symmetry symmetry LAVAL LAVAL SHINZOX SHINZOX ININI ININI

symmetry

symmetry

LAVAL

LAVAL

SHINZOX

SHINZOX

ININI

ININI

ININI

ININI

b d p q Dyslexia…

bd pq

• Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX: only mirror symmetry Definitions • Transformation, Group • Évariste Galois 1811, 1832. Symmetry: Property of invariance of an objet under a space transformation

Definitions Symmetric: Invariant under at least two transformations Asymmetric: Invariant under one transformation. Dissymetric: Lost of symmetry…

Transformation • Bijection which maps a geometric set in itself M f(M)=M’ • Affine transformation maps two points P and P’ such that: f(M) = P’ + O(PM) P P P’ f : positions O : vectors

Affine transformation preserves lines, planes, parallelism • Translation: O identity P P P P’ • Homothety: O(PM)=k. PM • Affinité: Homothety in one direction • Isometry: preserves distances P P • Simililarity: preserves ratios P P

Translation • Infinite periodic lattices

Homothety • Self-similar objects • Infinite fractals

Similitude Infinite fractal Logarithmic spiral (r=aebq) q -> q+q’ r -> re-bq’ q’

Isometries f(M) = P’ + O(PM) • Isometry ||O(u)||=||u|| distance-preserving map • Two types of isometry: • Affine isometry: f(M) • Transforms points. • Microscopic properties of crystals (electronic structure) • Helix of pitch P • Translation • Rotations • Reflections (a, Pa /2 p) • Linear isometry O(PM) • Transforms vectors (directions) • Macroscopic properties of crystals (response functions) 60° • Rotations • Reflections E ?

Linear isometry- 2 D ||O(u)|| = ||u|| • In the plane (2 D) • Rotations q • Determinant +1 • Eigenvalues eiq, e-iq • Reflections (reflections through an axis) q/2 • Determinant -1 • Eigenvalues -1, 1

Linear isometry - 3 D • ||O(u)|| = |l| ||u|| Eigenvalues |l | = 1 • l : 3 rd degree equation (real coefficients) ± 1, eiq, e-iq (det. = ± 1) • In space (3 D) : • det. = 1 • Direct symmetry • det. = -1 • Indirect symmetry Rotoreflections Rotations a) Rotation by angle q b) Roto-reflection q Improper rotation c) Inversion (p) d) Roto-inversion (p+q ) c) Reflection (0) q q q

Stereographic projection • To represent directions preserves angles on the sphere NN Direction OM M P’ O P P’ M’ P S P, projection of OM : Intersection of SM and equator • Conform transformation (preserves angles locally) but not affine

Main symmetry operations • Conventionally • Direct • Rotations (An) • Reflections (M) • Inversion (C) • Rotoinversion (An) • n-fold rotation An (2 p/n) • Represented by a polygon of same symmetry. . . . A 2 vertical A 2 horizontal • Indirect • • A 3 ~ M vertical M horizontal . . . A 5 A 4 • Locus of invariant points _ . . . • Symmetry element Rotoreflections (An) Reflection (M) Inversion (C) Rotoinversions (An) . . . _ . . M . . Inversion . . A 4

Difficulties… • Some symmetry are not intuitive • Reflection (mirrors) • Rotoinversion ‘’The ambidextrous universe’’ Why do mirrors reverse left and right but not top and bottom

Composition of symmetries • Two reflections with angle a = rotation 2 a M’M=A M 2 a a M’ • Euler construction AN 3 AN 1 p/N 1 Composition of two rotations = rotation A N 2 A N 1= A N 3 • No relation between N 1, N 2 et N 3 AN 2 p/N 2

Point group: definition • The set of symmetries of an object forms a group G • • • A and B G, AB G (closure) Associativity (AB)C=A(BC) Identity element E (1 -fold rotation) Invertibility A, A-1 No commutativity in general (rotation 3 D) 1 2 2 1 • Example: point groupe of a rectangular table (2 mm) Mx My A 2 • Multiplicity: number of elements 2 mm

Composition of rotations Constraints AN 1 p/N 1 AN 3 AN 2 p/N 2 234 Spherical triangle, angles verifies: 22 N (N>2), 233, 234, 235 Dihedral groups Multiaxial groups

Curie’s groups Cubic Hexagonal Tetragonal Trigonal Orthorhombic Monoclinic Triclinic . . . An 1 2 3 4 6 ¥ 32 422 622 ¥ 2 An A 2 222 Points groups • Sorted by Symmetry degree • Curie‘s limit groups • Chiral, propers _ An _ 1 _ 2=m _ 3 _ 4 _ 6=3/m • Impropers An/M ¥ /m 2/m 4/m 6/m 4 mm 6 mm An M 2 mm 3 m ¥m _ An M _ 3 m _ _ 42 m (4 m 2) 62 m (6 m 2) ¥ /mm An /MM’ mmm 4/mmm 6/mmm An An’ 23 432 _ m 3 _ 43 m ¥¥ _ An An’ _ m 3 m ¥/m¥/m • Centrosymmetric

23 _ m 3 Multiaxial groups 532 432 _ _ __ 43 m m 3 m 53 m Tétraèdre Octaèdre Icosaèdre Cube Dodécaèdre

Points group: Notations • Hermann-Mauguin (International notation - 1935) • Generators (not minimum) • Symmetry directions • Reflection ( - ): defined by the normal to the plane Primary Direction: higher-order symmetry Secondary directions : lower-order 4 2 2 mmm Notation réduite 4 mm m Tertiary directions : lowest-order • Schönflies : Cn, Dnh

Les 7 groupes limites de Pierre Curie Cône tournant Vecteur axial + polaire Cylindre tordu Tenseur axial d’ordre 2 Cylindre tournant Vecteur axial (H) Cône Vecteur polaire (E, F) Cylindre Tenseur polaire d’ordre 2 (susceptibilité) Sphère tournante Scalaire axial (chiralité) Sphère Scalaire polaire (pression, masse) ¥ ¥ 2 ¥ /m ¥m ¥ /mm ¥¥ ¥ /m