States Describe properties of physical (biological) systems Ecology: equilibrium populations in a biom Physics: energy (kinetic+potential) of an object Geometry: shape (congruence) of a figure

Transformations Ecology individuals die and are born Physics objects move Geometry figures are rotated & reflected are changes that happen to systems

SYMMETRIES are transformations that preserve states Ecology: evolution preserves equilibrium populations Physics: falling preserves energy Geometry: rotations & reflections preserve shape Consider the following quote where operation means transformation, thing means system, and appearance means state:

What is Symmetry? “A thing is symmetrical if one can subject it to a certain operation and it appears exactly the same as before.” …Herman Weyl An operational definition: Herman Weyl ( )

Rotation & Reflection Symmetries are geometric transformations that preserve the shape of regular m-gons (although they may move vertices and edges) admits 3 rotations and 3 reflections How many transformations does a pentagon have ? Construct a figure with only two symmetries ? admits 4 rotations and 4 reflections

Permutations can be used to describe transformations a Rotation (clockwise by 120 degrees) bc Reflection about vertical line through the vertex a Problem: Compute the other four permutations

Composition of transformations gives new transformations we observe that

Multiplication Table Problem: complete the following table

The Language of Symmetry Group Theory: Let G be a set of elements G={a,b,c, …}. Then the following axioms endow G with a group structure: (i) For any two elements a, b in G, ab and ba are also in G (Closure) (ii) Composition is associative: a(bc) =(ab)c (iii) G contains the identity e, with ae=ea=a for all a in G. (iv) For every a in G, it’s inverse exist:

Associativity Composition of transformations is associative because transformations are functions and composition of functions is always associative Problem: use the table to show that the set of symmetries of an equilateral triangle with composition forms a group Problem: use the associative property to prove that if A and B are m x m matrices and AB = identity matrix then B is the inverse of A

Matrix Groups Problem: for any integer m > 0 show that the set of m x m matrices with nonzero determinant forms a group under matrix multiplication and describe it as a group of transformations Problem: show that the following set of 3 x 3 matrices forms a subgroup of this group

Homomorphisms and Representations Definition: A homomorphism of a group G to a group H is a function f : G  H that satisfies f(ab) = f(a)f(b) whenever a and b are elements in G. If H is a group of m x m matrices it is called an m- dimensional representation of the group G. Problem: Show that the set of integers Z is a group under addition and that f is a representation of Z Problem: Construct 2 and 3 dimensional representations of the group of symmetries of an equilateral triangle

References about Groups and Symmetries Rosen, Joe, Symmetry Discovered, Concepts and Applications in Nature and Science, Dover, New York, 1998 Sternberg, Shlomo, Group Theory and Physics, Cambridge University Press, 1997 Grossman, Israel and Magnus, Wilhelm, Groups and their Graphs, New Mathematical Library published by The Mathematical Association of America, 1992

Growth as Transformations Consider an initial rectangle, with sides of lengths a < b, that is repeatedly transformed into a rectangle with sides of lengths b < a+b Sequences of such transformations characterize growth – of biological populations and snowflakes This transformation changes the shape – are any properties (states) ever preserved ? Yes - Golden Rectangles are preserved !

Transformation Geometry

Transformation Algebra Represent the lengths of the sides of a rectangle by a column ‘vector’ The geometric transformation induces an algebraic transformation Problem: construct the 2 x 2 matrix that represents this transformation

Fibonacci Sequence Leonardo of Pisa ( ), commonly called Fibonacci, proposed the following problem: A pair of rabbits is bought into a confined place. This pair and every other pair, begets one new pair in a month, starting in their second year. How many pairs will there be after one, two, …, months, assuming that no deaths occur ? Direct Solution 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … Transformation Solution

Phi = Golden Ratio The transformed rectangle is similar to the original if and only if Problem: Compute the other root of this Golden Ratio equation

Eigenvectors, Eigenvalues and the Golden Ratio We observe that and using the fact thatis also a solution of the Golden Equation are eigenvectors of the transformation since the transformation simply multiplies them by eigenvalues.

Invariant Subspaces The set of column vectors and are 1-dimensional vector subspaces that are invariant under since and the sets is a 2-dim vector space and Problem Show that every vector incan be expressed uniquely as a sumwith Problem Show that the k-fold transformation satisfies

Geometry of Invariant Subspaces subspace Problem: Show that the eigenspaces (lines) for T are orthogonal, this follows necessarily since the matrix that represents T is symmetric

Fundamental Theorem of Growth Theorem Every rectangle grows, through successive transformations by T, into a Golden Rectangle. Proof. This follows from Corollary 1 since as k increases and this vector represents a Golden Rectangle. Problem: Show that the ratios of line segments in a Pentacle (Brown page 101) all equal PHI (inscribe the Pentacle in a unit circle in the complex plane so the points are 5-th roots of unity and express the intersections of lines as convex combinations) Problem: How does the Golden Ratio describe the shape of Nautilus shells, the placement of leaves on Calamansi plants, and the shape of the humans? Pentacle

References about the Fibonacci Sequence, Golden Ratio and Growth 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 numbers, The Fibonacci Quarterly, volume 21, number 2, pages , May 1983 Thompson, D’Arcy Wentworth, Growth and Form, Vol. I and II, Cambridge University Press, Cambridge, 1952 Brown, Dan, The Da Vinci Code (special illustrated edition), Doubleday, New York, 2004 Afrizah Bte Mohd Hassan, Tan Chun Hsuan Joyce, Jenny Angggraini Njo, Kwok Kah Peck, Chia Ling Ling, Welcome Fibonacci Numbers, NUS Science Foundation Project Report

Tiling and Crystals tilings provide patterns used by artists and arise in tissues and beehives have discrete symmetry groups, consisting of translations, rotations, and reflections, that transform the (infinitely extended) tiling or crystal into itself crystals are common in nature and are synthesized to determine structures of compounds and proteins

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

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

Lattices The lattice spanned by linearly independent vectors u and v forms a group under the operations of vector addition.

Translation Symmetry Groups A latticeacts as a group of translations of the vector space as follows: fordefine the transformation by They constitute the translations in the full symmetry groups of a tile or a crystal.

Reciprocal Lattice Definition: The reciprocal lattice consists of such that Theorem 3. Ifis spanned by the column vectors of a matrix of a lattice all vectors w in is spanned by the columns of the transposed inverse of Problem: Show this using a direct computation.

Fundamental Theorem of X-Ray Crystallography Theorem If a crystal has translational symmetry group and if the crystal is illuminated with X-rays having wavelength from an incident direction described by a unit vector u, then the reflected X-rays ONLY occur in directions described by unit vectors v that satisfy the condition (we observe that in physics the lattice L has units of length and the reciprocal lattice has units of inverse length also called spatial frequency, the vector on the left is called the scattering vector) Proof. See the discussion on Bragg scattering in Jenkins and White, Fundamentals of Optics, or any other serious physics book.

Representations of the Symmetric Groups The symmetric groupis the group of permutations on n-objects Example the 1-dim trivial representation A m-dimensional representation of a group describes the group as a group of transformations on the vector space Example the 1-dim parity representation Example the n-dimensional representation by n x n permutation matrices

Irreducible Representations of two 1-dim, trivial and parity one 2-dim, matrices that rotate and reflect an equilateral triangle two 1-dim, trivial and parity two 3-dim matrices that rotate a cube (permute four diagonals) matrices that rotate and reflect a regular tetradedron one 2-dim given by composition where h is determined by permutation of the three cubic axes Theorem These are all since

Decomposition into Irreducible Representations Example the 3-dimensional permutation representation of decomposes into the sum of the 1-dimensional trivial irreducible representation and the unique 2-dimensional irreducible representation Example Carbon tetrachloride molecules are invariant under rotations and reflections – described by the tetradedron representation of and their displacement configuration states are described by a 15- dimensional representation that decomposes (Sternberg page 103) into the sum of one 1-dim irreducible representations one 2-dim irreducible representation one 3-dim irreducible cubic representation three copies of the 3-dim irreducible tetrahedral representation Vibrational eigenvalues are constrained (6 with multiplicities 1,2,3,3,3,3) and the symmetries can therefore be observed by spectroscopy. Problem Prove this assuming that vib. eig. are eig of a symmetric matrix H and symmetry means that p(g)H = Hp(g) for all g in Group

Lie Groups and their Representations In addition to the discrete groups there are groups, such as the group of ALL translations of space or ALL 3 x 3 matrices with nonzero determinant, that are described by continuous parameters, they are called Lie groups and they describe important symmetries of particles and fields in physics and even describe symmetries of differential equations that describe fluid flow and biological growth Irreducible representations of Lie groups determine physical observables, such as energy, mass, charge and more exotic quantities, in the same way that irreducible representations of discrete groups describe the vibrational eigenvalues (spectra) of molecules The irreducible representations of the most important Lie groups, the unitary groups, are exactly determined by the irreducible representations of the symmetric groups that we have discussed, this amazing result was discovered by Isaac Schur and later rediscovered by Herman Weyl – the father of symmetry