(
)
{\displaystyle \left(\right)}
{
}
{\displaystyle \left\{\right\}}
{
n
}
{\displaystyle \left\{n\right\}}
n
{\displaystyle n}
.
{\displaystyle .}
Notiert werden solche regelmäßigen Sterne mit Schläfli-Symbolen
{
n
/
k
}
{\displaystyle \left\{n/k\right\}}
n
{\displaystyle n}
k
{\displaystyle k}
Beispiel
Der Fünfstrahlstern ergibt sich, wenn beim Verbinden der fünf Eckpunkte jedes Mal
immer einer (beim
{
5
/
2
}
{\displaystyle \left\{5/2\right\}}
immer zwei Punkte (beim
{
5
/
3
}
{\displaystyle \left\{5/3\right\}}
keine vom Dreieck
keine vom Viereck
1 Pentagramm Fünfeck
{
5
/
2
}
{\displaystyle \left\{5/2\right\}}
{
5
/
3
}
{\displaystyle \left\{5/3\right\}}
{\displaystyle }
keine vom Sechseck
2 Heptagramme Siebeneck
{
7
/
2
}
{\displaystyle \left\{7/2\right\}}
{
7
/
5
}
{\displaystyle \left\{7/5\right\}}
{
7
/
3
}
{\displaystyle \left\{7/3\right\}}
{
7
/
4
}
{\displaystyle \left\{7/4\right\}}
1 Oktogramm Achteck
{
8
/
3
}
{\displaystyle \left\{8/3\right\}}
{
8
/
5
}
{\displaystyle \left\{8/5\right\}}
2 Ennea- Neuneck
{
9
/
2
}
{\displaystyle \left\{9/2\right\}}
{
9
/
7
}
{\displaystyle \left\{9/7\right\}}
{
9
/
4
}
{\displaystyle \left\{9/4\right\}}
{
9
/
5
}
{\displaystyle \left\{9/5\right\}}
1 DekagrammZehneck
{
10
/
3
}
{\displaystyle \left\{10/3\right\}}
{
10
/
7
}
{\displaystyle \left\{10/7\right\}}
4 Hendeka-Elfeck
{
11
/
2
}
{\displaystyle \left\{11/2\right\}}
{
11
/
9
}
{\displaystyle \left\{11/9\right\}}
{
11
/
3
}
{\displaystyle \left\{11/3\right\}}
{
11
/
8
}
{\displaystyle \left\{11/8\right\}}
{
11
/
4
}
{\displaystyle \left\{11/4\right\}}
{
11
/
7
}
{\displaystyle \left\{11/7\right\}}
{
11
/
5
}
{\displaystyle \left\{11/5\right\}}
{
11
/
6
}
{\displaystyle \left\{11/6\right\}}
1 Dodeka-Zwölfeck
{
12
/
5
}
{\displaystyle \left\{12/5\right\}}
{
12
/
7
}
{\displaystyle \left\{12/7\right\}}
5 Trideka-13-Eck
{
13
/
2
}
{\displaystyle \left\{13/2\right\}}
{
13
/
11
}
{\displaystyle \left\{13/11\right\}}
{
13
/
3
}
{\displaystyle \left\{13/3\right\}}
{
13
/
10
}
{\displaystyle \left\{13/10\right\}}
{
13
/
4
}
{\displaystyle \left\{13/4\right\}}
{
13
/
9
}
{\displaystyle \left\{13/9\right\}}
{
13
/
5
}
{\displaystyle \left\{13/5\right\}}
{
13
/
8
}
{\displaystyle \left\{13/8\right\}}
{
13
/
6
}
{\displaystyle \left\{13/6\right\}}
{
13
/
7
}
{\displaystyle \left\{13/7\right\}}
2 Tetradeka-14-Eck
{
14
/
3
}
{\displaystyle \left\{14/3\right\}}
{
14
/
11
}
{\displaystyle \left\{14/11\right\}}
{
14
/
5
}
{\displaystyle \left\{14/5\right\}}
{
14
/
9
}
{\displaystyle \left\{14/9\right\}}
3 Pentadeka-15-Eck
{
15
/
2
}
{\displaystyle \left\{15/2\right\}}
{
15
/
13
}
{\displaystyle \left\{15/13\right\}}
{
15
/
4
}
{\displaystyle \left\{15/4\right\}}
{
15
/
11
}
{\displaystyle \left\{15/11\right\}}
{
15
/
7
}
{\displaystyle \left\{15/7\right\}}
{
15
/
8
}
{\displaystyle \left\{15/8\right\}}
3 Hexadeka-16-Eck
{
16
/
3
}
{\displaystyle \left\{16/3\right\}}
{
16
/
13
}
{\displaystyle \left\{16/13\right\}}
{
16
/
5
}
{\displaystyle \left\{16/5\right\}}
{
16
/
11
}
{\displaystyle \left\{16/11\right\}}
{
16
/
7
}
{\displaystyle \left\{16/7\right\}}
{
16
/
9
}
{\displaystyle \left\{16/9\right\}}
7 Heptadeka-17-Eck
{
17
/
2
}
{\displaystyle \left\{17/2\right\}}
{
17
/
15
}
{\displaystyle \left\{17/15\right\}}
{
17
/
3
}
{\displaystyle \left\{17/3\right\}}
{
17
/
14
}
{\displaystyle \left\{17/14\right\}}
{
17
/
4
}
{\displaystyle \left\{17/4\right\}}
{
17
/
13
}
{\displaystyle \left\{17/13\right\}}
{
17
/
5
}
{\displaystyle \left\{17/5\right\}}
{
17
/
12
}
{\displaystyle \left\{17/12\right\}}
{
17
/
6
}
{\displaystyle \left\{17/6\right\}}
{
17
/
11
}
{\displaystyle \left\{17/11\right\}}
{
17
/
7
}
{\displaystyle \left\{17/7\right\}}
{
17
/
10
}
{\displaystyle \left\{17/10\right\}}
{
17
/
8
}
{\displaystyle \left\{17/8\right\}}
{
17
/
9
}
{\displaystyle \left\{17/9\right\}}
2 Oktodeka-18-Eck
{
18
/
5
}
{\displaystyle \left\{18/5\right\}}
{
18
/
13
}
{\displaystyle \left\{18/13\right\}}
{
18
/
7
}
{\displaystyle \left\{18/7\right\}}
{
18
/
11
}
{\displaystyle \left\{18/11\right\}}
8 Enneadeka-19-Eck
{
19
/
2
}
{\displaystyle \left\{19/2\right\}}
{
19
/
17
}
{\displaystyle \left\{19/17\right\}}
{
19
/
3
}
{\displaystyle \left\{19/3\right\}}
{
19
/
16
}
{\displaystyle \left\{19/16\right\}}
{
19
/
4
}
{\displaystyle \left\{19/4\right\}}
{
19
/
15
}
{\displaystyle \left\{19/15\right\}}
{
19
/
5
}
{\displaystyle \left\{19/5\right\}}
{
19
/
14
}
{\displaystyle \left\{19/14\right\}}
{
19
/
6
}
{\displaystyle \left\{19/6\right\}}
{
19
/
13
}
{\displaystyle \left\{19/13\right\}}
{
19
/
7
}
{\displaystyle \left\{19/7\right\}}
{
19
/
12
}
{\displaystyle \left\{19/12\right\}}
{
19
/
8
}
{\displaystyle \left\{19/8\right\}}
{
19
/
11
}
{\displaystyle \left\{19/11\right\}}
{
19
/
9
}
{\displaystyle \left\{19/9\right\}}
{
19
/
10
}
{\displaystyle \left\{19/10\right\}}
3 Ikosa-20-Eck
{
20
/
3
}
{\displaystyle \left\{20/3\right\}}
{
20
/
17
}
{\displaystyle \left\{20/17\right\}}
{
20
/
7
}
{\displaystyle \left\{20/7\right\}}
{
20
/
13
}
{\displaystyle \left\{20/13\right\}}
{
20
/
9
}
{\displaystyle \left\{20/9\right\}}
{
20
/
11
}
{\displaystyle \left\{20/11\right\}}
5 Ikosihen-21-Eck
{
21
/
2
}
{\displaystyle \left\{21/2\right\}}
{
21
/
19
}
{\displaystyle \left\{21/19\right\}}
{
21
/
4
}
{\displaystyle \left\{21/4\right\}}
{
21
/
17
}
{\displaystyle \left\{21/17\right\}}
{
21
/
5
}
{\displaystyle \left\{21/5\right\}}
{
21
/
16
}
{\displaystyle \left\{21/16\right\}}
{
21
/
8
}
{\displaystyle \left\{21/8\right\}}
{
21
/
13
}
{\displaystyle \left\{21/13\right\}}
{
21
/
10
}
{\displaystyle \left\{21/10\right\}}
{
21
/
11
}
{\displaystyle \left\{21/11\right\}}
0 4 Doikosagramme vom 22-Eck
10 Trikosagramme vom 23-Eck
0 3 Tetraikosagramme vom 24-Eck
{
p
,
q
}
{\displaystyle \left\{p,q\right\}}
{
3
,
3
}
{\displaystyle \left\{3,3\right\}}
selbstduale Tetraeder .
{
3
,
4
}
{\displaystyle \left\{3,4\right\}}
Oktaeder , die Inversion
{
4
,
3
}
{\displaystyle \left\{4,3\right\}}
Würfel .
{
3
,
5
}
{\displaystyle \left\{3,5\right\}}
Ikosaeder , die Inversion
{
5
,
3
}
{\displaystyle \left\{5,3\right\}}
Dodekaeder .
{
3
,
6
}
{\displaystyle \left\{3,6\right\}}
{
6
,
3
}
{\displaystyle \left\{6,3\right\}}
{
4
,
4
}
{\displaystyle \left\{4,4\right\}}
Das entscheidende Merkmal, worin sich das Schläfli-Symbol eines Platonischen Körpers
{
m
,
n
}
{\displaystyle \left\{m,n\right\}}
{
m
,
n
}
{\displaystyle \left\{m,n\right\}}
2
⋅
(
m
+
n
)
>
m
⋅
n
{\displaystyle 2\cdot (m+n)>m\cdot n}
2
⋅
(
m
+
n
)
=
m
⋅
n
{\displaystyle 2\cdot (m+n)=m\cdot n}
{
3
,
5
/
2
}
{\displaystyle \left\{3,5/2\right\}}
Große Ikosaeder , die Inversion
{
5
/
2
,
3
}
{\displaystyle \left\{5/2,3\right\}}
Große Sterndodekaeder .
{
5
,
5
/
2
}
{\displaystyle \left\{5,5/2\right\}}
Große Dodekaeder , die Inversion
{
5
/
2
,
5
}
{\displaystyle \left\{5/2,5\right\}}
Kleine Sterndodekaeder .
{
3
,
3
,
3
}
{\displaystyle \left\{3,3,3\right\}}
Pentachoron ,
{
4
,
3
,
3
}
{\displaystyle \left\{4,3,3\right\}}
Tesserakt ), das Duale
{
3
,
3
,
4
}
{\displaystyle \left\{3,3,4\right\}}
regulären 16-Zeller (Hexadekachor),
{
3
,
4
,
3
}
{\displaystyle \left\{3,4,3\right\}}
regulären 24-Zeller (Ikositetrachor).
{
3
,
3
,
3
,
3
}
{\displaystyle \left\{3,3,3,3\right\}}
{
3
4
}
{\displaystyle \left\{3^{4}\right\}}
5-Simplex
S
5
{\displaystyle {\mathfrak {S}}_{5}}
{
3
,
3
,
3
,
3
,
3
}
{\displaystyle \left\{3,3,3,3,3\right\}}
{
3
5
}
{\displaystyle \left\{3^{5}\right\}}
6-Simplex
S
6
{\displaystyle {\mathfrak {S}}_{6}}
{
3
d
−
1
}
{\displaystyle \left\{3^{d-1}\right\}}
d-Simplex
S
d
{\displaystyle {\mathfrak {S}}_{\;\!\!d}}
