The Schläfli symbol , named after the Swiss mathematician Ludwig Schläfli , is used in the form to describe regular polygons , polyhedra and other polyhedra, also in higher dimensions .
![\ left \ {p, q, r, \ dots \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cea406703465e1dda6a54a8ecd25a2a22d50fbe4)
If is a natural number , the symbol describes a regular polygon ( corner).
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![{p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60dc94eba00d06d7c095ade5f5a9e9a138a6efc1)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
If a fraction is not necessarily shortened , then it describes a star .
The symbol describes a paving by means of a regular corner, indicating how many such polygons meet at each corner .
![\ left \ {p, q \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b915248eca425e8034eea92c27e651f0eb69af4f)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![q](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)
The inversion of a Schläfli symbol provides the dual polygon for this.
Examples
denotes a corner![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
or denotes the pentagram of the pentagon
![Pentagram.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/1/12/Pentagram.svg/70px-Pentagram.svg.png)
or and or denote the two possible heptagrams of the heptagon and![{\ displaystyle \ left \ {7/5 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/249aa46f9ce8b565ecef186567d1ab23883eccd4)
![\ left \ {7/3 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2224a01d3d8518e80c363290007e2b18b70c5272)
![Obtuse heptagram.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Obtuse_heptagram.svg/70px-Obtuse_heptagram.svg.png)
![Acute heptagram.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/8/83/Acute_heptagram.svg/70px-Acute_heptagram.svg.png)
or and or denote the two possible Enneagrams from Neuneck and![{\ displaystyle \ left \ {9/7 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/988bc7834571a1b7cc491c53c8b669b98378a570)
![{\ displaystyle \ left \ {9/4 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2257d77c926449109db329dc002a516a965fd71)
![01 Neuneck-Stern-9-2-7.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0a/01_Neuneck-Stern-9-2-7.svg/70px-01_Neuneck-Stern-9-2-7.svg.png)
![01 Neuneck-Stern-9-4-5.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8d/01_Neuneck-Stern-9-4-5.svg/70px-01_Neuneck-Stern-9-4-5.svg.png)
or denotes the decagram of the decagon
![01-decagon-star-10-3-7.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/d/dd/01-Zehneck-Stern-10-3-7.svg/70px-01-Zehneck-Stern-10-3-7.svg.png)
or or or or denote the four possible hendekagrams from Elfeck and![{\ displaystyle \ left \ {11/9 \ right \}, \; \ left \ {11/3 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3a6d05761e8a775551263af1ef7923f724720e)
![{\ displaystyle \ left \ {11/8 \ right \}, \; \ left \ {11/4 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39ff92cabee946d327921c804e2f630d146126da)
![{\ displaystyle \ left \ {11/7 \ right \}, \; \ left \ {11/5 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54c1a5b57b760217fbc9748db18f9649cc8d1746)
![01 Elfeck-Stern-11-2-9.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c4/01_Elfeck-Stern-11-2-9.svg/70px-01_Elfeck-Stern-11-2-9.svg.png)
![01 Elfeck-Stern-11-3-8.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/a/af/01_Elfeck-Stern-11-3-8.svg/70px-01_Elfeck-Stern-11-3-8.svg.png)
![01 Elfeck-Stern-11-4-7.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/01_Elfeck-Stern-11-4-7.svg/70px-01_Elfeck-Stern-11-4-7.svg.png)
![01 Elfeck-Stern-11-5-6.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/9/93/01_Elfeck-Stern-11-5-6.svg/70px-01_Elfeck-Stern-11-5-6.svg.png)
or or or or or designate the five possible Tridekagramme from Dreizehneck and![{\ displaystyle \ left \ {13/11 \ right \}, \; \ left \ {13/3 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f55ad28eeef1e5a0116326fe37db2ae4b89b7121)
![{\ displaystyle \ left \ {13/10 \ right \}, \; \ left \ {13/4 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ae28ebea045f73ac8d1ff1d85a22c92bbe8411a)
![{\ displaystyle \ left \ {13/9 \ right \}, \; \ left \ {13/5 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ba03c47f7a0f0ffe8cf4a1e77b813c24892994)
![{\ displaystyle \ left \ {13/8 \ right \}, \; \ left \ {13/6 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dbb48c2b58a96ca29386572c070c3d13c9186ee)
![01-triangular-star-13-2-11.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a6/01-Dreizehneck-Stern-13-2-11.svg/70px-01-Dreizehneck-Stern-13-2-11.svg.png)
![01-triangular-star-13-3-10.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6d/01-Dreizehneck-Stern-13-3-10.svg/70px-01-Dreizehneck-Stern-13-3-10.svg.png)
![01-triangular-star-13-4-9.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/9/9d/01-Dreizehneck-Stern-13-4-9.svg/70px-01-Dreizehneck-Stern-13-4-9.svg.png)
![01-triangular-star-13-5-8.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/01-Dreizehneck-Stern-13-5-8.svg/70px-01-Dreizehneck-Stern-13-5-8.svg.png)
![01-triangular-star-13-6-7.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/9/97/01-Dreizehneck-Stern-13-6-7.svg/70px-01-Dreizehneck-Stern-13-6-7.svg.png)
or and or denote the two possible tetradecagrams of the fourteenth and![{\ displaystyle \ left \ {14/11 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11c3ca1356abd428920447071cff1d4a20a382ac)
![{\ displaystyle \ left \ {14/5 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e0690941ebc6c385dd4e76f8d72c666f44b6b2)
![01-fourteen-corner-star-14-3-11.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/2/28/01-Vierzehneck-Stern-14-3-11.svg/70px-01-Vierzehneck-Stern-14-3-11.svg.png)
![01-fourteen-corner-star-14-5-9.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/01-Vierzehneck-Stern-14-5-9.svg/70px-01-Vierzehneck-Stern-14-5-9.svg.png)
or or as well as or denote the three possible pentadecagrams from the fifteenth and![{\ displaystyle \ left \ {15/13 \ right \}, \; \ left \ {15/4 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/948e35bbcc08a3afb53fcecc2803cdebcbd5372e)
![{\ displaystyle \ left \ {15/11 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8820e6a07791829ef69437b6c198535146c7b8)
![{\ displaystyle \ left \ {15/7 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/035103a9597665edd008b7def8297dbf1ac398d1)
![01-fifteen-pointed-star-15-4-1.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7e/01-F%C3%BCnfzehneck-Stern-15-4-1.svg/70px-01-F%C3%BCnfzehneck-Stern-15-4-1.svg.png)
![01-fifteen-pointed-star-15-7-1.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b8/01-F%C3%BCnfzehneck-Stern-15-7-1.svg/70px-01-F%C3%BCnfzehneck-Stern-15-7-1.svg.png)
or or as well as or denote the three possible hexadecagrams of the sixteenth and![{\ displaystyle \ left \ {16/13 \ right \}, \; \ left \ {16/5 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe480ef08ea78c3394c3283977eac3563c92fda)
![{\ displaystyle \ left \ {16/11 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb19993ed09a7e654de02442d31dd260a9af445)
![{\ displaystyle \ left \ {16/7 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa3aa8d380391855bcbea75aeaf8296208335b6)
![01 hexagon-16-3-13.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/9/99/01_Sechzehneck-16-3-13.svg/70px-01_Sechzehneck-16-3-13.svg.png)
![01 Hexagon-16-5-11.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6b/01_Sechzehneck-16-5-11.svg/70px-01_Sechzehneck-16-5-11.svg.png)
![01 hexagon-16-7-9.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/4/42/01_Sechzehneck-16-7-9.svg/70px-01_Sechzehneck-16-7-9.svg.png)
or or or or or or or denote the seven possible heptadecagrams from the seventeenth corner and![{\ displaystyle \ left \ {17/15 \ right \}, \; \ left \ {17/3 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8dc3c80207e540bd604eaff909614bf4bdc3366)
![{\ displaystyle \ left \ {17/14 \ right \}, \; \ left \ {17/4 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63bfeb5fe16db5fb609c399d9886df694bb217f3)
![{\ displaystyle \ left \ {17/13 \ right \}, \; \ left \ {17/5 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaebb9a7baa021f462cf21ed04a898496edb80fa)
![{\ displaystyle \ left \ {17/12 \ right \}, \; \ left \ {17/6 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd212e0e845fe916b49cd8bee8ea7782b1554fd)
![{\ displaystyle \ left \ {17/11 \ right \}, \; \ left \ {17/7 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c14e22fab9ad635c402c8d0f3fb97d511a60391)
![{\ displaystyle \ left \ {17/10 \ right \}, \; \ left \ {17/8 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1777d63092756c6b5ba74d356b921e257e185e)
![01-Siebzehneck-Stern-17-2-15.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/8/80/01-Siebzehneck-Stern-17-2-15.svg/70px-01-Siebzehneck-Stern-17-2-15.svg.png)
![01-17-corner-star-17-3-14.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/6/69/01-Siebzehneck-Stern-17-3-14.svg/70px-01-Siebzehneck-Stern-17-3-14.svg.png)
![01-Seventeen-star-17-4-13.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/5/52/01-Siebzehneck-Stern-17-4-13.svg/70px-01-Siebzehneck-Stern-17-4-13.svg.png)
![01-Siebzehneck-Stern-17-6-11.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/01-Siebzehneck-Stern-17-6-11.svg/70px-01-Siebzehneck-Stern-17-6-11.svg.png)
![01-Siebzehneck-Stern-17-7-10.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/c/ca/01-Siebzehneck-Stern-17-7-10.svg/70px-01-Siebzehneck-Stern-17-7-10.svg.png)
![01-Seventeen-star-17-8-9.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/3/35/01-Siebzehneck-Stern-17-8-9.svg/70px-01-Siebzehneck-Stern-17-8-9.svg.png)
or or or or or or or or denote the eight possible Enneadekagrams from the nineteenth and![{\ displaystyle \ left \ {19/17 \ right \}, \; \ left \ {19/3 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a818908f8f5a7396814f1dcc401da3fbdb6d989)
![{\ displaystyle \ left \ {19/16 \ right \}, \; \ left \ {19/4 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/febc94cbce2c4d8e4287b9aa397272031eb44a34)
![{\ displaystyle \ left \ {19/15 \ right \}, \; \ left \ {19/5 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/032c6440ed1e3e337bd6cbd258d4bb54778852ff)
![{\ displaystyle \ left \ {19/14 \ right \}, \; \ left \ {19/6 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7016001e5cbec1c856227e249cd822b6f540766a)
![{\ displaystyle \ left \ {19/13 \ right \}, \; \ left \ {19/7 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cab71b09f76b2978d841d2177085249aa947b5df)
![{\ displaystyle \ left \ {19/12 \ right \}, \; \ left \ {19/8 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a09230c50f173b7ea333d350aba9d53c4b911eb0)
![{\ displaystyle \ left \ {19/11 \ right \}, \; \ left \ {19/9 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4ee05030bc2b26bd2dc242759ebfd4f814577c)
![01-nineteenth-star-19-2-17.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/a/ac/01-Neunzehneck-Stern-19-2-17.svg/70px-01-Neunzehneck-Stern-19-2-17.svg.png)
![01-nineteenth-star-19-3-16.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/4/40/01-Neunzehneck-Stern-19-3-16.svg/70px-01-Neunzehneck-Stern-19-3-16.svg.png)
![01-nineteenth-star-19-4-15.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b2/01-Neunzehneck-Stern-19-4-15.svg/70px-01-Neunzehneck-Stern-19-4-15.svg.png)
![01-nineteenth-star-19-6-13.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/2/25/01-Neunzehneck-Stern-19-6-13.svg/70px-01-Neunzehneck-Stern-19-6-13.svg.png)
![01-nineteenth-star-19-7-12.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/1/17/01-Neunzehneck-Stern-19-7-12.svg/70px-01-Neunzehneck-Stern-19-7-12.svg.png)
![01-nineteenth-star-19-8-11.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/8/80/01-Neunzehneck-Stern-19-8-11.svg/70px-01-Neunzehneck-Stern-19-8-11.svg.png)
![01-nineteenth-star-19-9-10.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7d/01-Neunzehneck-Stern-19-9-10.svg/70px-01-Neunzehneck-Stern-19-9-10.svg.png)
or or as well as or denote the three possible icons of the twenty- corner and![{\ displaystyle \ left \ {20/17 \ right \}, \; \ left \ {20/7 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b06d251010ffe4889bd306a92abfa91d3816e1e6)
![{\ displaystyle \ left \ {20/13 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67860181af0872571368ce3a131e464618523f6d)
![{\ displaystyle \ left \ {20/9 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67de5281877a1d06b94d0b6908c2e7c6a21eaa94)
![01 Zwanzigeck-Stern-20-3-17.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6c/01_Zwanzigeck-Stern-20-3-17.svg/70px-01_Zwanzigeck-Stern-20-3-17.svg.png)
![01 Zwanzigeck-Stern-20-7-13.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7d/01_Zwanzigeck-Stern-20-7-13.svg/70px-01_Zwanzigeck-Stern-20-7-13.svg.png)
![01 Zwanzigeck-Stern-20-9-11.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/01_Zwanzigeck-Stern-20-9-11.svg/70px-01_Zwanzigeck-Stern-20-9-11.svg.png)
: p is the number of corners of the polygon used; q is the number of polygons that meet at a corner
denotes the self-dual tetrahedron .
denotes the octahedron , the inversion the dual cube to the octahedron .
![\ left \ {4, 3 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1714062082d858a5ebce2c44be467ab3e9c0a855)
means the icosahedron , the inversion the dual for icosahedron dodecahedron .
![\ left \ {5, 3 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eca7c8ec1c2825f77de88ffb0013b94f37dfe19b)
The triangular tiling denotes the inversion the hexagonal tiling, which is dual to the triangular tiling.
![\ left \ {6, 3 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3a1b01c27a114bc1b24d340ad56005e15d4ca77)
describes the self-dual square tiling.
- The decisive feature in which the Schläfli symbol of a Platonic solid differs from that of a Platonic parquet is that it applies to a solid , but to a parquet .
![{\ displaystyle \ left \ {m, n \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74af2f81fbbd224f31a08c903a1afec7f20e709f)
![{\ displaystyle \ left \ {m, n \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74af2f81fbbd224f31a08c903a1afec7f20e709f)
![{\ displaystyle 2 \ cdot (m + n)> m \ cdot n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f99acba39be477eafba9525451fc3125f43dbe8)
![{\ displaystyle 2 \ cdot (m + n) = m \ cdot n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd43461f48b042f79df5091ef8c4bbdf06594dae)
denotes the large icosahedron , the inversion the large star dodecahedron dual to the large icosahedron .
![\ left \ {5/2, 3 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e74e402a83ddc58fbe65d7c8347f224862eaa70)
denotes the large dodecahedron , the inversion the small star dodecahedron , which is dual to the large dodecahedron .
![\ left \ {5/2, 5 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2a8de366464fc489abc37d0b859b53a07ddd27)
denotes the Pentachoron ,
the four-dimensional cube ( tesseract ), the dual to the regular 16-cell (Hexadekachor),
![\ left \ {3,3,4 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43b5aa7cff5d880177ee31873b7063442f96eaa4)
the regular 24-cell (Ikositetrachor),
the regular 120-cell , the dual to the regular 600-cell .
![\ left \ {3,3,5 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29905a0c3e0f6115bfe2274165fea71f822f430b)
literature
Web links