In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, 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 (mathematics), group ...
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 amb ...
that as an abstract group is a
dihedral group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
Dih
''n'' (for ''n'' ≥ 2).
Types
There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations:
Schönflies notation,
Coxeter notation, and
orbifold notation.
;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 (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...
about one axis (
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
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 groups.
Schönflies notation is used, with
Coxeter notation in brackets, and
orbifold notation 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 ''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 (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...
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 axis.
''D
nh'' is the symmetry group for a regular ''n''-sided
prism and also for a regular n-sided
bipyramid
In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise ...
. ''D
nd'' is the symmetry group for a regular ''n''-sided
antiprism
In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
, and also for a regular n-sided
trapezohedron. ''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 an abelian 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 non-identi ...
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 with quadrilateral faces, meaning it is a polyhedron with six Face (geometry), faces; it has eight Vertex (geometry), vertices and twelve Edge (geometry), edges. A ''rectangular cuboid'' (sometimes also calle ...
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 (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
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'',
,5 (*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
*
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 (mathematics), group ...
*
Cyclic symmetry in three dimensions
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