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The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p0736801.png" /> of symmetries of a polytope (cf. [[Polyhedron|Polyhedron]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p0736802.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p0736803.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p0736804.png" />, that is, the group of all motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p0736805.png" /> which send <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p0736806.png" /> to itself. A polytope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p0736807.png" /> is called regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p0736808.png" /> acts transitively on the set of its  "flag set of a polytopeflags" , that is, collections
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p0736809.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368010.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368011.png" />-dimensional closed face and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368012.png" />. The group of symmetries of a regular polytope is generated by reflections (see [[Reflection group|Reflection group]]). Its fundamental domain is a simplicial cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368013.png" /> whose vertex is the centre of the polytope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368014.png" />, and whose edges pass through the centres of the faces constituting some flag <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368015.png" />. By the same token the generating reflections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368016.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368017.png" /> have a natural enumeration: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368018.png" /> is the reflection relative to the hyperplane bounding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368019.png" /> which does not pass through the centre of the face <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368020.png" />. The generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368022.png" /> commute for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368023.png" />, and the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368024.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368025.png" /> — the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368026.png" />-dimensional (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368027.png" />-dimensional) faces of the polytope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368028.png" /> containing the face <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368029.png" /> (if it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368031.png" />). The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368032.png" /> is called the Schläfli symbol of the polytopes. The three-dimensional regular polytopes (Platonic solids) have the following Schläfli symbols: the [[Tetrahedron|tetrahedron]] — <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368033.png" />, the [[Cube|cube]] — <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368034.png" />, the [[Octahedron|octahedron]] — <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368035.png" />, the [[Dodecahedron|dodecahedron]] — <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368036.png" />, and the [[Icosahedron|icosahedron]] — <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368037.png" />.
+
The group $  \mathop{\rm Sym}  P $
 +
of symmetries of a polytope (cf. [[Polyhedron|Polyhedron]]) $  P $
 +
in an  $  n $-
 +
dimensional Euclidean space  $  E  ^ {n} $,
 +
that is, the group of all motions of $  E  ^ {n} $
 +
which send  $  P $
 +
to itself. A polytope $  P $
 +
is called regular if  $  \mathop{\rm Sym}  P $
 +
acts transitively on the set of its  "flag set of a polytopeflags" , that is, collections
  
The Schläfli symbol determines a regular polytope up to a similarity. Reversal of a Schläfli symbol corresponds to transition to the reciprocal polytope, whose vertices ly at the centres of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368038.png" />-dimensional faces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368039.png" />. Reciprocal polytopes have the same symmetry group.
+
$$
 +
= \{ \Gamma _ {0} \dots \Gamma _ {n-} 1 \}
 +
$$
  
All possible Schläfli symbols of regular polytopes can be obtained from the classification of finite reflection groups, by selecting those with a linear Coxeter graph. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368040.png" /> there are only 3 regular polytopes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368041.png" />: the simplex, the cube and the polytope reciprocal to the cube (the analogue of the octahedron). Their Schläfli symbols are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368044.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368045.png" />-dimensional space there are 6 regular polytopes: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368050.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368051.png" />.
+
where  $  \Gamma _ {k} $
 +
is a  $  k $-
 +
dimensional closed face and  $  \Gamma _ {k-} 1 \subset  \Gamma _ {k} $.
 +
The group of symmetries of a regular polytope is generated by reflections (see [[Reflection group|Reflection group]]). Its fundamental domain is a simplicial cone  $  K $
 +
whose vertex is the centre of the polytope  $  P $,  
 +
and whose edges pass through the centres of the faces constituting some flag  $  F $.
 +
By the same token the generating reflections  $  r _ {1} \dots r _ {n} $
 +
of the group  $  \mathop{\rm Sym}  P $
 +
have a natural enumeration: $  r _ {k} $
 +
is the reflection relative to the hyperplane bounding  $  K $
 +
which does not pass through the centre of the face  $  \Gamma _ {k-} 1 $.  
 +
The generators  $  r _ {k} $
 +
and  $  r _ {l} $
 +
commute for  $  | k - l | \geq  2 $,  
 +
and the order of  $  r _ {k} r _ {k+} 1 $
 +
is equal to $  p _ {k} $—
 +
the number of  $  k $-
 +
dimensional (or  $  ( k- 1 ) $-
 +
dimensional) faces of the polytope  $  \Gamma _ {k+} 1 $
 +
containing the face  $  \Gamma _ {k-} 2 $(
 +
if it is assumed that  $  \Gamma _ {n} = P $
 +
and $  \Gamma _ {-} 1 = \emptyset $).  
 +
The sequence  $  \{ p _ {1} \dots p _ {n-} 1 \} $
 +
is called the Schläfli symbol of the polytopes. The three-dimensional regular polytopes (Platonic solids) have the following Schläfli symbols: the [[Tetrahedron|tetrahedron]] —  $  \{ 3 , 3 \} $,
 +
the [[Cube|cube]] —  $  \{ 4 , 3 \} $,  
 +
the [[Octahedron|octahedron]] —  $  \{ 3 , 4 \} $,  
 +
the [[Dodecahedron|dodecahedron]] —  $  \{ 5 , 3 \} $,  
 +
and the [[Icosahedron|icosahedron]] —  $  \{ 3 , 5 \} $.
  
Each face of a regular polytope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368052.png" /> is also a regular polytope, the Schläfli symbol of which is the initial segment of the Schläfli symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368053.png" />. For example, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368054.png" />-dimensional face of the polytope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368055.png" /> has the Schläfli symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368056.png" />, that is, it is a [[Dodecahedron|dodecahedron]].
+
The Schläfli symbol determines a regular polytope up to a similarity. Reversal of a Schläfli symbol corresponds to transition to the reciprocal polytope, whose vertices ly at the centres of the  $  ( n- 1 ) $-
 +
dimensional faces of  $  P $.
 +
Reciprocal polytopes have the same symmetry group.
 +
 
 +
All possible Schläfli symbols of regular polytopes can be obtained from the classification of finite reflection groups, by selecting those with a linear Coxeter graph. For  $  n\geq  5 $
 +
there are only 3 regular polytopes in  $  E  ^ {n} $:
 +
the simplex, the cube and the polytope reciprocal to the cube (the analogue of the octahedron). Their Schläfli symbols are  $  \{ 3 \dots 3 \} $,
 +
$  \{ 4 , 3 \dots 3 \} $
 +
and  $  \{ 3 \dots 3 , 4 \} $.  
 +
In  $  4 $-
 +
dimensional space there are 6 regular polytopes:  $  \{ 3 , 3 , 3 \} $,
 +
$  \{ 4 , 3 , 3 \} $,
 +
$  \{ 3 , 3 , 4 \} $,
 +
$  \{ 3 , 4 , 3 \} $,
 +
$  \{ 5 , 3 , 3 \} $,
 +
and  $  \{ 3 , 3 , 5 \} $.
 +
 
 +
Each face of a regular polytope  $  P $
 +
is also a regular polytope, the Schläfli symbol of which is the initial segment of the Schläfli symbol of $  P $.  
 +
For example, a $  3 $-
 +
dimensional face of the polytope $  \{ 5 , 3 , 3 \} $
 +
has the Schläfli symbol $  \{ 5 , 3 \} $,  
 +
that is, it is a [[Dodecahedron|dodecahedron]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Regular polytopes" , Dover, reprint  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Multi-dimensional spaces" , Moscow  (1966)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Regular polytopes" , Dover, reprint  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Multi-dimensional spaces" , Moscow  (1966)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
A presentation of the polyhedron group is given by
 
A presentation of the polyhedron group is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368057.png" /></td> </tr></table>
+
$$
 +
< r _ {1} \dots r _ {n}  \mid  ( r _ {k} r _ {l} )  ^ {2} = 1 \ \
 +
\textrm{ for }  | k- l | \geq  2 ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073680/p07368058.png" /></td> </tr></table>
+
$$
 +
( r _ {k} r _ {k+} 1 )  ^ {2}  = p _ {k} \  \textrm{ for }  k = 1 \dots n- 1 > .
 +
$$
  
 
This shows that this group is a [[Coxeter group|Coxeter group]].
 
This shows that this group is a [[Coxeter group|Coxeter group]].

Latest revision as of 08:06, 6 June 2020


The group $ \mathop{\rm Sym} P $ of symmetries of a polytope (cf. Polyhedron) $ P $ in an $ n $- dimensional Euclidean space $ E ^ {n} $, that is, the group of all motions of $ E ^ {n} $ which send $ P $ to itself. A polytope $ P $ is called regular if $ \mathop{\rm Sym} P $ acts transitively on the set of its "flag set of a polytopeflags" , that is, collections

$$ F = \{ \Gamma _ {0} \dots \Gamma _ {n-} 1 \} $$

where $ \Gamma _ {k} $ is a $ k $- dimensional closed face and $ \Gamma _ {k-} 1 \subset \Gamma _ {k} $. The group of symmetries of a regular polytope is generated by reflections (see Reflection group). Its fundamental domain is a simplicial cone $ K $ whose vertex is the centre of the polytope $ P $, and whose edges pass through the centres of the faces constituting some flag $ F $. By the same token the generating reflections $ r _ {1} \dots r _ {n} $ of the group $ \mathop{\rm Sym} P $ have a natural enumeration: $ r _ {k} $ is the reflection relative to the hyperplane bounding $ K $ which does not pass through the centre of the face $ \Gamma _ {k-} 1 $. The generators $ r _ {k} $ and $ r _ {l} $ commute for $ | k - l | \geq 2 $, and the order of $ r _ {k} r _ {k+} 1 $ is equal to $ p _ {k} $— the number of $ k $- dimensional (or $ ( k- 1 ) $- dimensional) faces of the polytope $ \Gamma _ {k+} 1 $ containing the face $ \Gamma _ {k-} 2 $( if it is assumed that $ \Gamma _ {n} = P $ and $ \Gamma _ {-} 1 = \emptyset $). The sequence $ \{ p _ {1} \dots p _ {n-} 1 \} $ is called the Schläfli symbol of the polytopes. The three-dimensional regular polytopes (Platonic solids) have the following Schläfli symbols: the tetrahedron — $ \{ 3 , 3 \} $, the cube — $ \{ 4 , 3 \} $, the octahedron — $ \{ 3 , 4 \} $, the dodecahedron — $ \{ 5 , 3 \} $, and the icosahedron — $ \{ 3 , 5 \} $.

The Schläfli symbol determines a regular polytope up to a similarity. Reversal of a Schläfli symbol corresponds to transition to the reciprocal polytope, whose vertices ly at the centres of the $ ( n- 1 ) $- dimensional faces of $ P $. Reciprocal polytopes have the same symmetry group.

All possible Schläfli symbols of regular polytopes can be obtained from the classification of finite reflection groups, by selecting those with a linear Coxeter graph. For $ n\geq 5 $ there are only 3 regular polytopes in $ E ^ {n} $: the simplex, the cube and the polytope reciprocal to the cube (the analogue of the octahedron). Their Schläfli symbols are $ \{ 3 \dots 3 \} $, $ \{ 4 , 3 \dots 3 \} $ and $ \{ 3 \dots 3 , 4 \} $. In $ 4 $- dimensional space there are 6 regular polytopes: $ \{ 3 , 3 , 3 \} $, $ \{ 4 , 3 , 3 \} $, $ \{ 3 , 3 , 4 \} $, $ \{ 3 , 4 , 3 \} $, $ \{ 5 , 3 , 3 \} $, and $ \{ 3 , 3 , 5 \} $.

Each face of a regular polytope $ P $ is also a regular polytope, the Schläfli symbol of which is the initial segment of the Schläfli symbol of $ P $. For example, a $ 3 $- dimensional face of the polytope $ \{ 5 , 3 , 3 \} $ has the Schläfli symbol $ \{ 5 , 3 \} $, that is, it is a dodecahedron.

References

[1] H.S.M. Coxeter, "Regular polytopes" , Dover, reprint (1973)
[2] B.A. Rozenfel'd, "Multi-dimensional spaces" , Moscow (1966) (In Russian)

Comments

A presentation of the polyhedron group is given by

$$ < r _ {1} \dots r _ {n} \mid ( r _ {k} r _ {l} ) ^ {2} = 1 \ \ \textrm{ for } | k- l | \geq 2 ; $$

$$ ( r _ {k} r _ {k+} 1 ) ^ {2} = p _ {k} \ \textrm{ for } k = 1 \dots n- 1 > . $$

This shows that this group is a Coxeter group.

References

[a1] H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1990)
How to Cite This Entry:
Polyhedron group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polyhedron_group&oldid=48235
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article