Symmetry Tom Suk Symmetry in 3 D Reflection

Symmetry Tomáš Suk

Symmetry in 3 D • Reflection symmetry – plane of symmetry • Rotational symmetry – axis of symmetry, combination of more axes? • Translational symmetry • Others

Rotational symmetry in 3 D • 1 axis of n-fold rotational symmetry – pyramids C 1, C 2, C 3, C 4, … • 1 axis of n-fold symmetry + n perpendicular axes of 2 -fold symmetry – prism D 1, D 2, D 3, D 4, … • Symmetrical polyhedra T, O, I

Rotational symmetry in 3 D Tetrahedron T – 4 axes of 3 -fold symmetry, – 3 axes of 2 -fold symmetry, total fold number 12

Rotational symmetry in 3 D Cube + Octahedron O – 3 axes of 4 -fold symmetry, – 4 axes of 3 -fold symmetry, – 6 axes of 2 -fold symmetry, total fold number 24

Rotational symmetry in 3 D Dodecahedron + icosahedron I – 6 axes of 5 -fold symmetry, – 10 axes of 3 -fold symmetry, – 15 axes of 2 -fold symmetry, total fold number 60

Infinite rotational symmetry in 3 D • 1 axis of ∞-fold rotational symmetry – conic C∞ e. g. bottle • 1 axis of ∞-fold symmetry + ∞ axes of 2 fold symmetry – cylinder D∞ • ∞ axes of ∞-fold symmetry – sphere K=O(3)

Rotation + reflection in 3 D • • Cn – n-fold rotational symmetry Cnh – Cn + horizontal reflection plane Cnv – Cn + n vertical reflection planes Dn – dihedral symmetry Dnh – Dn + horizontal reflection plane Dnd – Dn + S 2 n – rotation & reflection C 1 h=C 1 v D 1=C 2 D 1 h=C 2 v D 1 d=C 2 h

Rotation + reflection in 3 D Cnv – Pyramidal symmetry Dnh – Prismatic symmetry Dnd – Antiprismatic symmetry

Rotation + reflection in 3 D • • T – chiral tetrahedral symmetry Th – pyritohedral symmetry - 3 planes Td – full tetrahedral symmetry - 6 planes O – chiral octahedral symmetry Oh – full octahedral symmetry - 9 planes I – chiral icosahedral symmetry Ih – full icosahedral symmetry - 15 planes

Rotation + reflection in 3 D Group T Group O No reflection symmetry

Rotation + reflection in 3 D • • C 1, C 2, C 3, C 4, … C 1 h, C 2 h, C 3 h, C 4 h, … C 1 v, C 2 v, C 3 v, C 4 v, … C∞v D 1, D 2, D 3, D 4, … D 1 h, D 2 h, D 3 h, D 4 h, … D∞h D 1 d, D 2 d , D 3 d , D 4 d , … S 2, S 4, S 6, S 8, … T, Th, Td, O, Oh, I, Ih, K

Central symmetry • 1 D: f(x)=f(-x) reflection • 2 D: f(x, y)=f(-x, -y) 2 -fold rotational symmetry • 3 D: f(x, y, z)=f(-x, -y, -z) reflection & rotation by 180° = group S 2

Rotation + reflection + translation in 3 D • • 7 crystal systems 32 point groups 14 Bravais lattices 230 space groups

Crystal systems Fold number Point groups • • triclinic n=1 monoclinic n=2 orthorhombic n=2 trigonal n=3 tetragonal n=4 cubic n=3, 4 hexagonal n=6 C 1, S 2 C 2, C 1 h, C 2 h C 2 v, D 2 h C 3, C 3 v, D 3 d , S 6 C 4, C 4 h, C 4 v , D 4 h , D 2 d, S 4 T, Th, Td, O, Oh C 6, C 3 h, C 6 h , C 6 v, D 6, D 3 h, D 6 h

Space groups - Schönflies symbols • • • triclinic C 11, S 21 monoclinic C 21 -3, C 1 h 1 -4, C 2 h 1 -6 orthorhombic. C 2 v 1 -22, D 21 -9, D 2 h 1 -28 trigonal C 31 -4, C 3 v 1 -6, D 31 -7, D 3 d 1 -6 , S 61 -2 tetragonal C 41 -6, C 4 h 1 -6, C 4 v 1 -12, D 41 -10, D 4 h 1 -20, D 2 d 1 -12, S 41 -2 • cubic T 1 -5, Th 1 -7, Td 1 -6, O 1 -8, Oh 1 -10 • hexagonal C 61 -6, C 3 h 1, C 6 h 1 -2, C 6 v 1 -4, D 61 -6, D 3 h 1 -4, D 6 h 1 -4

Bravais lattices

Crystals - examples Gypsum: Crystal system monoclinic Space group C 2 h 6 = C 2/c Aquamarine: Crystal system hexagonal Space group D 6 h 4 = P 63 /mmc

Helical symmetry Examples: Screw, DNA – double helix Infinite helical symmetry 2 -fold helical symmetry

Helical symmetry Rotation & translation • Infinite helical symmetry • n-fold helical symmetry • Non-repeating helical symmetry

Symmetries Thank you for your attention