This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dih_n ( n ≥ 2 ). A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... The symmetry group of an object (e. ... This article may be confusing for some readers, and should be edited to enhance clarity. ...

See also point groups in two dimensions. In geometry, a point group in two dimensions is an isometry group in two dimensions that leaves the origin fixed, or correspondingly, an isometry group of a circle. ...

Chiral:

• D_n (22n) of order 2n - dihedral symmetry (abstract group D_n)

Achiral: This article may be confusing for some readers, and should be edited to enhance clarity. ...

• D_nh (*22n) of order 4n - prismatic symmetry (abstract group D_n × C₂)
• D_nd (or D_nv) (2*n) of order 4n - antiprismatic symmetry (abstract group D_2n)

For a given n, all three have n-fold rotational symmetry about one axis (rotation by an angle of 360°/n does not change the object), and 2-fold about a perpendicular axis, hence about n of those. For n = ∞ they correspond to three frieze groups. Schönflies notation is used, and, in parentheses, Conway's orbifold notation. The term horizontal (h) is used with respect to a vertical axis of rotation. Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ... Rotation of a planar figure around a point Rotation of a planar body is the movement when points of the body travel in circular trajectories around a fixed point called the center of rotation. ... A frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). ... Arthur Moritz Schönflies (April 17, 1853 Landsberg an der Warthe(Gorzów) – May 27, 1928) was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. ... Conways orbifold notation, introduced by John Conway, applies, among other things, to three classes of symmetry types: wallpaper groups, frieze groups and point groups in three dimensions. ...

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 in 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 C_nv of the same order.

With reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis we have D_nh (*22n). Figures with the axes of symmetry drawn in. ...

D_nd (or D_nv) has vertical mirror planes between the horizontal rotation axes, not through them. As a result the vertical axis is a 2n-fold rotoreflection axis.

D_nh is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. D_nd is the symmetry group for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. D_n is the symmetry group of a partially rotated prism. In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... The trapezohedron is the dual polyhedron of the corresponding antiprism. ...

n=1 is not included because the three symmetries are equal to other ones:

• D₁ and C₂: group of order 2 with a single 180° rotation
• D_1h and C_2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane
• D_1d and C_2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that plane

For n=2 there is not one main axes and two additional axes, but there are three equivalent ones.

• D₂ (222) of order 4 is one of the three symmetry group types with the Klein four-group as abstract group. It has three perpendicular 2-fold rotation axes. It is the symmetry group of a cuboid with an S written on two opposite faces, in the same orientation.
• D_2h (*222) of order 8 is the symmetry group of a cuboid
• D_2d (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 scaled in the direction of a line connecting the midpoints of two opposite edges (D_2d is a subgroup of T_d, by scaling we reduce the symmetry).

Dnh (*22n): This article is about the mathematical group. ... In anatomy, the cuboid bone is a bone in the foot. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ...

D_5h (*225): Geometric prism images I generated these images myself for the prism (geometry) page. ...

Pentagrammic prism

Pentagrammic antiprism

D_4d (2*4): Image File history File links Pentagrammic_prism. ... Image File history File links Pentagrammic_prism. ... In geometry, the pentagrammic prism is the first in an infinite set of concave prisms formed by square sides and two regular star polygon caps, in this case two pentagrams. ... Image File history File links Download high resolution version (639x640, 15 KB) Summary Uniform pentagrammic prism 3. ... Image File history File links Download high resolution version (639x640, 15 KB) Summary Uniform pentagrammic prism 3. ...

Snub square antiprism

D_5d (2*5):

Pentagonal antiprism

Pentagrammic crossed antiprism

5-sided trapezohedron

D_17d (*22(17)): Download high resolution version (835x666, 53 KB)Pentagonal antiprism, made by me using POV-Ray, see image:poly. ... Download high resolution version (835x666, 53 KB)Pentagonal antiprism, made by me using POV-Ray, see image:poly. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... Image File history File links Download high resolution version (640x640, 16 KB) Summary Pentagrammic cross-antiprism 3. ... Image File history File links Download high resolution version (640x640, 16 KB) Summary Pentagrammic cross-antiprism 3. ... Download high resolution version (452x773, 37 KB)Pentagonal antiprism, made by me using POV-Ray, see image:poly. ... Download high resolution version (452x773, 37 KB)Pentagonal antiprism, made by me using POV-Ray, see image:poly. ... The trapezohedron is the dual polyhedron of the corresponding antiprism. ...

Heptadecagonal antiprism