Schläfli icon

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 . ${\displaystyle \left\{p,q,r,\dots \right\}}$

If is a natural number , the symbol describes a regular polygon ( corner). ${\displaystyle p}$ ${\displaystyle {p}}$ ${\displaystyle p}$

If a fraction is not necessarily shortened , then it describes a star . ${\displaystyle p}$

The symbol describes a paving by means of a regular corner, indicating how many such polygons meet at each corner . ${\displaystyle \left\{p,q\right\}}$ ${\displaystyle p}$ ${\displaystyle q}$

The inversion of a Schläfli symbol provides the dual polygon for this.

Examples

Regular polygons

${\displaystyle \left\{n\right\}}$denotes a corner ${\displaystyle n}$ ${\displaystyle .}$

Stars

${\displaystyle \left\{5/2\right\}}$or denotes the pentagram of the pentagon ${\displaystyle \left\{5/3\right\}}$ ${\displaystyle .}$

${\displaystyle \left\{7/2\right\}}$or and or denote the two possible heptagrams of the heptagon and ${\displaystyle \left\{7/5\right\}}$ ${\displaystyle \left\{7/3\right\}}$ ${\displaystyle \left\{7/4\right\}}$ ${\displaystyle .}$

${\displaystyle \left\{9/2\right\}}$or and or denote the two possible Enneagrams from Neuneck and ${\displaystyle \left\{9/7\right\}}$ ${\displaystyle \left\{9/4\right\}}$ ${\displaystyle \left\{9/5\right\}}$ ${\displaystyle .}$

${\displaystyle \left\{10/3\right\}}$or denotes the decagram of the decagon ${\displaystyle \left\{10/7\right\}}$ ${\displaystyle .}$

${\displaystyle \left\{11/2\right\}}$or or or or denote the four possible hendekagrams from Elfeck and ${\displaystyle \left\{11/9\right\},\;\left\{11/3\right\}}$ ${\displaystyle \left\{11/8\right\},\;\left\{11/4\right\}}$ ${\displaystyle \left\{11/7\right\},\;\left\{11/5\right\}}$ ${\displaystyle \left\{11/6\right\}}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle .}$

${\displaystyle \left\{13/2\right\}}$or or or or or designate the five possible Tridekagramme from Dreizehneck and ${\displaystyle \left\{13/11\right\},\;\left\{13/3\right\}}$ ${\displaystyle \left\{13/10\right\},\;\left\{13/4\right\}}$ ${\displaystyle \left\{13/9\right\},\;\left\{13/5\right\}}$ ${\displaystyle \left\{13/8\right\},\;\left\{13/6\right\}}$ ${\displaystyle \left\{13/7\right\}}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle .}$

${\displaystyle \left\{14/3\right\}}$or and or denote the two possible tetradecagrams of the fourteenth and ${\displaystyle \left\{14/11\right\}}$ ${\displaystyle \left\{14/5\right\}}$ ${\displaystyle \left\{14/9\right\}}$ ${\displaystyle .}$

${\displaystyle \left\{15/2\right\}}$or or as well as or denote the three possible pentadecagrams from the fifteenth and ${\displaystyle \left\{15/13\right\},\;\left\{15/4\right\}}$ ${\displaystyle \left\{15/11\right\}}$ ${\displaystyle \left\{15/7\right\}}$ ${\displaystyle \left\{15/8\right\}}$ ${\displaystyle ,}$ ${\displaystyle .}$

${\displaystyle \left\{16/3\right\}}$or or as well as or denote the three possible hexadecagrams of the sixteenth and ${\displaystyle \left\{16/13\right\},\;\left\{16/5\right\}}$ ${\displaystyle \left\{16/11\right\}}$ ${\displaystyle \left\{16/7\right\}}$ ${\displaystyle \left\{16/9\right\}}$ ${\displaystyle ,}$ ${\displaystyle .}$

${\displaystyle \left\{17/2\right\}}$or or or or or or or denote the seven possible heptadecagrams from the seventeenth corner and ${\displaystyle \left\{17/15\right\},\;\left\{17/3\right\}}$ ${\displaystyle \left\{17/14\right\},\;\left\{17/4\right\}}$ ${\displaystyle \left\{17/13\right\},\;\left\{17/5\right\}}$ ${\displaystyle \left\{17/12\right\},\;\left\{17/6\right\}}$ ${\displaystyle \left\{17/11\right\},\;\left\{17/7\right\}}$ ${\displaystyle \left\{17/10\right\},\;\left\{17/8\right\}}$ ${\displaystyle \left\{17/9\right\}}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle .}$

${\displaystyle \left\{19/2\right\}}$or or or or or or or or denote the eight possible Enneadekagrams from the nineteenth and ${\displaystyle \left\{19/17\right\},\;\left\{19/3\right\}}$ ${\displaystyle \left\{19/16\right\},\;\left\{19/4\right\}}$ ${\displaystyle \left\{19/15\right\},\;\left\{19/5\right\}}$ ${\displaystyle \left\{19/14\right\},\;\left\{19/6\right\}}$ ${\displaystyle \left\{19/13\right\},\;\left\{19/7\right\}}$ ${\displaystyle \left\{19/12\right\},\;\left\{19/8\right\}}$ ${\displaystyle \left\{19/11\right\},\;\left\{19/9\right\}}$ ${\displaystyle \left\{19/10\right\}}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle ,}$ ${\displaystyle .}$

${\displaystyle \left\{20/3\right\}}$or or as well as or denote the three possible icons of the twenty- corner and ${\displaystyle \left\{20/17\right\},\;\left\{20/7\right\}}$ ${\displaystyle \left\{20/13\right\}}$ ${\displaystyle \left\{20/9\right\}}$ ${\displaystyle \left\{20/11\right\}}$ ${\displaystyle ,}$ ${\displaystyle .}$

Platonic solids

${\displaystyle \left\{p,q\right\}}$: p is the number of corners of the polygon used; q is the number of polygons that meet at a corner

${\displaystyle \left\{3,3\right\}}$denotes the self-dual tetrahedron .

${\displaystyle \left\{3,4\right\}}$denotes the octahedron , the inversion the dual cube to the octahedron . ${\displaystyle \left\{4,3\right\}}$

${\displaystyle \left\{3,5\right\}}$means the icosahedron , the inversion the dual for icosahedron dodecahedron . ${\displaystyle \left\{5,3\right\}}$

Platonic parquets

${\displaystyle \left\{3,6\right\}}$The triangular tiling denotes the inversion the hexagonal tiling, which is dual to the triangular tiling. ${\displaystyle \left\{6,3\right\}}$

${\displaystyle \left\{4,4\right\}}$ 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 . ${\displaystyle \left\{m,n\right\}}$ ${\displaystyle \left\{m,n\right\}}$ ${\displaystyle 2\cdot (m+n)>m\cdot n}$ ${\displaystyle 2\cdot (m+n)=m\cdot n}$

Kepler-Poinsot body

${\displaystyle \left\{3,5/2\right\}}$denotes the large icosahedron , the inversion the large star dodecahedron dual to the large icosahedron . ${\displaystyle \left\{5/2,3\right\}}$

${\displaystyle \left\{5,5/2\right\}}$denotes the large dodecahedron , the inversion the small star dodecahedron , which is dual to the large dodecahedron . ${\displaystyle \left\{5/2,5\right\}}$

Four-dimensional bodies

${\displaystyle \left\{3,3,3\right\}}$denotes the Pentachoron ,

${\displaystyle \left\{4,3,3\right\}}$the four-dimensional cube ( tesseract ), the dual to the regular 16-cell (Hexadekachor), ${\displaystyle \left\{3,3,4\right\}}$

${\displaystyle \left\{3,4,3\right\}}$the regular 24-cell (Ikositetrachor),

${\displaystyle \left\{5,3,3\right\}}$the regular 120-cell , the dual to the regular 600-cell . ${\displaystyle \left\{3,3,5\right\}}$

literature

• Harold Scott MacDonald Coxeter : Regular Polytopes .

Web links

• Eric W. Weisstein : Schläfli symbol . In: MathWorld (English).