The rotation $z\mapsto e^{\pi i/3}z$ has order exactly $6$ in the isometry group of the plane

Proved

CurveSymmetry.sixtyDegree_isometry_order

by carlok · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebraic-geometrycurve-symmetry-leanzariski-topology

Let $\rho:\mathbb{C}\to\mathbb{C}$ , $\rho(z)=e^{\pi i/3}z$ , be the rotation through $60^\circ$ about the origin, regarded as an element of the group of isometries of the Euclidean plane $\mathbb{C}$ under composition.

Then

\operatorname{ord}(\rho)=6,

that is, $\rho^6=\mathrm{id}$ and $\rho^k\ne\mathrm{id}$ for $1\le k\le5$ .

This is the rotation of order six in the quintic example of equation (2) of the note. The statement concerns the isometry itself, not only its coefficient $e^{\pi i/3}$ as a root of unity.

Formalization Note: $\rho$ is the affine isometry $z\mapsto az+b$ with $a=e^{\pi i/3}$ , of modulus one, and $b=0$ , as an element of the group ℂ ≃ᵢ ℂ.

Preamble

import Definitions.Def_CurveSymmetry_01_RealLoci
import Definitions.Def_CurveSymmetry_02_FamilyBasics
import Definitions.Def_CurveSymmetry_03_IsometriesAndCharts
import Definitions.Def_CurveSymmetry_04_ProjectiveClosure
import Mathlib.Algebra.MvPolynomial.Nilpotent
import Mathlib.Algebra.MvPolynomial.NoZeroDivisors
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Pi
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.Complex.OperatorNorm
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Analysis.Normed.Affine.MazurUlam
import Mathlib.Data.Complex.Basic
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.MvPolynomial.Homogeneous
import Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic
import Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.Tactic
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Compactification.OnePoint.ProjectiveLine

open CurveSymmetry
set_option autoImplicit false

Formal statement

theorem CurveSymmetry.sixtyDegree_isometry_order :
    orderOf (affineDirectIsometry sixtyDegreeCoefficient 0 sixtyDegree_norm) = 6 := by sorry

Source

C. Perassi, Sharp symmetry bounds for real algebraic curves (note), equation (2), p. 1, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/QuinticExample.lean (C. Perassi)

Human review

Endorsed by Shuze Chen · Oct 7, 2026
Confirmed by the moderator at approval.
Endorsed by carlok · Oct 7, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

The rotation z↦eπi/3zz\mapsto e^{\pi i/3}zz↦eπi/3z has order exactly 666 in the isometry group of the plane

The rotation $z\mapsto e^{\pi i/3}z$ has order exactly $6$ in the isometry group of the plane