Triangular prism

For the optical prism, see Triangular prism (optics).

In geometry, a triangular prism or three-sided prism is a type of prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.

If the sides are squares, it is called a uniform polyhedron. In general the sides can be congruent rectangles.

Equivalently, it is a pentahedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms. All cross-sections parallel to the base faces are the same triangle.

A right triangular prism is semiregular if the base faces are equilateral triangles, and the other three faces are squares.

A general right triangular prism can have rectangular sides.

The dual of a triangular prism is a 3-sided bipyramid.

The symmetry group of a right 3-sided prism with regular base is D_3h of order 12. The rotation group is D₃ of order 6.

The symmetry group does not contain inversion.

Volume

The volume of any prism is the product of the area of the base and the distance between the two base faces. In this case the base is a triangle so we simply need to compute the area of the triangle and multiply this by the length of the prism:

See also

• Prism (geometry)
• Semiregular E-polytope

Inversion in a point

In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*. In other words, the vector from X to P is the same as the vector from P to X*.

The formula for the inversion in P is

x*=2a−x

where a, x and x* are the position vectors of P, X and X* respectively.

This mapping is an isometric involutive affine transformation which has exactly one fixed point, which is P.

In odd-dimensional Euclidean space it does not preserve orientation, it is an indirect isometry.

Geometrically in 3D it amounts to rotation about an axis through P by an angle of 180°, combined with reflection in the plane through P which is perpendicular to the axis; the result does not depend on the orientation (in the other sense) of the axis. Notations for the type of operation, or the type of group it generates, are , C_i, S₂, and 1×. The group type is one of the three symmetry group types in 3D without any pure rotational symmetry, see cyclic symmetries with n=1.

The following point groups in three dimensions contain inversion:

• C_nh and D_nh for even n
• S_2n and D_nd for odd n
• T_h, O_h, and I_h

Closely related to inverse in a point is reflection in respect to a plane, which can be thought of as a "inversion in a plane".

Inversion with respect to the origin

Inversion with respect to the origin corresponds to additive inversion of the position vector, and also to scalar multiplication by −1. The operation commutes with every other linear transformation, but not with translation. "Inversion" without indicating "in a point", "in a line" or "in a plane", means this inversion, also called parity transformation.

Rotation group

In mechanics and geometry, the rotation group is the group of all rotations about the origin of 3-dimensional Euclidean space R³ under the operation of composition.

By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation (i.e. handedness) of space. A length-preserving transformation which reverses orientation is called an improper rotation.

Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Moreover, the rotation group has a natural manifold structure for which the group operations are smooth; so it is in fact a Lie group. The rotation group is often denoted SO(3) for reasons explained below.

Properties

Besides just preserving length, rotations also preserve the angles between vectors. This follows from the fact that the standard dot product between two vectors u and v can be written purely in terms of length:

Hence, any length-preserving transformation in R³ preserves the dot product, and thus the angle between vectors. It is a quick check that every rotation maps an orthonormal basis of R³ to another orthonormal basis.

It should be noted that rotations are often defined as linear transformations that preserve the inner product on R³. By the above argument, this is equivalent to requiring them to preserve length.

Another important property of the rotation group is that it is nonabelian. That is, the order in which rotations are composed makes a difference. For example, a quarter turn around the positive x-axis followed by a quarter turn around the positive y-axis is a different rotation than the one obtained by first rotating around y and then x.

Orthogonal and rotation matrices

Main articles: Orthogonal matrix and Rotation matrix

Like any linear transformation, a rotation can always be represented by a matrix. Let R be a given rotation. With respect to the standard basis (e₁,e₂,e₃) of R³ the columns of R are given by (Re₁,Re₂,Re₃). Since the standard basis is orthonormal, the columns of R form another orthonormal basis. This orthonormality condition can be expressed in the form

where R^T denotes the transpose of R and I is the 3 × 3 identity matrix. Matrices for which this property holds are called orthogonal matrices. The group of all 3 × 3 orthogonal matrices is denoted O(3).

In addition to preserving length, rotations must also preserve orientation. A matrix will preserve or reverse orientation according to whether the determinant of the matrix is positive or negative. For an orthogonal matrix R, note that det R^T = det R implies (det R)² = 1 so that det R = ±1. The subgroup of orthogonal matrices with determinant +1 is called the special orthogonal group, denoted SO(3).

Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to matrix multiplication, the rotation group is isomorphic to the special orthogonal group SO(3).

Improper rotations correspond to orthogonal matrices with determinant −1, and they do not form a group because the product of two improper rotations is a proper rotation.

Axis of rotation

Every rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of R³ which is called the axis of rotation (this is Euler's rotation theorem). Each rotation acts as a normal 2-dimensional rotation in the plane orthogonal to this axis. Since every 2-dimensional rotation can be represented by an angle φ, an arbitrary 3-dimensional rotation can be specified by an axis of rotation together with an angle of rotation about this axis. (Technically, one needs to specify an orientation for the axis and whether the rotation is taken to be clockwise or counterclockwise with respect to this orientation).

For example, counterclockwise rotation about the positive z-axis by angle φ is given by

Given a unit vector n in R³ and an angle φ, let R(φ, n) represent a counterclockwise rotation about the axis through n (with orientation determined by n). Then

• R(0, n) is the identity transformation for any n
• R(φ, n) = R(−φ, −n)
• R(π + φ, n) = R(π − φ, −n)

Using these properties one can show that any rotation can be represented by a unique angle φ in the range 0 ≤ φ ≤ π and a unit vector n such that

• n is arbitrary if φ = 0
• n is unique if 0 < φ < π
• n is unique up to a sign if φ = π (that is, the rotations R(π, ±n) are identical)

Topology

Consider the solid ball in R³ of radius π (that is, all points of R³ of distance π or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotation through angles between 0 and -π correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one remaining issue is that the two rotations through π and through -π are the same. So we identify (or "glue together") antipodal points on the surface of the ball. After this identification, we arrive at a topological space homeomorphic to the rotation group.

Indeed, the ball with antipodal surface points identified is a smooth manifold, and this manifold is diffeomorphic to the rotation group. It is also diffeomorphic to the real 3-dimensional projective space, so the latter can also serve as a topological model for the rotation group.

These identifications illustrate that SO(3) is connected but not simply connected. As to the latter, in the ball with antipodal surface points identified, consider the path running from the "north pole" straight through the center down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how you deform the loop, the start and end point have to remain antipodal, or else the loop will "break open". In terms of rotations, this loop represents a continuous sequence of rotations about the z-axis starting and ending at the identity rotation (i.e. a series of rotation through an angle φ where φ runs from 0 to 2π).

Surprisingly, if you run through the path twice (so that φ runs from 0 to 4π), i.e. from north pole down to south pole, jump back up to the north pole and run again down to the south pole, you get a closed loop which can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second half of the path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems.

The same argument can be performed in general, and it shows that the fundamental group of SO(3) is cyclic of order 2. In physics applications, the non-triviality of the fundamental group allows for the existence of objects known as spinors, and is an important tool in the development of the spin-statistics theorem.

The universal cover of SO(3) is a Lie group called Spin(3). The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S³ and can be understood as the group of unit quaternions (i.e. those with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The resulting map

S³ → SO(3)

is a surjective homomorphism of Lie groups, with kernel {±1}.

Representations of rotations

We have seen that there are a variety of ways to represent rotations:

• as orthogonal matrices with determinant 1,
• by axis and rotation angle
• via the unit quaternions (see quaternions and spatial rotations) and the map S³ → SO(3).

Another method is to specify an arbitrary rotation by a sequence of rotations about some fixed axes. See:

• Euler angles

See charts on SO(3) for further discussion.

Generalizations

The rotation group generalizes quite naturally to n-dimensional Euclidean space, Rⁿ. The group of all proper and improper rotations in n dimensions is called the orthogonal group, O(n), and the subgroup of proper rotations is called the special orthogonal group, SO(n).

In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature. However, one can still define generalized rotations which preserve this inner product. Such generalized rotations are known as Lorentz transformations and the group of all such transformations is called the Lorentz group.

The rotation group SO(3) can be described as a subgroup of E⁺(3), the Euclidean group of direct isometries of R³. This larger group is the group of all motions of a rigid body: each of these is a combination of a rotation about an arbitrary axis and a translation along the axis, or put differently, a combination of an element of SO(3) and an arbitrary translation.

In general, the rotation group of an object is the symmetry group within the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.

Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

Notation

There are two competing notations for the dihedral group associated to a polygon with n sides. In geometry the group is denoted D_n, while in algebra the same group is denoted by D_2n to indicate the number of elements.

In this article, D_n (and sometimes Dih_n) refers to the symmetries of a regular polygon with n sides.

Definition

Elements

The six reflection symmetries of a regular hexagon.

A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. The associated rotations and reflections make up the dihedral group D_n. The following picture shows the effect of the sixteen elements of D₈ on a stop sign:

The first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections.

Group structure

As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry. This operation gives the symmetries of a polygon the algebraic structure of a finite group.

The composition of these two reflections is a rotation.

The following Cayley table shows the effect of composition in the group D₃ (the symmetries of an equilateral triangle). R₀ denotes the identity; R₁ and R₂ denote counterclockwise rotations by 120 and 240 degrees; and S₀, S₁, and S₂ denote reflections across the three lines shown in the picture to the right.

	R0	R1	R2	S0	S1	S2
R0	R0	R1	R2	S0	S1	S2
R1	R1	R2	R0	S1	S2	S0
R2	R2	R0	R1	S2	S0	S1
S0	S0	S2	S1	R0	R2	R1
S1	S1	S0	S2	R1	R0	R2
S2	S2	S1	S0	R2	R1	R0

For example, S₂S₁ = R₁ because the reflection S₁ followed by the reflection S₂ results in a 120-degree rotation. (This is the normal backwards order for composition.) Note that the composition operation is not commutative.

In general, the group D_n has elements R₀,...,R_n−1 and S₀,...,S_n−1, with composition given by the following formulas:

In all cases, addition and subtraction of subscripts should be performed using modular arithmetic with modulus n.

[edit] Matrix representation

The symmetries of this pentagon are linear transformations.

If we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of D_n as matrices, with composition being matrix multiplication.

For example, the elements of the group D₄ can be represented by the following eight matrices:

In general, the matrices for elements of D_n have the following form:

and

The first matrix is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk ⁄ n. The second matrix is a reflection across a line that makes an angle of πk ⁄ n with the x-axis.

[edit] Small dihedral groups

For n = 1 we have Dih₁. This notation is rarely used except in the framework of the series, because it is equal to Z₂. For n = 2 we have Dih₂, the Klein four-group. Both are exceptional within the series:

• they are abelian; for all other values of n the group Dih_n is not abelian
• they are not subgroups of the symmetric group S_n, corresponding to the fact that 2n > n ! for these n.

The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups stand for the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element.

Dih1	Dih2	Dih3	Dih4	Dih5	Dih6	Dih7

The dihedral group as symmetry group in 2D and rotation group in 3D

An example of abstract group Dih_n, and a common way to visualize it, is the group D_n of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. D_n consists of n rotations of multiples of 360°/n about the origin, and reflections across n lines through the origin, making angles of multiples of 180°/n with each other. This is the symmetry group of a regular polygon with n sides (for n ≥3, and also for the degenerate case n = 2, where we have a line segment in the plane).

Dihedral group D_n is generated by a rotation r of order n and a reflection f of order 2 such that

frf = r ^{− 1} (in geometric terms: in the mirror a rotation looks like an inverse rotation)

In matrix form, an anti-clockwise rotation and a reflection in the x-axis are given by

(in terms of complex numbers: multiplication by and complex conjugation).

By setting

and defining and for we can write the product rules for D_n as

(Compare coordinate rotations and reflections.)

The dihedral group D₂ is generated by the rotation r of 180 degrees, and the reflection f across the x-axis. The elements of D₂ can then be represented as {e, r, f, rf}, where e is the identity or null transformation and rf is the reflection across the y-axis.

The four elements of D₂ (x-axis is vertical here)

D₂ is isomorphic to the Klein four-group.

If the order of D_n is greater than 4, the operations of rotation and reflection in general do not commute and D_n is not abelian; for example, in D₄, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees:

D₄ is nonabelian (x-axis is vertical here)

Thus, beyond their obvious application to problems of symmetry in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups.

The 2n elements of D_n can be written as e, r, r²,...,rⁿ⁻¹, f, r f, r² f,...,rⁿ⁻¹ f. The first n listed elements are rotations and the remaining n elements are axis-reflections (all of which have order 2). The product of two rotations or two reflections is a rotation; the product of a rotation and a reflection is a reflection.

So far, we have considered D_n to be a subgroup of O(2), i.e. the group of rotations (about the origin) and reflections (across axes through the origin) of the plane. However, notation D_n is also used for a subgroup of SO(3) which is also of abstract group type Dih_n: the proper symmetry group of a regular polygon embedded in three-dimensional space (if n ≥ 3). Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group (in analogy to tetrahedral, octahedral and icosahedral group, referring to the proper symmetry groups of a regular tetrahedron, octahedron, and icosahedron respectively).

Equivalent definitions and properties

Further equivalent definitions of Dih_n are:

• The automorphism group of the graph consisting only of a cycle with n vertices (if n ≥ 3).
• The group with presentation

or

(Indeed the only finite groups that can be generated by two elements of order 2 are the dihedral groups and the cyclic groups)

From the second presentation follows that Dih_n belongs to the class of coxeter groups.

• The semidirect product of cyclic groups Z_n and Z₂, with Z₂ acting on Z_n by inversion (thus, Dih_n always has a normal subgroup isomorphic to Z_n ):

is isomorphic to Dih_n if φ(0) is the identity and φ(1) is inversion.

If we consider Dih_n (n ≥ 3) as the symmetry group of a regular n-gon and number the polygon's vertices, we see that Dih_n is a subgroup of the symmetric group S_n.

The properties of the dihedral groups Dih_n with n ≥ 3 depend on whether n is even or odd. For example, the center of Dih_n consists only of the identity if n is odd, but if n is even the center has two elements, namely the identity and the element r^{n / 2} (with D_n as a subgroup of O(2), this is inversion; since it is scalar multiplication by −1, it is clear that it commutes with any linear transformation).

For odd n, abstract group Dih_2n is isomorphic with the direct product of Dih_n and Z₂.

In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones.

All the reflections are conjugate to each other in case n is odd, but they fall into two conjugacy classes if n is even. If we think of the isometries of a regular n-gon: for odd n there are rotations in the group between every pair of mirrors, while for even n only half of the mirrors can be reached from one by these rotations.

If m divides n, then Dih_n has n / m subgroups of type Dih_m, and one subgroup Z_m. Therefore the total number of subgroups of Dih_n (n ≥ 1), is equal to d (n) + σ (n), where d (n) is the number of positive divisors of n and σ (n) is the sum of the positive divisors of n. See List of small groups for the cases n ≤ 8.

Examples of automorphism groups

Dih₉ has 18 inner automorphisms. As 2D isometry group D₉, the group has mirrors at 20° intervals. The 18 inner automorphisms provide rotation of the mirrors by multiples of 20°, and reflections. As isometry group these are all automorphisms. As abstract group there are in addition to these, 36 outer automorphisms, e.g. multiplying angles of rotation by 2.

Dih₁₀ has 10 inner automorphisms. As 2D isometry group D₁₀, the group has mirrors at 18° intervals. The 10 inner automorphisms provide rotation of the mirrors by multiples of 36°, and reflections. As isometry group there are 10 more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors 18° with respect to the inner automorphisms. As abstract group there are in addition to these 10 inner and 10 outer automorphisms, 20 more outer automorphisms, e.g. multiplying rotations by 3.

Compare the values 6 and 4 for Euler's totient function, the multiplicative group of integers modulo n for n = 9 and 10, respectively. This triples and doubles the number of automorphisms compared with the two automorphisms as isometries (keeping the order of the rotations the same or reversing the order).

In general, the automorphism group of Dih_n is isomorphic to the affine group Aff(Z/nZ).

Infinite dihedral group

In addition to the finite dihedral groups, there is the infinite dihedral group Dih_∞. Every dihedral group is generated by a rotation r and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer n such that rⁿ is the identity, and we have a finite dihedral group of order 2n. If the rotation is not a rational multiple of a full rotation, then there is no such n and the resulting group has infinitely many elements and is called Dih_∞. It has presentations

and is isomorphic to a semidirect product of Z and Z₂, and to the free product Z₂ * Z₂. It is the automorphism group of the graph consisting of a path infinite to both sides. Correspondingly, it is the isometry group of Z (see also symmetry groups in one dimension).

Generalized dihedral group

For any abelian group H, the generalized dihedral group of H, written Dih(H), is the semidirect product of H and Z₂, with Z₂ acting on H by inverting elements. I.e., with φ(0) the identity and φ(1) inversion.

Thus we get:

(h₁, 0) * (h₂, t₂) = (h₁ + h₂, t₂)

(h₁, 1) * (h₂, t₂) = (h₁ - h₂, 1 + t₂)

for all h₁, h₂ in H and t₂ in Z₂.

(Writing Z₂ multiplicatively, we have (h₁, t₁) * (h₂, t₂) = (h₁ + t₁h₂, t₁t₂) .)

Note that (h, 0) * (0,1) = (h,1), i.e. first the inversion and then the operation in H. Also (0, 1) * (h, t) = (- h, 1 + t); indeed (0,1) inverts h, and toggles t between "normal" (0) and "inverted" (1) (this combined operation is its own inverse).

The subgroup of Dih(H) of elements (h, 0) is a normal subgroup of index 2, isomorphic to H, while the elements (h, 1) are all their own inverse.

The conjugacy classes are:

• the sets {(h,0 ), (-h,0 )}
• the sets {(h + k + k, 1) | k in H }

Thus for every subgroup M of H, the corresponding set of elements (m,0) is also a normal subgroup. We have:

Dih(H) / M = Dih ( H / M )

Examples:

• Dih_n = Dih(Z_n)
  • For even n there are two sets {(h + k + k, 1) | k in H }, and each generates a normal subgroup of type Dih_{n / 2}. As subgroups of the isometry group of the set of vertices of a regular n-gon they are different: the reflections in one subgroup all have two fixed points, while none in the other subgroup has (the rotations of both are the same). However, they are isomorphic as abstract groups.
  • For odd n there is only one set {(h + k + k, 1) | k in H }
• Dih_∞ = Dih(Z); there are two sets {(h + k + k, 1) | k in H }, and each generates a normal subgroup of type Dih_∞. As subgroups of the isometry group of Z they are different: the reflections in one subgroup all have a fixed point, the mirrors are at the integers, while none in the other subgroup has, the mirrors are in between (the translations of both are the same: by even numbers). However, they are isomorphic as abstract groups.
• Dih(S¹), or orthogonal group O(2,R), or O(2): the isometry group of a circle, or equivalently, the group of isometries in 2D that keep the origin fixed. The rotations form the circle group S¹, or equivalently SO(2,R), also written SO(2), and R/Z ; it is also the multiplicative group of complex numbers of absolute value 1. In the latter case one of the reflections (generating the others) is complex conjugation. There are no proper normal subgroups with reflections. The discrete normal subgroups are cyclic groups of order n for all positive integers n. The quotient groups are isomorphic with the same group Dih(S¹).
• Dih(Rⁿ ): the group of isometries of Rⁿ consisting of all translations and inversion in all points; for n = 1 this is the Euclidean group E(1); for n > 1 the group Dih(Rⁿ ) is a proper subgroup of E(n ), i.e. it does not contain all isometries.
• H can be any subgroup of Rⁿ, e.g. a discrete subgroup; in that case, if it extends in n directions it is a lattice.
  • Discrete subgroups of Dih(R² ) which contain translations in one direction are of frieze group type and 22.
  • Discrete subgroups of Dih(R² ) which contain translations in two directions are of wallpaper group type p1 and p2.
  • Discrete subgroups of Dih(R³ ) which contain translations in three directions are space groups of the triclinic crystal system.

Dih(H) is Abelian, with the semidirect product a direct product, if and only if all elements of H are their own inverse:

• Dih(Z₁) = Dih₁ = Z₂
• Dih(Z₂) = Dih₂ = Z₂ × Z₂ (Klein four-group)
• Dih(Dih₂) = Dih₂ × Z₂ = Z₂ × Z₂ × Z₂

etc.

Topology

Dih(Rⁿ ) and its dihedral subgroups are disconnected topological groups. Dih(Rⁿ ) consists of two connected components: the identity component isomorphic to Rⁿ, and the component with the reflections. Similarly O(2) consists of two connected components: the identity component isomorphic to the circle group, and the component with the reflections.

For the group Dih_∞ we can distinguish two cases:

• Dih_∞ as the isometry group of Z
• Dih_∞ as a 2-dimensional isometry group generated by a rotation by an irrational number of turns, and a reflection

Both topological groups are totally disconnected, but in the first case the (singleton) components are open, while in the second case they are not. Also, the first topological group is a closed subgroup of Dih(R) but the second is not a closed subgroup of O(2).

Symmetry group

The symmetry group of an object (image, signal, etc., e.g. in 1D, 2D or 3D) is the group of all isometries under which it is invariant with composition as the operation. It is a subgroup of the isometry group of the space concerned.

Introduction

(If not stated otherwise, we consider symmetry groups in Euclidean geometry here, but the concept may also be studied in wider contexts, see below.)

The "objects" may be geometric figures, images and patterns, such as a wallpaper pattern. The definition can be made more precise by specifying what is meant by image or pattern, e.g. a function of position with values in a set of colors. For symmetry of e.g. 3D bodies one may also want to take physical composition into account. The group of isometries of space induces a group action on objects in it.

The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries (like reflections, glide reflections and improper rotations) under which the figure is invariant. The subgroup of orientation-preserving isometries (i.e. translations, rotations and compositions of these) which leave the figure invariant is called its proper symmetry group. The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientation-reversing isometries under which it is invariant).

Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O(n) by choosing the origin to be a fixed point. The proper symmetry group is a subgroup of the special orthogonal group SO(n) then, and therefore also called rotation group of the figure.

Discrete symmetry groups come in three types: (1) finite point groups, which include only rotations, reflections, inversion and rotoinversion - they are in fact just the finite subgroups of O(n), (2) infinite lattice groups, which include only translations, and (3) infinite space groups which combines elements of both previous types, and may also include extra transformations like screw axis and glide reflection. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups.

Two geometric figures are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of the Euclidean group E(n) (the isometry group of Rⁿ), where two subgroups H₁, H₂ of a group G are conjugate, if there exists g ∈ G such that H₁=g^-1H₂g. For example:

• two 3D figures have mirror symmetry, but with respect to a different mirror plane
• two 3D figures have 3-fold rotational symmetry, but with respect to a different axis
• two 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction

Sometimes a broader concept of "same symmetry type" is used, resulting in e.g. 17 wallpaper groups.

When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This excludes for example in 1D the group of translations by a rational number. A "figure" with this symmetry group is non-drawable and up to arbitrarily fine detail homogeneous, without being really homogeneous.

One dimension

The isometry groups in 1D where for all points the set of images under the isometries is topologically closed are:

• the trivial group C₁
• the groups of two elements generated by a reflection in a point; they are isomorphic with C₂
• the infinite discrete groups generated by a translation; they are isomorphic with Z
• the infinite discrete groups generated by a translation and a reflection in a point; they are isomorphic with the generalized dihedral group of Z, Dih(Z), also denoted by D_∞ (which is a semidirect product of Z and C₂).
• the group generated by all translations (isomorphic with R); this group cannot be the symmetry group of a "pattern": it would be homogeneous, hence could also be reflected. However, a uniform 1D vector field has this symmetry group.
• the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group of R, Dih(R).

See also symmetry groups in one dimension.

Two dimensions

Up to conjugacy the discrete point groups in 2 dimensional space are the following classes:

• cyclic groups C₁, C₂, C₃, C₄,... where C_n consists of all rotations about a fixed point by multiples of the angle 360°/n
• dihedral groups D₁, D₂, D₃, D₄,... where D_n (of order 2n) consists of the rotations in C_n together with reflections in n axes that pass through the fixed point.

C₁ is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, for example the letter F. C₂ is the symmetry group of the letter Z, C₃ that of a triskelion, C₄ of a swastika, and C₅, C₆ etc. are the symmetry groups of similar swastika-like figures with five, six etc. arms instead of four.

D₁ is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry, for example the letter A. D₂, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle, and D₃, D₄ etc. are the symmetry groups of the regular polygons.

The actual symmetry groups in each of these cases have two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors.

The remaining isometry groups in 2D with a fixed point, where for all points the set of images under the isometries is topologically closed are:

• the special orthogonal group SO(2) consisting of all rotations about a fixed point; it is also called the circle group S¹, the multiplicative group of complex numbers of absolute value 1. It is the proper symmetry group of a circle and the continuous equivalent of C_n. There is no figure which has as full symmetry group the circle group, but for a vector field it may apply (see the 3D case below).
• the orthogonal group O(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle. It is also called Dih(S¹) as it is the generalized dihedral group of S¹.

For non-bounded figures, the additional isometry groups can include translations; the closed ones are:

• the 7 frieze groups
• the 17 wallpaper groups
• for each of the symmetry groups in 1D, the combination of all symmetries in that group in one direction, and the group of all translations in the perpendicular direction
• ditto with also reflections in a line in the first direction

Three dimensions

Up to conjugacy the set of 3D point groups consists of 7 infinite series, and 7 separate ones. In crystallography they are restricted to be compatible with the discrete translation symmetries of a crystal lattice. This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups (27 from the 7 infinite series, and 5 of the 7 others).

See point groups in three dimensions.

The continuous symmetry groups with a fixed point include those of:

• cylindrical symmetry without a symmetry plane perpendicular to the axis, this applies for example often for a bottle
• cylindrical symmetry with a symmetry plane perpendicular to the axis
• spherical symmetry

For objects and scalar fields the cylindrical symmetry implies vertical planes of reflection. However, for vector fields it does not: in cylindrical coordinates with respect to some axis, has cylindrical symmetry with respect to the axis if and only if A_ρ,A_φ, and A_z have this symmetry, i.e., they do not depend on φ. Additionally there is reflectional symmetry if and only if A_φ = 0.

For spherical symmetry there is no such distinction, it implies planes of reflection.

The continuous symmetry groups without a fixed point include those with a screw axis, such as an infinite helix. See also subgroups of the Euclidean group.

Symmetry groups in general

In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Once we know what kind of mathematical structure we are concerned with, we should be able to pinpoint what mappings preserve the structure. Conversely, specifying the symmetry can define the structure, or at least clarify what we mean by an invariant, geometric language in which to discuss it; this is one way of looking at the Erlangen programme.

For example, automorphism groups of certain models of finite geometries are not "symmetry groups" in the usual sense, although they preserve symmetry. They do this by preserving families of point-sets rather than point-sets (or "objects") themselves.

Like above, the group of automorphisms of space induces a group action on objects in it.

For a given geometric figure in a given geometric space, consider the following equivalence relation: two automorphisms of space are equivalent if and only if the two images of the figure are the same (here "the same" does not mean something like e.g. "the same up to translation and rotation", but it means "exactly the same"). Then the equivalence class of the identity is the symmetry group of the figure, and every equivalence class corresponds to one isomorphic version of the figure.

There is a bijection between every pair of equivalence classes: the inverse of a representative of the first equivalence class, composed with a representative of the second.

In the case of a finite automorphism group of the whole space, its order is the order of the symmetry group of the figure multiplied by the number of isomorphic versions of the figure.

Examples:

• Isometries of the Euclidean plane, the figure is a rectangle: there are infinitely many equivalence classes; each contains 4 isometries.
• The space is a cube with Euclidean metric; the figures include cubes of the same size as the space, with colors or patterns on the faces; the automorphisms of the space are the 48 isometries; the figure is a cube of which one face has a different color; the figure has a symmetry group of 8 isometries, there are 6 equivalence classes of 8 isometries, for 6 isomorphic versions of the figure.

Compare Lagrange's theorem (group theory) and its proof.

Bipyramid

An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal pyramid and its mirror image base-to-base.

The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.

The face-transitive bipyramids are the dual polyhedra of the uniform prisms and will generally have isosceles triangle faces.

Three bipyramids can be made out of all equilateral triangles, the octahedron (tetragonal bipyramid), which counts among the Platonic solids, and the triangular and pentagonal bipyramids, which count among the Johnson solids.

A bipyramid can be projected on a sphere or globe as n equally spaced lines of longitude going from pole to pole, and bisected by a line around the equator.

Bipyramid faces, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry D_nh.

Forms

1. Triangular bipyramid - 6 faces - dual triangular prism
2. Tetragonal bipyramid (octahedron is a special case) - 8 faces - dual cube
3. Pentagonal bipyramid - 10 faces - dual pentagonal prism
4. Hexagonal bipyramid - 12 faces - dual hexagonal prism
5. Heptagonal bipyramid - 14 faces - dual heptagonal prism
6. Octagonal bipyramid - 16 faces - dual octagonal prism
7. Enneagonal bipyramid - 18 faces - dual enneagonal prism
8. Decagonal bipyramid - 20 faces - dual decagonal prism

• ...n-agonal bipyramid - 2n faces - dual n-agonal prism

3

4

5

6

8

10

Symmetry groups

If the base is regular and the line through the apexes intersects the base at its center, the symmetry group of the n-agonal bipyramid has dihedral symmetry D_nh of order 4n, except in the case of a regular octahedron, which has the larger octahedral symmetry group O_h of order 48, which has three versions of D_4h as subgroups. The rotation group is D_n of order 2n, except in the case of a regular octahedron, which has the larger symmetry group O of order 24, which has three versions of D₄ as subgroups.

Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges. So the regular polyhedra — the Platonic solids and Kepler-Poinsot polyhedra — are arranged into dual pairs, with the exception of the regular tetrahedron which is self-dual.

Duality is also sometimes called reciprocity or polarity.

Kinds of duality

There are many kinds of duality. The kinds most relevant to polyhedra are:

• Polar reciprocity
• Topological duality
• Abstract duality

Polar reciprocation

Duality is most commonly defined in terms of polar reciprocation about a concentric sphere. Here, each vertex (pole) is associated with a face plane (polar plane or just polar) so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere

x² + y² + z² = r²,

the vertex

(x₀,y₀,z₀)

is associated with the plane

x₀x + y₀y + z₀z = r².

The vertices of the dual, then, are the poles reciprocal to the face planes of the original, and the faces of the dual lie in the polars reciprocal to the vertices of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual.

Notice that the exact form of the dual will depend on what sphere we reciprocate with respect to; as we move the sphere around, the dual form distorts. The choice of center (of the sphere) is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will necessarily intersect at a single point, and this is usually taken to be the center. Failing that a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents) can be used.

If a polyhedron has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since traditional "Euclidean" space never reaches infinity, the projective equivalent, called extended Euclidean space, must be formed by adding the required 'plane at infinity'. Some theorists prefer to stick to Euclidean space and say that there is no dual. Meanwhile Wenninger (1983) found a way to represent these infinite duals, in a manner suitable for making models (of some finite portion!).

The concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged; in fact it is often mistakenly taken to be a particular version of the same. Projective polarity works well enough for convex polyhedra. But for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. See for example Grünbaum & Shepherd (1988), and Gailiunas & Sharp (2005). Wenninger (1983) also discusses some issues on the way to deriving his infinite duals.

Canonical duals

Any convex polyhedron can be distorted into a canonical form, in which a midsphere or intersphere exists tangent to every edge, such that the average position of these points is the center of the sphere, and this form is unique up to congruences.

If we reciprocate such a polyhedron about its intersphere, the dual polyhedron will share the same edge-tangency points and so must also be canonical; it is the canonical dual, and the two together form a canonical dual compound.

Topological duality

We can distort a dual polyhedron such that it can no longer be obtained by reciprocating the original in any sphere; in this case we can say that the two polyhedra are still topologically dual.

It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph on the sphere or on a flat plane, and the corresponding graph formed by the dual of this polyhedron is its dual graph.

Abstract duality

Duality of a pair of abstract polyhedra is a particular relationship between two partially-ordered sets, each representing the elements (faces, edges, etc) of a polyhedron. Such a 'poset' may in turn be represented in a Hasse diagram. The diagram of the dual polyhedron is obtained by turning the diagram upside-down.

Dorman Luke construction

For a uniform polyhedron, the face of the dual polyhedron may be found from the original polyhedron's vertex figure using the Dorman Luke construction. This construction was originally described by Cundy & Rollett (1961) and later generalised by Wenninger (1983).

As an example, here is the vertex figure (red) of the cuboctahedron being used to derive a face (blue) of the rhombic dodecahedron.

Before beginning the construction, the vertex figure ABCD is (in this case) obtained by cutting each connected edge at its mid-point.

Dorman Luke's construction then proceeds:

1. Draw the circumcircle (tangent to every corner).
2. Draw lines tangent to the circumcircle at each corner A, B, C, D.
3. Mark the points E, F, G, H, where each line meets the adjacent line.
4. The polygon EFGH is a face of the dual polyhedron.

The size of the vertex figure was chosen so that its circumcircle lies on the intersphere of the cuboctahedron, which also becomes the intersphere of the dual rhombic dodecahedron.

Dorman Luke's construction can only be used where a polyhedron has such an intersphere and the vertex figure is cyclic, i.e. for uniform polyhedra.

Dual polytopes

Duality can be generalized to n-dimensional space and dual polytopes.

The vertices of one polytope correspond to the (n − 1)-dimensional elements, or facets, of the other, and the j points that define a (j − 1)-dimensional element will correspond to j hyperplanes that intersect to give a (n − j)-dimensional element. The dual of a honeycomb can be defined similarly.