Polyhedron group
The group of symmetries of a polytope (cf. Polyhedron)
in an
-dimensional Euclidean space
, that is, the group of all motions of
which send
to itself. A polytope
is called regular if
acts transitively on the set of its "flag set of a polytopeflags" , that is, collections
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where is a
-dimensional closed face and
. The group of symmetries of a regular polytope is generated by reflections (see Reflection group). Its fundamental domain is a simplicial cone
whose vertex is the centre of the polytope
, and whose edges pass through the centres of the faces constituting some flag
. By the same token the generating reflections
of the group
have a natural enumeration:
is the reflection relative to the hyperplane bounding
which does not pass through the centre of the face
. The generators
and
commute for
, and the order of
is equal to
— the number of
-dimensional (or
-dimensional) faces of the polytope
containing the face
(if it is assumed that
and
). The sequence
is called the Schläfli symbol of the polytopes. The three-dimensional regular polytopes (Platonic solids) have the following Schläfli symbols: the tetrahedron —
, the cube —
, the octahedron —
, the dodecahedron —
, and the icosahedron —
.
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 -dimensional faces of
. 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 there are only 3 regular polytopes in
: the simplex, the cube and the polytope reciprocal to the cube (the analogue of the octahedron). Their Schläfli symbols are
,
and
. In
-dimensional space there are 6 regular polytopes:
,
,
,
,
, and
.
Each face of a regular polytope is also a regular polytope, the Schläfli symbol of which is the initial segment of the Schläfli symbol of
. For example, a
-dimensional face of the polytope
has the Schläfli symbol
, 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
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This shows that this group is a Coxeter group.
References
[a1] | H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1990) |
Polyhedron group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polyhedron_group&oldid=48235