hyperbolic Tilings

• $\{4,3,6\}$ - Order-6 cubic honeycomb

Square Tiling Cells

Euclidean Tilings

• $\{4,4,2\}$ - Square tiling dichoron

Paracompact Hyperbolic Tilings

• $\{4,4,3\}$ - Order-3 square tiling honeycomb
• $\{4,4,4\}$ - Order-4 square tiling honeycomb

Pentagonal Faces

Dodecahedral Cells

Spherical Tilings

• $\{5,3,2\}$ - Dodecahedral dichoron
• $\{5,3,3\}$ - Hecatonicosachoron
• $\{5,3,\frac{5}{2}\}$ - Grand hecatonicosachoron

Compact Hyperbolic Tilings

• $\{5,3,4\}$ - Order-4 dodecahedral honeycomb
• $\{5,3,5\}$ - Order-5 dodecahedral honeycomb

Paracompact Hyperbolic Tilings

• $\{5,3,6\}$ - Order-6 dodecahedral honeycomb

Great Dodecahedral Cells

• $\{5, \frac {5}{2}, 3\}$ - Great grand hecatonicosachoron
• $\{5,\frac{5}{2},5\}$ - Great hecatonicosachoron

Pentagrammal Faces

Great Stellated Dodecahedral Cells

• $\{\frac{5}{2}, 3, 3\}$ - Great grand stellated hecatonicosachoron
• $\{\frac{5}{2},3,5\}$ - Great stellated hecatonicosachoron

Small Stellated Dodecahedral Cells

• $\{\frac {5}{2}, 5, 3\}$ - Small stellated hecatonicosachoron
• $\{\frac{5}{2},5,\frac{5}{2}\}$ - Grand stellated hecatonicosachoron

Hexagonal Faces

Hexagonal Tiling Cells

Euclidean Tilings

• $\{6,3,2\}$ - Hexagonal tiling dichoron

Paracompact Hyperbolic Tilings

• $\{6,3,3\}$ - Order-3 hexagonal tiling honeycomb
• $\{6,3,4\}$ - Order-4 hexagonal tiling honeycomb
• $\{6,3,5\}$ - Order-5 hexagonal tiling honeycomb
• $\{6,3,6\}$ - Order-6 hexagonal tiling honeycomb

5-Dimensional

Square Hosohedral Cells

Cubic Hosochoric Tera

Euclidean Tilings

• $\{2,4,3,4\}$ - Cubic honeycomb hosoteron

Tetrahedral Cells

Pentachoral Tera

Spherical Tilings

• $\{3,3,3,3\}$ - Hexateron
• $\{3,3,3,4\}$ - Pentacross

Compact Hyperbolic Tilings

• $\{3,3,3,5\}$ - Order-5 pentachoric tetracomb

Hexadecachoral Tera

Euclidean Tilings

• $\{3,3,4,3\}$ - Hexadecachoric tetracomb

Octahedral Cells

Icositetrachoral Tera

Euclidean Tilings

• $\{3,4,3,3\}$ - Icositetrachoric tetracomb

Paracompact Hyperbolic Tilings

• $\{3,4,3,4\}$ - Icositetrachoric tetracomb

Cubic Cells

Tesseractic Tera

Spherical Tilings

• $\{4,3,3,3\}$ - Penteract

Euclidean Tilings

• $\{4,3,3,4\}$ - Tesseractic tetracomb

Compact Hyperbolic Tilings

• $\{4,3,3,5\}$ - Order-5 tesseractic tetracomb

Cubic Honeycomb Tera

Euclidean Tilings

• $\{4,3,4,2\}$ - Cubic honeycomb diteron

Paracompact Hyperbolic Tilings

• $\{4,3,4,3\}$ - Order-3 cubic honeycomb honeycomb

Dodecahedral Cells

Hecatonicosachoric Tera

Compact Hyperbolic Tilings

• $\{5, 3, 3, 3\}$ - Order-3 hecatonicosachoric tetracomb
• $\{5,3,3,4\}$ - Order-4 hecatonicosachoric tetracomb
• $\{5,3,3,5\}$ - Order-5 hecatonicosachoric tetracomb

See Also

• List of Truncated Shapes by Schläfli Symbol
• List of Rectified Shapes by Schläfli Symbol