SPS Lecture: Symmetry in Art and Biology sublime creations MIND ART NATURE BIOLOGY perceptual dualities

Rotational/Reflection Symmetry Algae, starfish, sand stars: Hanh, p. 39, 168, 168. Bowl: Amratia, BC, Hahn, p. 3

Translational Symmetry Wristband: Mesin, Ukrain, ca B. C., may show the dentinal structure of mammoth tusks, Hahn, p. 3 Honeycomb: exhibiting hexagonal tiling, Hahn, p. 70

Planar Tiles, Devlin, 1997, p. 164

The Moors used all 17 Wallpaper Patterns, first discovered by the ancient Egyptians, to decorate the Alhambra in Granada, Spain

Rigid Transformations n RigidTransformations T : Plane  Plane such that distance(T(a),T(b)) = distance(a,b), a,b in Plane n Transform every figure X subset Plane into a congruent figure T(X) = { T(p) | p in X } n Form a group with multiplication defined by composition ToS(a) =T(S(a)), a group is closed under multiplication, contains the identity I, contains the inverse of each element, and multiplication is associative: (ToS)oU = To(SoU) n Is a continuous group (or Lie group): can be parameterized (locally) by real numbers, this can be seen since every rigid transformation can be specified by a rotation about a specified point followed by a translation

Symmetry Groups n X is invariant under T if T(X) = X n The set of transformations under which a figure X is invariant forms the symmetry group of X n Sym(general figure) = { I } n Sym(isoceles triangle a,b,c; d(a,b) not = d(b,c)) = { I, A } where A is the flip transformation A(a) = a, A(b) = c, A(c) = b n Sym(equilateral triangle) = { I, R, RoR, A, B, C } where R is rotation by 120 degrees and A, B, C are flips such that A(a) = a, B(b) = b, C(c) = c n Sym(circle) = { R_angle | addition modulo 360 degrees} U {F_angle | addition modulo 180 degrees }

Wallpaper Patterns n F : Plane  {red, yellow, green, blue} n Figures X_red = { p | F(p) = red}, X_yellow, etc. n Sym(F) = { T | T(X_c} = X_c } for every color c n Lattice group: generated by two linearly independent translation vectors n Admissible colorings: symmetry groups are discrete and contain a lattice subgroup n Two colorings are equivalent if they have the same symmetry groups. n Wallpaper pattern: equivalence class of admissible planar colorings

Plato, Devlin, 1997, p

Spatial Tiles, Devlin, 1997, p. 165 Pomegranate seeds grow to rhombic dodecahedrons (green) from spheres in a bicubic lattice There are 230 three-dimensional ‘wallpaper patterns’ classified by crystallographers in the 19-th century

Golden Ratio It appears in natures angles, e.g. plantains, pinecones, peacocks and snails, and Nautilus: Hahn, p. 175, 454, 455, 456. Its remarkable number theoretic properties give it optimal circle partitioning properties, Lawton 1.

Growth Processes of a population of immortal cells that divide at the end of each calendar year is described by iterating a linear transformation # cells at least one year old# cells Fibbonaci sequence, whose ratio of successive terms approaches the golden ratio

Projection, Devlin, 1997, p. 130

Durer, Devlin, 1997, p. 168

Desargues’ Theorem, Devlin, 1997, p

Cross Ratio abcd Preserved under projective transformations Implied by Desargues’ Theorem Symmetry is group if linear fractional transformations related to string theories

Peron Frobenius Theorem Nonnegative matrices (mild assumptions) have unique positive eigenvectors Implied by cross ratio invariance Implies that general growth processes yield symmetries (stationary equilibria) Similar limits of spatial growth patterns yield fractal tiles with amazing properties, Lawton 2 More general transformation groups (conformal) are related to biological growth, Thompson

Penrose Aperiodic Tiling Exhibits five fold local rotational symmetry, ratio of fat to thin tiles approaches the Golden Ratio, Devlin, p. 168

References Devlin, Keith, Mathematics: The Science of Patterns, Scientific American Library, Division of HPHLP, New York, 1997 Grossman, Israel and Magnus, Wilhelm, Groups and their Graphs, The Mathematical Association of America, 1992 Hahn, Werner, Symmetry as a Developmental Principle in Nature and Art, World Scientific, Singapore, 1998 Lawton, Wayne, Kronecker’s theorem and rational approximation of algebraic, The Fibonacci Quarterly, volume 21, number 2, pages , May 1983 Lawton, Wayne and Resnikoff, Howard, Fractal tiling for multiple mirror telescopes, U. S. Patent 4,904,073, 27, February 1990 Thompson, D’Arcy Wentworth, Growth and Form, Vol. I and II, Cambridge University Press, Cambridge, 1952