wheels on wheels: how mathematics draws symmetrical flowers stefana.r.vutova penyo.m.michev patrons: john.rosenthal david.brown

introduction epicycles definition: a circle the center of which moves on the circumference of a larger circle parametric equation:

parameters relative ratio of the radii of the circles relative rate at which each one rotates relative direction in which they rotate phase differences between each rotation, that is the relative initial starting positions

changing the parameters changing the relative size of the radii of the circles: does not change symmetry

changing the parameters all rotating in the same direction (1,7,13) (1,7,-11) does not change symmetry changing the direction of one circle

adding a phase : changing the parameters does not change symmetry opening the curves by using phase changes is a way of seeing features that are otherwise hidden from view

changing the parameters relative rate of rotation (frequency) (5, 17, 31) (11, 25, 43) (irrational ratio) changes symmetry

modular arithmetic and symmetry definition: a system of arithmetic for integers where numbers “wrap around” after they reach a certain value – the modulus. two integers a and b are said to be congruent modulo m if their difference (a-b) is an integer multiple of m. This is expressed mathematically as: a ≡ b (mod m)

complex notation adopting complex notation is the key to unfolding symmetry in epicycle curves:

q prime to m symmetry behavior of the parametric equation when we increase time by the term represents an angle by which the function rotates

q prime to m symmetry what this means: if we divide the cartesian plane into m sectors, then the function will trace a certain pattern every q th sector, and if q is prime to m, then eventually all m sectors will be filled and the function will produce m-fold symmetry if we pick frequencies (3,11,-21) with congruence relation 3 mod(8) congruence

GCD symmetry (q not prime m) if we again look at the behavior of our parametric function as time is advanced by, we see that when q is not relatively prime to m things change: the term is no longer in reduced form, which means that the curve will trace its pattern in less than m sectors, or in other words the angle by which it advances is increased, in effect reducing the symmetry

GCD symmetry (q not prime m) if we have then a set of frequencies all congruent to 4 mod(24) we will not see 24-fold symmetry, but rather 24/GCD(4, 24) = 6-fold: frequencies (4,28,-52) with congruence relation 4 mod(24)

k-multiplication symmetry behavior of the function when all frequencies are multiplied by some integer k

k-multiplication symmetry both functions produce the same curves g(s) requires k-times less time to trace out the particular pattern introducing a new variable

k-multiplication symmetry original set: k-multiplied set: does not change the symmetry (1,15,-27) (2,30,-54)

restating conjectures 1. frequencies all congruent to q mod(m) where q is relatively prime to m produce m-fold symmetry 2. multiplying a set of frequencies does not change the symmetry 3. if q is not prime to m, then the symmetry displayed is m/GCD(q, m)-fold

conflict ? the contradiction: choose 2 mod(14) congruence statement 2 claims: statement 3 claims: same congruence, different symmetry (2,16,-26) (2,30,-54)

standing wave analogy what is a standing wave: A standing wave is a pattern of constructive and destructive interference amongst incident and reflected waves that travel through it. These standing wave patterns represent the lowest energy vibration modes of an object, that is they are favored because they result in highest amplitude output for least amount of energy. all harmonic frequencies are integer multiples of the fundamental

finding the greatest symmetry steps: 1. search for the largest possible m 2. search for GCD of a, b, c, q and m, divide by it 3. search for GCD of q and m, divide by it 4. applying steps 1-3 will produce the m which will determine the symmetry displayed by a given set of coefficients given a set: (2,30,-54) largest m = 28 GCD(a,b,c,q,m) = 2 (2,30,-54)  (1,15,-27) m=28  m/GCD=14 q=1, m=14 prime congruence

c onclusions so far: rigorous mathematical proof of GCD symmetry future work: finding a mathematical proof for the steps required to find the actual symmetry given a set of coefficients