In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, dihedral symmetry in three dimensions is one of three infinite sequences of
point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometrie ...
which have a
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
that as an abstract group is a
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
Dih
''n'' (for ''n'' ≥ 2).
Types
There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations:
Schönflies notation The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the ...
,
Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram ...
, and
orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advanta ...
.
;Chiral:
*''D
n'',
'n'',2sup>+, (22''n'') of order 2''n'' – dihedral symmetry or para-n-gonal group (abstract group:
''Dihn'').
;Achiral:
*''D
nh'',
'n'',2 (*22''n'') of order 4''n'' – prismatic symmetry or full ortho-n-gonal group (abstract group: ''Dih
n'' × ''Z''
2).
*''D
nd'' (or ''D
nv''),
''n'',2+ (2*''n'') of order 4''n'' – antiprismatic symmetry or full gyro-n-gonal group (abstract group: ''Dih''
2''n'').
For a given ''n'', all three have ''n''-fold
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
about one axis (
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
by an angle of 360°/''n'' does not change the object), and 2-fold rotational symmetry about a perpendicular axis, hence about ''n'' of those. For ''n'' = ∞, they correspond to three
Frieze group
In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetri ...
s.
Schönflies notation The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the ...
is used, with
Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram ...
in brackets, and
orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advanta ...
in parentheses. The term horizontal (h) is used with respect to a vertical axis of rotation.
In 2D, the symmetry group ''D
n'' includes reflections in lines. When the 2D plane is embedded horizontally in a 3D space, such a reflection can either be viewed as the restriction to that plane of a reflection through a vertical plane, or as the restriction to the plane of a rotation about the reflection line, by 180°. In 3D, the two operations are distinguished: the group ''D
n'' contains rotations only, not reflections. The other group is
pyramidal symmetry
In three dimensional geometry, there are four infinite series of point groups in three dimensions (''n''≥1) with ''n''-fold rotational or reflectional symmetry about one axis (by an angle of 360°/''n'') that does not change the object.
They are ...
''C
nv'' of the same order, 2''n''.
With
reflection symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...
in a plane perpendicular to the ''n''-fold rotation axis, we have ''D
nh'',
(*22''n'').
''D
nd'' (or ''D
nv''),
''n'',2+ (2*''n'') has vertical mirror planes between the horizontal rotation axes, not through them. As a result, the vertical axis is a 2''n''-fold
rotoreflection
In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
axis.
''D
nh'' is the symmetry group for a regular ''n''-sided
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
and also for a regular n-sided
bipyramid. ''D
nd'' is the symmetry group for a regular ''n''-sided
antiprism
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass o ...
, and also for a regular n-sided
trapezohedron
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...
. ''D
n'' is the symmetry group of a partially rotated prism.
''n'' = 1 is not included because the three symmetries are equal to other ones:
*''D''
1 and ''C''
2: group of order 2 with a single 180° rotation.
*''D''
1''h'' and ''C''
2''v'': group of order 4 with a reflection in a plane and a 180° rotation about a line in that plane.
*''D''
1''d'' and ''C''
2''h'': group of order 4 with a reflection in a plane and a 180° rotation about a line perpendicular to that plane.
For ''n'' = 2 there is not one main axis and two additional axes, but there are three equivalent ones.
*''D''
2,
,2sup>+, (222) of order 4 is one of the three symmetry group types with the
Klein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...
as abstract group. It has three perpendicular 2-fold rotation axes. It is the symmetry group of a
cuboid
In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...
with an S written on two opposite faces, in the same orientation.
*''D''
2''h'',
,2 (*222) of order 8 is the symmetry group of a cuboid.
*''D''
2''d'',
,2+ (2*2) of order 8 is the symmetry group of e.g.:
**A square cuboid with a diagonal drawn on one square face, and a perpendicular diagonal on the other one.
**A regular
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
scaled in the direction of a line connecting the midpoints of two opposite edges (''D''
2''d'' is a subgroup of
''Td''; by scaling, we reduce the symmetry).
Subgroups
For ''D
nh'',
,2 (*22n), order 4n
* ''C
nh'',
+,2 (n*), order 2n
* ''C
nv'',
,1 (*nn), order 2n
* ''D
n'',
,2sup>+, (22n), order 2n
For ''D
nd'',
n,2+ (2*n), order 4n
* ''S''
2''n'',
n+,2+ (n×), order 2n
* ''C
nv'',
+,2 (n*), order 2n
* ''D
n'',
,2sup>+, (22n), order 2n
''D
nd'' is also subgroup of ''D''
2''nh''.
Examples
''D
nh'',
'n'' (*22''n''):
''D''
5''h'',
(*225):
''D''
4''d'',
,2+ (2*4):
''D''
5''d'',
0,2+ (2*5):
''D''
17''d'',
4,2+ (2*17):
See also
*
List of spherical symmetry groups
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.
This a ...
*
Point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometrie ...
*
Cyclic symmetry in three dimensions
In three dimensional geometry, there are four infinite series of point groups in three dimensions (''n''≥1) with ''n''-fold rotational or reflectional symmetry about one axis (by an angle of 360°/''n'') that does not change the object.
They are ...
References
*
*
N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 11: ''Finite symmetry groups'', 11.5 Spherical Coxeter groups
*
External links
Graphic overview of the 32 crystallographic point groups– form the first parts (apart from skipping ''n''=5) of the 7 infinite series and 5 of the 7 separate 3D point groups
{{DEFAULTSORT:Dihedral Symmetry In Three Dimensions
Symmetry
Euclidean symmetries
Group theory