Symmetry Tom Suk Symmetry in art harmonious or

Symmetry Tomáš Suk

Symmetry in art - harmonious or aesthetically pleasing proportionality and balance Symmetry in physics - quantum symmetry, more general Symmetry in chemistry, geology, biology - only approximate Symmetry in mathematics - today's theme

Symmetry in mathematics Function f(x) is symmetric, if there is such geometric transformation G that f(x)= f(G(x)) f(x) is then symmetric with respect to G x – coordinates in n-dimensional space

Symmetry in mathematics f(x)= f(G(x)) Two limit cases: f(x) constant, G(x) arbitrary – uninteresting f(x) arbitrary, G(x) identity – no symmetry function →← image function infinite →← finite support

Symmetry in mathematics f(x)= f(G(x)) More than one G(x): They create a group operation is composition neutral element is identity Groups: Øfinite Øcountable Øcardinality of continuum

Class of symmetry • Set of similar groups, they differ by a parameter only • Each group is applicable on different function Terminological remark: symmetry group ≠ symmetric group

Symmetry in 1 D • Reflection symmetry σ, C 2, D 1 - also mirror symmetry f(x-a)=f(a-x) • Translational symmetry Z f(x)=f(x+kλ) • Reflection and translation D∞

Symmetry in 1 D • Scale symmetry → fractals f(x)=f(bkx)

Symmetry in 2 D • Reflection symmetry = axial symmetry • Rotational symmetry – fold number n C 1, C 2, C 3, C 4, … rotation by 360°/n • Dihedral symmetry – reflection + rotation D 1, D 2, D 3, D 4, … • Circular symmetry D∞ =O(2) – reflection + rotation by arbitrary angle

Rotational symmetry in 2 D Examples C 3 D 1 D 5 C 3 D 6 C 4 D 24 D∞

Rotation + reflection in 2 D • C 1, C 2, C 3, C 4, … • D 1, D 2, D 3, D 4, … D ∞ C 1 - No symmetry

Frieze symmetry • f(x, y), x – infinite support, y – finite support • Frieze - long stretch of painted, sculpted or even calligraphic decoration

Frieze symmetry p 111 (translation only) p 1 a 1 (translation + glide reflection) p 1 m 1 (translation + horizontal line reflection + glide reflection) pm 11 (translation + vertical line reflection) p 112 (translation + 180° rotation) pma 2 (translation + 180° rotation + vertical line reflection + glide reflection) pmm 2 (translation + 180° rotation + horizontal line reflection + vertical line reflection + glide reflection)

Glide reflection • Translation & reflection f(x, y)=f(x+λ, y) f(x, y)=f(x+λ/2, -y)

Wallpaper symmetry • Also ornamental symmetry • Translation in >1 directions + reflection + rotation • Fold number 1, 2, 3, 4, 6 • 17 groups: p 1, p 2, pm, pg, cm, pmg, pgg, cmm, p 4 m, p 4 g, p 3 m 1, p 31 m, p 6 m

Wallpaper symmetry

Wallpaper symmetry Example: p 6 m

Wallpaper symmetry Example: p 4 g Maurits Cornelis Escher: Angels and devils

Similarity Symmetry • Scaling & rotation • Scaling & translation

Similarity Symmetry Nautilus pompilius