1. symmetry

3. LAVAL

5. SHINZOX

7. ININI

11. b d p q Dyslexia…

12.  

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

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

15. 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’ f : positions O : vectors P

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

17. Translation Infinite periodic lattices

18. Self-similar objects Infinite fractals Homothety

19. Similitude  ’ ’’ r -> re -b  ’ e -b  ’ Infinite fractal Logarithmic spiral (r=ae b  )

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

21. Linear isometry- 2D In the plane (2D) || O(u) || = || u || Rotations Reflections (reflections through an axis)   Determinant +1 Eigenvalues e i , e -i  Determinant -1 Eigenvalues -1, 1

22. Linear isometry - 3D   c) Inversion (  ) d) Roto-inversion (  ) c) Reflection (0) In space (3D) : || O(u) || = |  || u || Eigenvalues |  = 1  : 3rd degree equation (real coefficients) ±1, e i , e -i  (det. = ± 1) Rotations Rotoreflections det. = 1 Direct symmetry det. = -1 Indirect symmetry a) Rotation by angle  b) Roto-reflection  Improper rotation

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

24. Main symmetry operations Conventionally Rotations (A n ) Reflections (M) Inversion (C) Rotoinversion (A n ) Indirect Rotoreflections (A n ) Reflection (M) Inversion (C) Rotoinversions (A n )............... A 2 verticalA 2 horizontal A3A3 A4A4 A5A5. M vertical.. Inversion. M horizontal M.... A4A4. Direct n-fold rotation A n (2  /n) Represented by a polygon of same symmetry. ~ _ _ Symmetry element Locus of invariant points

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

26. Composition of symmetries Two reflections with angle  = rotation 2  Composition of two rotations = rotation M’M=A M M’   AN1AN1 AN2AN2 AN3AN3  /N 1  /N 2 AN2AN1=AN3AN2AN1=AN3 Euler construction No relation between N 1, N 2 et N 3

27. 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 3D) Example: point groupe of a rectangular table (2mm)  1 2 1 2 MxMx MyMy A2A2 2mm Multiplicity: number of elements

28. Composition of rotations AN1AN1 AN2AN2 AN3AN3  /N 1  /N 2 Spherical triangle, angles verifies: 22N (N>2), 233, 234, 235 Dihedral groups Multiaxial groups 234 Constraints

29. Points groups Sorted by Symmetry degree Curie‘s limit groups Chiral, propers Impropers Centrosymmetric m343mm3m  /m  346=3/m2=m1 32422622222 _____ 34621 4/m6/m2/m 3m4mm6mm2mm 3m42m (4m2) __ _ 62m (6m2) _ _ 4/ mmm 6/ mmm 43223 ___   /m  2  m  /mm    Triclinic Monoclinic Orthorhombic Trigonal Tetragonal Hexagonal Cubic Curie’s groups... A n A n’ A n A n A 2 A n A n /M A n M A n M A n /MM’ A n A n’ _ _ _

30. Multiaxial groups 23432 532 m3 _ 43m _ m3m _ 53m _ _ Tétraèdre Octaèdre Cube Icosaèdre Dodécaèdre

31. Points group: Notations Schönflies : C n, D n, D nh 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 Tertiary directions : lowest-order 4 2 24 2 2 mmm 4 m mm Notation réduite

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