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 ...
, 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 groups 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 brackets, and, in parentheses,
orbifold notation.
Types
;Chiral:
*''C
n'',
sup>+, (''nn'') of order ''n'' - ''n''-fold rotational symmetry - acro-n-gonal group (abstract group
''Zn''); 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](https://upload.wikimedia.org/wikipedia/commons/3/37/S_shaped_packing.jpeg)
*''C
nh'',
+,2">+,2 (''n''*) of order 2''n'' - prismatic symmetry or ortho-n-gonal group (abstract group
''Zn'' × ''Dih
1''); 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. It has
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 ...
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, quadrilat ...
.
*''S
2n'',
+,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 ''S
2'' (1×), also denoted by ''C
i''; this is
inversion symmetry
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
.
''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
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 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
See also
*
Dihedral symmetry in three dimensions
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih''n'' (for ''n'' ≥ 2).
Types
Ther ...
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
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Symmetry