Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun)...

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symmetry

Transcript of Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun)...

Page 1: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

symmetry

Page 2: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

symmetry

Page 3: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

LAVAL

Page 4: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

LAVAL

Page 5: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

ININI

Page 6: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

ININI

Page 7: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

ININI

Page 8: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

ININI

Page 9: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

b dp qDyslexia…

Page 10: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

β δ π θ

Page 11: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

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

Page 12: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

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)

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P P’

f : positions O : vectors

P

Page 13: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Affine transformation

•  Translation: O identity

•  Homothety: O(PM)=k.PM

•  (also): 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

Page 14: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Translation

•  Infinite periodic lattices

Page 15: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

•  Self-similar objects •  Infinite fractals

Homothety

Page 16: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Similitude

θ  -> θ+θ’

θ’

r -> re-bθ’

e-bθ’

Infinite fractal Logarithmic spiral (r=aebθ)

Page 17: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Isometries

•  Isometry ||O(u)||=||u|| distance-preserving map

•  Helix of pitch P

(α, Pα /2π)

•  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)

Page 18: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Linear isometry- 2D

•  In the plane (2D) ||O(u)|| = ||u||

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θθ

θθ

cossinsincos

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− θθ

θθ

cossinsincos

•  Rotations •  Reflections (reflections by an axis)

θ θ/2

•  Determinant +1 •  Eigenvalues eiθ, e-iθ •  Determinant -1

•  Eigenvalues -1, 1

Page 19: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Linear isometry - 3D

θ θ

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

•  In space (3D) : •  ||O(u)|| = |λ| ||u|| Eigenvalues |λ | = 1

•  λ : 3rd degree equation (real coefficients)

±1, eiθ, e-iθ (det. = ± 1)

Rotations Rotoreflections

•  det. = 1 •  Direct symmetry

•  det. = -1 •  Indirect symmetry

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

Page 20: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

O

N

M

P

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

Page 21: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Main symmetry operations

•  Conventionally

•  Rotations (An) •  Reflections (M) •  Inversion (C)

•  Rotoinversion (An)

•  Indirect •  Rotoreflections (An) •  Reflection (M) •  Inversion (C) •  Rotoinversions (An)

. . .

. . .

. . . .

. .

.

. .

A2 vertical A2 horizontal A3 A4 A5

.

M vertical

. .

Inversion

.

M horizontal M

. . . .

A4

.

•  Direct •  n-fold rotation An (2π/n) •  Represented by a polygon of same symmetry.

~ _

_

•  Symmetry element •  Locus of invariant points

Page 22: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Difficulties…

•  Some symmetry are not intuitive

•  Reflection (mirrors) •  Rotoinversion

‘’The ambidextrous universe’’ Why do mirrors reverse left and right but not top and bottom

Page 23: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Composition of symmetries •  Two reflections with angle α = rotation 2α

Composition of two rotations = rotation

M’M=A M

M’ α

AN1 AN2 AN3

π/N1 π/N2

AN2AN1=AN3

•  Euler construction

•  No relation between N1, N2 et N3

Page 24: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

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)

* E Mx My A2

E E Mx My A2

Mx Mx E A2 MyMy My A2 E MxA2 A2 My Mx E

≠ 1

2 1

2

Mx

My A2

2mm •  Multiplicity: number of elements

Page 25: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Composition of rotations

AN1 AN2 AN3

π/N1 π/N2

Spherical triangle, angles verifies:

ππππ

>++321 NNN

1111

321>++

NNN

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

234

Constraints

Page 26: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Points groups

•  Sorted by Symmetry degree

•  Curie‘s limit groups

•  Chiral, propers

•  Impropers

•  Centrosymmetric

m3 43m m3m ∞ /m ∞/m

3 4 6=3/m 2=m 1

32 422 622 222 _ _ _ _ _

3 4 6 2 1

4/m 6/m 2/m

3m 4mm 6mm 2mm

3m 42m (4m2) _ _ _ 62m (6m2) _ _

4/ mmm 6/ mmm mmm

432 23 _ _ _

∞ /m

∞ 2

∞ m

∞ /mm

∞ ∞

Triclin

ic

Mon

oclin

ic

Ort

horh

ombi

c

Trigon

al

Tetr

agon

al

Hex

agon

al

Cubi

c

Curie’s

grou

ps

...

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’

_

_

_

Page 27: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Multiaxial groups

23 432 532

m3 _

43m _

m3m _

53m _ _

Tétraèdre Octaèdre

Cube

Icosaèdre

Dodécaèdre

Page 28: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Points group: Notations

•  Schönflies : Cn, Dn, Dnh

•  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 2 m m m

4 m m m Notation

réduite

Page 29: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Les 7 groupes limites de Pierre Curie

∞ /m ∞ /m

∞ 2

∞ /m

∞ /mm

∞ ∞

∞ m

Rotating cone

Twisted cylindre

Rotating cylinder

Cone

Cylindrer

Chiral sphere

Sphere

axial + polar vectors

Axial tensor order 2

axial vector (H)

Polar vector (E, F)

Polar tensor ordre 2 (susceptibility)

Axial scalar (chiralité)

Polar scalar (pression, masse)

Page 30: Lect1 2016 Orientation eng - Université Paris-SaclayDefinitions • Symmetry: • From greak (sun) ‘’with" (metron) "measure" • Same etymology as "commensurate" • Until mid-XIX:

Definitions Symmetric: Invariant under at least two

transformations

Asymmetric: Invariant under one transformation. Dissymmetric: Lost of symmetry…