In three dimensional

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 ...

, there are four infinite series 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 ...

(''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 the finite

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 ...

s on a

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

. For ''n'' = ∞ they correspond to four

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 symmetrie ...

s. Schönflies notation is used. The terms horizontal (h) and vertical (v) imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are

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 diagra ...

in brackets, and, in parentheses,

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 adva ...

Types

;Chiral: *''C_n'', sup>+, (''nn'') of order ''n'' - ''n''-fold rotational symmetry - acro-n-gonal group (abstract group ''Z_n''); for ''n''=1: no symmetry (

trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...

) ;Achiral: S shaped packing

*''C_nh'', ⁺,2 (''n''*) of order 2''n'' - prismatic symmetry or ortho-n-gonal group (abstract group ''Z_n'' × ''Dih₁''); for ''n''=1 this is denoted by ''C_s'' (1*) and called

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 the ...

, also

bilateral symmetry Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, take the face of a human being which has a ...

. It has

with respect to a plane perpendicular to the ''n''-fold rotation axis. *''C_nv'', (*''nn'') of order 2''n'' - pyramidal symmetry or full acro-n-gonal group (abstract group ''Dih_n''); in biology ''C_2v'' is called biradial symmetry. For ''n''=1 we have again ''C_s'' (1*). It has vertical mirror planes. This is the symmetry group for a regular ''n''-sided

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...

. *''S_2n'', ⁺,2n⁺ (''n''×) of order 2''n'' - gyro-n-gonal group (not to be confused with

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

s, for which the same notation is used; abstract group ''Z_2n''); It has 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, also called 2''n''-fold improper rotation axis, i.e., the symmetry group contains a combination of a reflection in the horizontal plane and a rotation by an angle 180°/n. Thus, like ''D_nd'', it contains a number of improper rotations without containing the corresponding rotations. ** for ''n''=1 we have ''S₂'' (1×), also denoted by ''C_i''; this is inversion symmetry. ''C_2h'', ,2⁺(2*) and ''C_2v'', (*22) of order 4 are two of the three 3D symmetry group types with the

Klein four-group In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one ...

as abstract group. ''C_2v'' applies e.g. for a rectangular tile with its top side different from its bottom side.

Frieze groups

In the limit these four groups represent Euclidean plane

s as C_∞, C_∞h, C_∞v, and S_∞. Rotations become translations in the limit. Portions of the infinite plane can also be cut and connected into an infinite cylinder.

Examples

References

* * ''On Quaternions and Octonions'', 2003, John Horton Conway and Derek A. Smith * ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, * Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

* Norman Johnson (mathematician), N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 11: ''Finite symmetry groups'', 11.5 Spherical Coxeter groups {{DEFAULTSORT:Cyclic Symmetries Symmetry

Types

Frieze groups

Examples

See also

References