Math 2.0

Group theory and generalizations Definition

Crystallographic group

Builds on Euclidean groupEuclidean spaceEuclidean isometryCompact spaceTranslationOrthogonal group

Definition

Let n≥1n \ge 1 and let E(n)E(n) be the Euclidean group of all isometries of ℝn\R^n. Each element of E(n)E(n) is a map x↦Ax+bx \mapsto Ax + b with AA in the orthogonal group O(n)O(n) and b∈ℝnb \in \R^n, and E(n)E(n) carries the topology of O(n)×ℝnO(n) \times \R^n. A subgroup Γ\Gamma of E(n)E(n) is a crystallographic group of dimension nn if:

  1. Γ\Gamma is a discrete subgroup of E(n)E(n).
  2. The quotient space ℝn/Γ\R^n / \Gamma of orbits is compact.

Remarks

Example: the group ℤn\Z^n of translations x↦x+bx \mapsto x + b with b∈ℤnb \in \Z^n. It is discrete, since distinct elements differ by a translation of length at least 11. The quotient ℝn/ℤn\R^n / \Z^n is the image of the compact cube [0,1]n[0, 1]^n, so it is compact; it is the nn-dimensional torus.

Non-example: the translations (x,y)↦(x+k,y)(x, y) \mapsto (x + k, y) of ℝ2\R^2 with k∈ℤk \in \Z. This group is discrete, but the quotient is an infinite cylinder, which is not compact, so clause 2 fails.

Equivalently, Γ\Gamma acts properly discontinuously on ℝn\R^n with a compact fundamental domain. Crystallographic groups of dimension 33 are called space groups, and those of dimension 22 are the wallpaper groups.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Compact space, concept, no evidence levelCompact spaceconceptEuclidean isometry, concept, no evidence levelEuclidean isometryconceptEuclidean space, object, no evidence levelEuclidean spaceobjectIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionTranslation, concept, no evidence levelTranslationconceptEuclidean group, object, no evidence levelEuclidean groupobjectTranspose, definition, no evidence levelTransposedefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectCrystallographic group, definition, no evidence levelCrystallographic groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Compact space concept
  2. Euclidean isometry concept
  3. Euclidean space object
  4. Identity matrix object
  5. Matrix definition
  6. Translation concept
  7. Euclidean group object
  8. Transpose definition
  9. Orthogonal matrix definition
  10. Orthogonal group object
  11. Crystallographic group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Space_group
About this page
TagGTHWVEB1NZ
URIm2://GTHWVEB1NZ
Kinddefinition
WritingW0 stub
MSC20H15, 52C22
DefinesCrystallographic group
UsesEuclidean group, Euclidean space, Euclidean isometry, Compact space, Translation, Orthogonal group
Changed2 November 2026

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