Completeness of the space-group catalogue (Fedorov–Schoenflies)

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LeanEval.Geometry.SpaceGroupsProblem.SpaceGroupCatalog.spaceGroupCatalog_complete

by Gabewhigham · Sep 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

crystallographydiscrete-geometrygroup-theory

Throughout, $E=\mathbb R^3$ with its Euclidean metric, $G_1,\dots,G_{230}$ denote the groups of the explicit catalogue spaceGroupCatalog (the group $G_n$ realises the space-group type with ITA number $n$ ; see the definition SpaceGroupCatalog), $G\sim G'$ means that some invertible affine map $\varphi$ of $E$ satisfies $\varphi G\varphi^{-1}=G'$ , and $G\sim_+G'$ means that such a $\varphi$ exists with $\det\varphi_{\mathrm{lin}}>0$ .

Every crystallographic group of $\mathbb R^3$ is orientation-preserving affinely equivalent to a member of the catalogue:

\forall\,G\ \text{crystallographic},\quad \exists\, n\in\{1,\dots,230\}:\ G_n\sim_+ G.

This is the existence half of the classification of three-dimensional space groups: every space group belongs to one of the $230$ types tabulated in the International Tables. It is the deep part of the theorem (Bieberbach's structure theory, the $73$ arithmetic classes of finite subgroups of $\mathrm{GL}_3(\mathbb Z)$ , and the computation of the non-symmorphic extensions).

Preamble

import Mathlib
import Definitions.Def_LeanEval_SpaceGroups_Definitions
import Definitions.Def_SpaceGroupCatalog

Formal statement

namespace LeanEval
namespace Geometry
namespace SpaceGroupsProblem
namespace SpaceGroupCatalog

theorem spaceGroupCatalog_complete :
    ∀ G : CrystallographicGroup 3, ∃ i : Fin 230, AffOPEquivalent (spaceGroupCatalog i) G.1 := by
  sorry

end SpaceGroupCatalog
end SpaceGroupsProblem
end Geometry
end LeanEval

Source

International Tables for Crystallography, Vol. A (T. Hahn, ed.), Section 1.4 and Table 1.4.1 (230 space-group types, 219 affine classes, 11 enantiomorphic pairs, 65 Sohncke types); classification due to E. S. Fedorov (1891) and A. Schoenflies (1891). Catalogue data: standard settings of ITA Vol. A via Hall symbols (S. R. Hall, Acta Cryst. A37 (1981) 517-525).

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