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: )sin(),cos()sin(),cos()sin(),cos()( ctctCbtbtBatatAtf

•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•

qbqmbnitnAtf

CeBeAe

tiytx

ctctCbtbtBatatAtf

jjjj

jj

ictibtiat

, and ; re whe, exp)(

)()(

)sin(),cos()sin(),cos()sin(),cos()(

1

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

•q prime to m symmetry•

m

iqtf

m

iqibitqmbA

mtiqmbA

mtinA

mtf

jjj

jj

jj

2exp

2exp2expexp

2exp

2exp

2

• behavior of the parametric equation when we increase time by

the termrepresents an angle by which the function rotates

m

iq 2exp

m

2

•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 qth 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 relation3 mod(8) congruence

•GCD symmetry•(q not prime m)

l

iptf

m

iqtf

mtf

mqGCDccpq

2exp

2exp

2

, where clm and :set can we

• 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:

m

2

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

m

iq 2exp

•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 relation4 mod(24)

•k-multiplication symmetry•

kiskqmkbAsg

qbqmbnitnAtf

jjj

jjjj

jj

re whe, exp:)(

, and ; re whe, exp)(

1

1

• behavior of the function when all frequencies are multiplied by some integer k

•k-multiplication symmetry•

k

stt 0

• both functions produce the same curves

itqmbA

itqmbAtf

jj

jj

exp

exp 0

itqmbA

itk

kqmkbA

iskqmkbAsg

jj

jj

jj

exp

exp

exp

0

• 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) congruencestatement 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 patternof constructive and destructive interference amongst incident and reflected waves thattravel through it. These standing wave patterns represent the lowest energy vibrationmodes of an object, that is they are favored because they result in highest amplitudeoutput 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=14q=1, m=14

prime congruence

•conclusions•

• 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