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wheels • on • wheels : how mathematics • draws • symmetrical • flowers

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

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wheels • on • wheels : how mathematics • draws • symmetrical • flowers

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  1. wheels•on•wheels:howmathematics•draws•symmetrical•flowers stefana.r.vutova penyo.m.michev patrons: john.rosenthal david.brown

  2. •introduction• •epicycles •definition: acircle the center of which moves on the circumference of a larger circle •parametricequation:

  3. •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

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

  5. •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

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

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

  8. •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 aand 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)

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

  10. •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

  11. •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 relation 3 mod(8) congruence

  12. •GCD symmetry•(qnot 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 tomthings 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

  13. •GCD symmetry•(qnot prime m) • if we have then a set of frequencies all congruent to 4mod(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)

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

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

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

  17. • 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. ifq is not prime to m, then the symmetry displayed is m/GCD(q,m)-fold

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

  19. •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

  20. •finding the greatest symmetry• • steps: given a set: (2,30,-54) largestm= 28 1.search for the largest possible m GCD(a,b,c,q,m) = 2 (2,30,-54)  (1,15,-27) m=28  m/GCD=14 2.search for GCD of a, b, c, qand m, divide by it q=1, m=14 prime congruence 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

  21. •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

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