Symmetry : orientation, position

Symmetry: definition

« definition » of symmetry :

From Greek (sun) ‘’with" (metron) "measure" Same etymology as "commensurate" ( Until mid-XIX: only mirror symmetry)

=> Property of invariance of an objet under a space transformation ’

‘’Transformation => Group (Evariste Galois, 1811-1832)’’

Transformation

• Bijection which maps a geometric set in itself

M f(M)=M’

• Affine transformation defined by P, P’ and O such that:

f(M) = P’ + O(PM)

& x'# &a# &txx t yx tzx #& x# P’ $ ! $ ! $ !$ ! P P $ y'! = $b! + $txy t yy tzy !$ y! f : positions $ z'! $ c ! $t t t !$ z ! % " % " % xz yz zz "% " O : vectors preserves lines, planes, parallelism Affine transformation

• Translation: O identity P’ P P • Homothety: O(PM)=k.PM

P P

• Affinité: Homothety in one direction

P P

• 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=aebθ)

θ -> θ+θ’ r -> re-bθ’

-bθ’ e θ’ 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)

• Translation • Helix of pitch P • Rotations • Reflections (α, Pα /2π)

• Linear isometry O(PM) • Transforms vectors (directions) • Macroscopic properties of crystals (response functions) 60° • Rotations E ? • Reflections Linear isometry – 2D

||O(u)|| = ||u|| • In the plane (2D)

• Rotations • Reflections (reflections by an axis)

θ θ/2

&cosθ − sinθ # &cos θ sinθ # $ ! $ ! % sinθ cosθ " % sinθ − cosθ " • Determinant +1 • Determinant -1 • Eigenvalues eiθ, e-iθ • Eigenvalues -1, 1 Linear isometry - 3D

• ||O(u)|| = |λ| ||u|| Eigenvalues |λ | = 1 • In space (3D) : • λ : 3rd degree equation (real coefficients) ±1, eiθ, e-iθ (det. = ± 1) • det. = 1 • det. = -1 • Direct symmetry • Indirect symmetry

Rotations Rotoreflections

a) Rotation by angle θ θ θ b) Roto-reflection θ Improper rotation

c) Inversion (π) θ d) Roto-inversion (π+θ ) c) Reflection (0) Stereographic projection

• To represent directions preserves angles on the sphere

N

Direction OM M

O P’ P P’ M’ P

P, projection of OM : Intersection of SM and equator S

• Conform transformation (preserves angles locally) but not affine Main symmetry operations

• Conventionally

• Direct • Rotations (A ) n • n-fold rotation An (2π/n) • Reflections (M) • Represented by a polygon of same symmetry. • Inversion (C) _

• Rotoinversion (An) ......

A2 vertical A2 horizontal A3 A4 A5

• Indirect ~ • Rotoreflections (An) • Symmetry element • Reflection (M) • Locus of invariant points • Inversion (C) _

• Rotoinversions (An) ......

M vertical M horizontal M Inversion A4 Some 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 α = rotation 2α M M’M=A 2α M’ α • Euler construction

A AN3 N2 AN1

π/N2 π/N1 Composition of two rotations = rotation

AN2AN1=AN3

• No relation between N1, N2 et N3 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) 1 2 • Identity element E (1-fold rotation) • Invertibility A, A-1 ≠ • No commutativity in general (rotation 3D) 2 1 • Example: point group of a rectangular table (2mm)

Mx

* E Mx My A2

E E Mx My A2 My Mx Mx E A2 My

My My A2 E Mx A2 A2 A2 My Mx E

• Multiplicity: number of elements 2mm Composition of rotations Constraints

AN2 AN1 AN3

π/N2 π/N1

234

Spherical triangle, angles verifies: π π π 1 1 1 + + > π + + >1 N1 N2 N3 N1 N2 N3

22N (N>2), 233, 234, 235 Dihedral groups Multiaxial groups

Point

... groups groups Monoclinic Triclinic Cubic Orthorhombic Trigonal Tetragonal Hexagonal Curie’s

An 1 2 3 4 6 ∞

An A2 222 32 422 622 ∞ 2 _ An _ _ _ _ _ 1 2=m 3 4 6=3/m

∞ An /M /m 2/m 4/m 6/m

An M 2mm 3m 4mm 6mm ∞ m _ An M _ _ _ _ _ 3m 42m (4m2) 62m (6m2)

∞/mm An /MM’ mmm 4/ mmm 6/ mmm

An A n’ 23 432 ∞ ∞ _ An A n’ _ _ _ m3 43m m3m ∞ /m∞ /m Multiaxial groups

23 432 532

_ _ _ __ m3 43m m3m 53m

Tétraèdre Octaèdre Icosaèdre

Cube Dodécaèdre Point groups: 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 Notation 4 m m m m m réduite m

Tertiary directions : lowest-order

• Schönflies : Cn, Dn, Dnh