Every group of the space-group catalogue is crystallographic

Proved

LeanEval.Geometry.SpaceGroupsProblem.SpaceGroupCatalog.spaceGroupCatalog_isCrystallographic

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 $\det\varphi_{\mathrm{lin}}>0$ .

Each catalogue group is a crystallographic group in the sense of the LeanEval definitions: for all $n$ ,

G_n \text{ is discrete and contains translations by three linearly independent vectors.}

Here discrete means that for every $x\in E$ and $\varepsilon>0$ only finitely many $g\in G_n$ satisfy $\operatorname{dist}(gx,x)\le\varepsilon$ . This certifies that the catalogue consists of genuine space groups, so that it can serve as a list of representatives in the classification.

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_isCrystallographic :
    ∀ i : Fin 230, IsCrystallographicGroup (spaceGroupCatalog i) := 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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