List of Regular Shapes by Schläfli Symbol

The following is a list of shapes by their Schläfli symbol

## 0-Dimensional

* (
)

{\displaystyle ()}

 - Point

## 1-Dimensional

* {
}

{\displaystyle \{\}}

 - Line segment

## 2-Dimensional

### Spherical Tilings

* {
1
}

{\displaystyle \{1\}}

 - Monogon
* {
2
}

{\displaystyle \{2\}}

 - Digon
* {
3
}

{\displaystyle \{3\}}

 - Triangle
* {
4
}

{\displaystyle \{4\}}

 - Square
* {
5
}

{\displaystyle \{5\}}

 - Pentagon
* {

5
2

}

{\displaystyle \{\frac{5}{2}\}}

 - Pentagram
* {
6
}

{\displaystyle \{6\}}

 - Hexagon
* {
7
}

{\displaystyle \{7\}}

 - Heptagon
* {

7
2

}

{\displaystyle \{\frac{7}{2}\}}

 - Heptagram
* {

7
3

}

{\displaystyle \{\frac{7}{3}\}}

 - Great heptagram
* {
8
}

{\displaystyle \{8\}}

 - Octagon
* {

8
3

}

{\displaystyle \{\frac{8}{3}\}}

 - Octagram
* {
9
}

{\displaystyle \{9\}}

 - Enneagon
* {

9
2

}

{\displaystyle \{\frac{9}{2}\}}

 - Enneagram
* {

9
4

}

{\displaystyle \{\frac{9}{4}\}}

 - Great enneagram
* {
10
}

{\displaystyle \{10\}}

 - Decagon
* {

10
3

}

{\displaystyle \{\frac{10}{3}\}}

 - Decagram

### Euclidean Tilings

* {

ℵ
0

}

{\displaystyle \{\aleph\_0\}}

 - Apeirogon

## 3-Dimensional

### Monogonal Faces

* {
1
,
2
}

{\displaystyle \{1,2\}}

 - Monogonal dihedron

### Digonal Faces

#### Spherical Tilings

* {
2
,
1
}

{\displaystyle \{2,1\}}

 - Monogonal hosohedron
* {
2
,
2
}

{\displaystyle \{2,2\}}

 - Digonal hosohedron
* {
2
,
3
}

{\displaystyle \{2,3\}}

 - Trigonal hosohedron
* {
2
,
4
}

{\displaystyle \{2,4\}}

 - Square hosohedron
* {
2
,
5
}

{\displaystyle \{2,5\}}

 - Pentagonal hosohedron
* {
2
,
6
}

{\displaystyle \{2,6\}}

 - Hexagonal hosohedron
* {
2
,
7
}

{\displaystyle \{2,7\}}

 - Heptagonal hosohedron
* {
2
,
8
}

{\displaystyle \{2,8\}}

 - Octagonal hosohedron

#### Euclidean Tilings

* {
2
,

ℵ
0

}

{\displaystyle \{ 2,\aleph\_0 \}}

 - Apeirogonal hosohedron

### Triangular Faces

#### Spherical Tilings

* {
3
,
2
}

{\displaystyle \{3,2\}}

 - Trigonal dihedron
* {
3
,
3
}

{\displaystyle \{3, 3\}}

 - Tetrahedron
* {
3
,
4
}

{\displaystyle \{3, 4\}}

 - Octahedron
* {
3
,
5
}

{\displaystyle \{3, 5\}}

 - Icosahedron
* {
3
,

5
2

}

{\displaystyle \{ 3, \frac{5}{2} \}}

 - Great icosahedron

#### Euclidean Tilings

* {
3
,
6
}

{\displaystyle \{3,6\}}

 - Triangular tiling

#### Compact Hyperbolic Tilings

* {
3
,
7
}

{\displaystyle \{3,7\}}

 - Order-7 triangular tiling
* {
3
,
8
}

{\displaystyle \{3,8\}}

 - Order-8 triangular tiling

#### Paracompact Hyperbolic Tilings

* {
3
,

ℵ
0

}

{\displaystyle \{3,\aleph\_0\}}

 - Infinite-order triangular tiling

### Square Faces

#### Spherical Tilings

* {
4
,
2
}

{\displaystyle \{4,2\}}

 - Square dihedron
* {
4
,
3
}

{\displaystyle \{4,3\}}

 - Cube

#### Euclidean Tilings

* {
4
,
4
}

{\displaystyle \{4,4\}}

 - Square tiling

#### Compact Hyperbolic Tilings

* {
4
,
5
}

{\displaystyle \{4,5\}}

 - Order-5 square tiling
* {
4
,
6
}

{\displaystyle \{4,6\}}

 - Order-6 square tiling

#### Paracompact Hyperbolic Tilings

* {
4
,

ℵ
0

}

{\displaystyle \{4,\aleph\_0\}}

 - Infinite-order square tiling

### Pentagonal Faces

#### Spherical Tilings

* {
5
,
2
}

{\displaystyle \{5,2\}}

 - Pentagonal dihedron
* {
5
,
3
}

{\displaystyle \{5, 3\}}

 - Dodecahedron
* {
5
,

5
2

}

{\displaystyle \{ 5, \frac{5}{2} \}}

 - Great dodecahedron

#### Compact Hyperbolic Tilings

* {
5
,
4
}

{\displaystyle \{5,4\}}

 - Order-4 pentagonal tiling
* {
5
,
5
}

{\displaystyle \{5,5\}}

 - Order-5 pentagonal tiling

#### Paracompact Hyperbolic Tilings

* {
5
,

ℵ
0

}

{\displaystyle \{5,\aleph\_0\}}

 - Infinite-order pentagonal tiling

### Pentagrammic Faces

* {

5
2

,
3
}

{\displaystyle \{\frac {5}{2}, 3\}}

 - Great stellated dodecahedron
* {

5
2

,
5
}

{\displaystyle \{\frac {5}{2}, 5\}}

 - Small stellated dodecahedron

### Hexagonal Faces

#### Spherical Tilings

* {
6
,
2
}

{\displaystyle \{6,2\}}

 - Hexagonal dihedron

#### Euclidean Tilings

* {
6
,
3
}

{\displaystyle \{6,3\}}

 - Hexagonal tiling

#### Compact Hyperbolic Tilings

* {
6
,
4
}

{\displaystyle \{6,4\}}

 - Order-4 hexagonal tiling

#### Paracompact Hyperbolic Tilings

* {
6
,

ℵ
0

}

{\displaystyle \{6,\aleph\_0\}}

 - Infinite-order hexagonal tiling

### Heptagonal Faces

#### Spherical Tilings

* {
7
,
2
}

{\displaystyle \{7,2\}}

 - Heptagonal dihedron

#### Compact Hyperbolic Tilings

* {
7
,
3
}

{\displaystyle \{7,3\}}

 - Order-3 heptagonal tiling

### Octagonal Faces

#### Spherical Tilings

* {
8
,
2
}

{\displaystyle \{8,2\}}

 - Octagonal dihedron

#### Compact Hyperbolic Tilings

* {
8
,
3
}

{\displaystyle \{8,3\}}

 - Order-3 octagonal tiling

### Apeirogonal Faces

#### Euclidean Tilings

* {

ℵ
0

,
2
}

{\displaystyle \{ \aleph\_0, 2 \}}

 - Apeirogonal dihedron

#### Paracompact Hyperbolic Tilings

* {

ℵ
0

,
3
}

{\displaystyle \{\aleph\_0,3\}}

 - Order-3 apeirogonal tiling
* {

ℵ
0

,
4
}

{\displaystyle \{\aleph\_0,4\}}

 - Order-4 apeirogonal tiling
* {

ℵ
0

,
5
}

{\displaystyle \{\aleph\_0,5\}}

 - Order-5 apeirogonal tiling
* {

ℵ
0

,
6
}

{\displaystyle \{\aleph\_0,6\}}

 - Order-6 apeirogonal tiling
* {

ℵ
0

,

ℵ
0

}

{\displaystyle \{\aleph\_0,\aleph\_0\}}

 - Infinite-order apeirogonal tiling

## 4-Dimensional

### Digonal Faces

#### Trigonal hosohedral cells

##### Spherical Tilings

* {
2
,
3
,
2
}

{\displaystyle \{2,3,2\}}

 - Trigonal dihedral hosochoron
* {
2
,
3
,
3
}

{\displaystyle \{2,3,3\}}

 - Tetrahedral hosochoron
* {
2
,
3
,
4
}

{\displaystyle \{2,3,4\}}

 - Octahedral hosochoron
* {
2
,
3
,
5
}

{\displaystyle \{2,3,5\}}

 - Icosahedral hosochoron

##### Euclidean Tilings

* {
2
,
3
,
6
}

{\displaystyle \{2,3,6\}}

 - Triangular tiling hosochoron

#### Square hosohedral cells

##### Spherical Tilings

* {
2
,
4
,
2
}

{\displaystyle \{2,4,2\}}

 - Square dihedral hosochoron
* {
2
,
4
,
3
}

{\displaystyle \{2,4,3\}}

 - Cubic hosochoron

##### Euclidean Tilings

* {
2
,
4
,
4
}

{\displaystyle \{2,4,4\}}

 - Square tiling hosochoron

#### Pentagonal hosohedral cells

* {
2
,
5
,
2
}

{\displaystyle \{2,5,2\}}

 - Pentagonal dihedral hosochoron
* {
2
,
5
,
3
}

{\displaystyle \{2,5,3\}}

 - Dodecahedral hosochoron

#### Hexagonal hosohedral cells

##### Euclidean Tilings

* {
2
,
6
,
3
}

{\displaystyle \{2,6,3\}}

 - Hexagonal tiling hosochoron

### Triangular Faces

#### Tetrahedral Cells

##### Spherical Tilings

* {
3
,
3
,
2
}

{\displaystyle \{3,3,2\}}

 - Tetrahedral dichoron
* {
3
,
3
,
3
}

{\displaystyle \{3,3,3\}}

 - Pentachoron
* {
3
,
3
,
4
}

{\displaystyle \{3, 3, 4\}}

 - Hexadecachoron
* {
3
,
3
,
5
}

{\displaystyle \{3,3,5\}}

 - Hexacosichoron
* {
3
,
3
,

5
2

}

{\displaystyle \{3, 3, \frac{5}{2}\}}

 - Grand hexacosichoron

##### Paracompact Hyperbolic Tilings

* {
3
,
3
,
6
}

{\displaystyle \{3,3,6\}}

 - Order-6 tetrahedral honeycomb

#### Octahedral Cells

##### Spherical Tilings

* {
3
,
4
,
2
}

{\displaystyle \{3,4,2\}}

 - Octahedral dichoron
* {
3
,
4
,
3
}

{\displaystyle \{3,4,3\}}

 - Icositetrachoron

##### Paracompact Hyperbolic Tilings

* {
3
,
4
,
4
}

{\displaystyle \{3,4,4\}}

 - Order-4 octahedral honeycomb

#### Icosahedral Cells

##### Spherical Tilings

* {
3
,
5
,
2
}

{\displaystyle \{3,5,2\}}

 - Icosahedral dichoron
* {
3
,
5
,

5
2

}

{\displaystyle \{3, 5, \frac {5}{2}\}}

 - Faceted hexacosichoron

##### Compact Hyperbolic Tilings

* {
3
,
5
,
3
}

{\displaystyle \{3,5,3\}}

 - Order-3 icosahedral honeycomb

#### Great Icosahedral Cells

* {
3
,

5
2

,
5
}

{\displaystyle \{3, \frac {5}{2}, 5\}}

 - Great faceted hexacosichoron

#### Triangular Tiling Cells

##### Euclidean Tilings

* {
3
,
6
,
2
}

{\displaystyle \{3,6,2\}}

 - Order-2 triangular tiling honeycomb

##### Paracompact Hyperbolic Tilings

* {
3
,
6
,
3
}

{\displaystyle \{3,6,3\}}

 - Order-3 triangular tiling honeycomb

### Square Faces

#### Cubic Cells

##### Spherical Tilings

* {
4
,
3
,
2
}

{\displaystyle \{4,3,2\}}

 - Cubic dichoron
* {
4
,
3
,
3
}

{\displaystyle \{4, 3, 3\}}

 - Tesseract

##### Euclidean Tilings

* {
4
,
3
,
4
}

{\displaystyle \{4,3,4\}}

 - Cubic honeycomb

##### Compact Hyperbolic Tilings

* {
4
,
3
,
5
}

{\displaystyle \{4,3,5\}}

 - Order-5 cubic honeycomb

##### Paracompact Hyperbolic Tilings

* {
4
,
3
,
6
}

{\displaystyle \{4,3,6\}}

 - Order-6 cubic honeycomb

#### Square Tiling Cells

##### Euclidean Tilings

* {
4
,
4
,
2
}

{\displaystyle \{4,4,2\}}

 - Square tiling dichoron

##### Paracompact Hyperbolic Tilings

* {
4
,
4
,
3
}

{\displaystyle \{4,4,3\}}

 - Order-3 square tiling honeycomb
* {
4
,
4
,
4
}

{\displaystyle \{4,4,4\}}

 - Order-4 square tiling honeycomb

### Pentagonal Faces

#### Dodecahedral Cells

##### Spherical Tilings

* {
5
,
3
,
2
}

{\displaystyle \{5,3,2\}}

 - Dodecahedral dichoron
* {
5
,
3
,
3
}

{\displaystyle \{5,3,3\}}

 - Hecatonicosachoron
* {
5
,
3
,

5
2

}

{\displaystyle \{5,3,\frac{5}{2}\}}

 - Grand hecatonicosachoron

##### Compact Hyperbolic Tilings

* {
5
,
3
,
4
}

{\displaystyle \{5,3,4\}}

 - Order-4 dodecahedral honeycomb
* {
5
,
3
,
5
}

{\displaystyle \{5,3,5\}}

 - Order-5 dodecahedral honeycomb

##### Paracompact Hyperbolic Tilings

* {
5
,
3
,
6
}

{\displaystyle \{5,3,6\}}

 - Order-6 dodecahedral honeycomb

#### Great Dodecahedral Cells

* {
5
,

5
2

,
3
}

{\displaystyle \{5, \frac {5}{2}, 3\}}

 - Great grand hecatonicosachoron
* {
5
,

5
2

,
5
}

{\displaystyle \{5,\frac{5}{2},5\}}

 - Great hecatonicosachoron

### Pentagrammal Faces

#### Great Stellated Dodecahedral Cells

* {

5
2

,
3
,
3
}

{\displaystyle \{\frac{5}{2}, 3, 3\}}

 - Great grand stellated hecatonicosachoron
* {

5
2

,
3
,
5
}

{\displaystyle \{\frac{5}{2},3,5\}}

 - Great stellated hecatonicosachoron

#### Small Stellated Dodecahedral Cells

* {

5
2

,
5
,
3
}

{\displaystyle \{\frac {5}{2}, 5, 3\}}

 - Small stellated hecatonicosachoron
* {

5
2

,
5
,

5
2

}

{\displaystyle \{\frac{5}{2},5,\frac{5}{2}\}}

 - Grand stellated hecatonicosachoron

### Hexagonal Faces

#### Hexagonal Tiling Cells

##### Euclidean Tilings

* {
6
,
3
,
2
}

{\displaystyle \{6,3,2\}}

 - Hexagonal tiling dichoron

##### Paracompact Hyperbolic Tilings

* {
6
,
3
,
3
}

{\displaystyle \{6,3,3\}}

 - Order-3 hexagonal tiling honeycomb
* {
6
,
3
,
4
}

{\displaystyle \{6,3,4\}}

 - Order-4 hexagonal tiling honeycomb
* {
6
,
3
,
5
}

{\displaystyle \{6,3,5\}}

 - Order-5 hexagonal tiling honeycomb
* {
6
,
3
,
6
}

{\displaystyle \{6,3,6\}}

 - Order-6 hexagonal tiling honeycomb

## 5-Dimensional

### Square Hosohedral Cells

#### Cubic Hosochoric Tera

##### Euclidean Tilings

* {
2
,
4
,
3
,
4
}

{\displaystyle \{2,4,3,4\}}

 - Cubic honeycomb hosoteron

### Tetrahedral Cells

#### Pentachoral Tera

##### Spherical Tilings

* {
3
,
3
,
3
,
3
}

{\displaystyle \{3,3,3,3\}}

 - Hexateron
* {
3
,
3
,
3
,
4
}

{\displaystyle \{3,3,3,4\}}

 - Pentacross

##### Compact Hyperbolic Tilings

* {
3
,
3
,
3
,
5
}

{\displaystyle \{3,3,3,5\}}

 - Order-5 pentachoric tetracomb

#### Hexadecachoral Tera

##### Euclidean Tilings

* {
3
,
3
,
4
,
3
}

{\displaystyle \{3,3,4,3\}}

 - Hexadecachoric tetracomb

### Octahedral Cells

#### Icositetrachoral Tera

##### Euclidean Tilings

* {
3
,
4
,
3
,
3
}

{\displaystyle \{3,4,3,3\}}

 - Icositetrachoric tetracomb

##### Paracompact Hyperbolic Tilings

* {
3
,
4
,
3
,
4
}

{\displaystyle \{3,4,3,4\}}

 - Icositetrachoric tetracomb

### Cubic Cells

#### Tesseractic Tera

##### Spherical Tilings

* {
4
,
3
,
3
,
3
}

{\displaystyle \{4,3,3,3\}}

 - Penteract

##### Euclidean Tilings

* {
4
,
3
,
3
,
4
}

{\displaystyle \{4,3,3,4\}}

 - Tesseractic tetracomb

##### Compact Hyperbolic Tilings

* {
4
,
3
,
3
,
5
}

{\displaystyle \{4,3,3,5\}}

 - Order-5 tesseractic tetracomb

#### Cubic Honeycomb Tera

##### Euclidean Tilings

* {
4
,
3
,
4
,
2
}

{\displaystyle \{4,3,4,2\}}

 - Cubic honeycomb diteron

##### Paracompact Hyperbolic Tilings

* {
4
,
3
,
4
,
3
}

{\displaystyle \{4,3,4,3\}}

 - Order-3 cubic honeycomb honeycomb

### Dodecahedral Cells

#### Hecatonicosachoric Tera

##### Compact Hyperbolic Tilings

* {
5
,
3
,
3
,
3
}

{\displaystyle \{5, 3, 3, 3\}}

 - Order-3 hecatonicosachoric tetracomb
* {
5
,
3
,
3
,
4
}

{\displaystyle \{5,3,3,4\}}

 - Order-4 hecatonicosachoric tetracomb
* {
5
,
3
,
3
,
5
}

{\displaystyle \{5,3,3,5\}}

 - Order-5 hecatonicosachoric tetracomb

## See Also

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