The 219 affine-class representatives are pairwise affinely inequivalent

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LeanEval.Geometry.SpaceGroupsProblem.SpaceGroupCatalog.spaceGroupCatalog_affine_irredundant

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

Let $n_1<\dots<n_{219}$ be the ITA numbers other than the second members $78,95,96,145,153,154,170,172,179,181,213$ of the enantiomorphic pairs. Then

G_{n_k}\sim G_{n_l}\ \Longrightarrow\ k=l .

In other words, the $219$ affine space-group types are pairwise distinct: no invertible affine map (of either orientation) conjugates two of these groups onto each other. Together with completeness, this gives the count of $219$ affine classes.

Formalization note. The list $n_1,\dots,n_{219}$ is affineRepIndex.

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_affine_irredundant :
    ∀ k l : Fin 219,
      AffinelyEquivalent (spaceGroupCatalog (affineRepIndex k)) (spaceGroupCatalog (affineRepIndex l)) →
        k = l := 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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