The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. This is strictly true for the mathematical formalism, but in the physical world quasicrystals occur with other symmetries, such as 5-fold. Rotation is the movement of a body in such a way that the distance between a certain fixed point and any given point of that body remains constant. ... Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... Quartz crystal A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ... Quasicrystals are a peculiar form of solid in which the atoms of the solid are arranged in a seemingly regular, yet non-repeating structure. ...

In mathematical terms, a crystal is modeled as a discrete lattice, generated by a list of independent finite translations. Because we insist on a lower bound on the spacing between lattice points, any rotational symmetry of the lattice must belong to a finite group. The force of the theorem is that not all finite groups are compatible with a discrete lattice; in any dimension, we will have only a finite number of compatible groups. See lattice for other meanings of this term, both within and without mathematics. ... Translation is an activity comprising the interpretation of the meaning of a text in one language—the source text—and the production of a new, equivalent text in another language—called the target text, or the translation. ... In mathematics, a finite group is a group which has finitely many elements. ...

Dimensions 2 and 3

The special cases of 2D (wallpaper groups) and 3D (space groups) are most heavily used in applications, and we can treat them together. We will prove the restriction first geometrically using lattices properties, then algebraically using matrix theory. Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical device used to describe and classify repetitive designs on two-dimensional surfaces, such as walls. ... The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ... Matrix theory is a branch of mathematics which focuses on the study of matrices. ...

Lattice proof

We might call this a proof in the style of Busby Berkeley, with lattice vectors rather than pretty ladies dancing and swirling in geometric patterns. A rotation symmetry in these dimensions must move a lattice point to a succession of other lattice points in the same plane, generating a regular polygon of coplanar lattice points. Thus we now confine our attention to the plane. Busby Berkeley (November 29, 1895–March 14, 1976), born William Berkeley Enos in Los Angeles, California, was a highly influential Hollywood movie director and musical choreographer. ... The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ... In mathematics, groups are often used to describe symmetries of objects. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ...

Lattices restrict polygons
 Compatible: 6-fold (3-fold), 4-fold (2-fold)
 Incompatible: 8-fold, 5-fold

Consider a lattice built from equilateral triangles. That is, the lattice basis vectors are two sides of an equilateral triangle, and all other displacements are sums of integer multiples of these. With 60° angles at each vertex, six of these triangles exactly fit (sum to 360°) around every lattice point, demonstrating 6-fold rotation symmetry. Instead building from squares, the vertex angles are 90°, four fit around each lattice point, and the rotation symmetry is 4-fold. These examples also exhibit 3-fold and 2-fold symmetry. Thus the possibilities included by the theorem exist. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

Now consider an 8-fold rotation, and the vectors between adjacent points of the polygon. If a displacement exists between any two lattice points, then that same displacement is repeated everywhere in the lattice. So collect all the edge displacements to begin at a single lattice point. The edge vectors become radial vectors, and their 8-fold symmetry implies a regular octagon of lattice points around the collection point. But this is impossible, because the new octagon is about 80% smaller than the original. The significance of the shrinking is that it is unlimited. Because the same construction can be repeated with the new octagon, and again and again until the distance between lattice points is as small as we like, no discrete lattice can have 8-fold symmetry. The same argument applies to any k-fold rotation, for k greater than 6.

Shrinking also eliminates 5-fold symmetry. Consider a regular pentagon of lattice points. If it exists, then we can take every other edge displacement and (head-to-tail) assemble a 5-point star, with the last edge returning to the starting point. The vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, but about 60% smaller than the original.

Matrix proof

Now consider matrix properties. The sum of the diagonal elements of a matrix is called the trace of the matrix. In 2D and 3D every rotation is a planar rotation, and the trace is a function of the angle alone. For a 2D rotation, the trace is 2 cos θ; for a 3D rotation, 1 + 2 cos θ. For the square matrix section, see square matrix. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...

Examples

• Consider a 60° (360°/6) rotation matrix with respect to an orthonormal basis in 2D.

The trace is precisely 1, an integer.

• Consider a 45° (360°/8) rotation matrix.

The trace is 2/√2, not an integer.

Using a lattice basis, neither orthogonality nor unit length is guaranteed, only independence. However, the trace is the same with respect to any basis. (Similarity transforms preserve trace.) In a lattice basis, because the rotation must map lattice points to lattice points, each matrix entry — and hence the trace — must be an integer. Thus, for example, wallpaper and crystals cannot have 8-fold rotational symmetry. The only possibilities are multiples of 60°, 90°, 120°, and 180°, corresponding to 6-, 4-, 3-, and 2-fold rotations. A rotation matrix is a matrix that generalizes the concept of a rotation of a set of points around a certain axis, through some arbitrary angle . ... In mathematics, an orthonormal basis of an inner product space V(i. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Several equivalence relations in mathematics are called similarity. ... In mathematics, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain. ...

Example

• Consider a 60° (360°/6) rotation matrix with respect to the oblique lattice basis for a tiling by equilateral triangles.

The trace is still 1. The determinant (always +1 for a rotation) is also preserved.

The general crystallographic restriction on rotations does not guarantee that a rotation will be compatible with a specific lattice. For example, a 60° rotation will not work with a square lattice; nor will a 90° rotation work with a rectangular lattice. Oblique case For the punctuation mark /, see slash (punctuation) In geometry, oblique describes an angle that is not a right angle. ... In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects are considered as well. ... In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

Higher dimensions

When the dimension of the lattice rises to four or more, rotations need no longer be planar; the 2D proof is inadequate. However, restrictions still apply, though more symmetries are permissible. This is of interest, not just for mathematics, but for the physics of quasicrystals under the cut-and-project theory. In this view, a 3D quasicrystal with 5-fold rotation symmetry might be the projection of a slab cut from a 4D lattice.

Example

• Consider a 4D rotation matrix with simultaneous rotation in two 2D subspaces.

This is a rotation both by 90° (in the first two dimensions) and by 180° (in the last two).

To state the restriction for all dimensions, it is convenient to shift attention away from rotations alone and concentrate on the integer matrices. We say that a matrix A has order k when its k-th power (but no lower), A^k, equals the identity. Thus a 6-fold rotation matrix in the equilateral triangle basis is an integer matrix with order 6. Let Ord_N denote the set of integers that can be the order of an N×N integer matrix. For example, Ord₂ = {1, 2, 3, 4, 6}. We wish to state an explicit formula for Ord_N. In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

Define a function ψ based on Euler's totient function φ; it will map positive integers to non-negative integers. For an odd prime, p, and a positive integer, k, set ψ(p^k) equal to the totient function value, φ(p^k), which in this case is p^k−p^k−1. Do the same for ψ(2^k) when k > 1. Set ψ(2) and ψ(1) to 0. Using the fundamental theorem of arithmetic, we can write any other positive integer uniquely as a product of prime powers, m = ∏_i p_i^k_i; set ψ(m) = ∑_i ψ(p_i^k_i). In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. ...

The crystallographic restriction in general form states that Ord_N consists of those positive integers m such that ψ(m) ≤ N.

Smallest dimension for a given order

m	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
ψ(m)	0	0	2	2	4	2	6	4	6	4	10	4	12	6	6

Note that these additional symmetries do not allow a planar slice to have, say, 8-fold rotation symmetry. In the plane, the 2D restrictions still apply. Thus the cuts used to model quasicrystals necessarily have thickness.

Further note that integer matrices include the complete point group, not just rotations. For example, a reflection is also symmetry of order 2. Insisting on determinant +1 trims the group to proper rotations. In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ... In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ...

See also

• Crystallographic point group
• Crystallography
• Wallpaper group

In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical device used to describe and classify repetitive designs on two-dimensional surfaces, such as walls. ...

External links

• The crystallographic restriction
• Bamberg, Cairns, Kilminster. The crystallographic restriction, permutations, and Goldbach's conjecture