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Governor’s School for the Sciences

Governor’s School for the Sciences. Mathematics. Day 12. MOTD: Pierre Fermat. 1601 to 1665 (France) Lawyer and Judge Worked in number theory Most famous for ‘Fermat’s Last Theorem’: x n + y n = z n only has integer solutions for n=2

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Governor’s School for the Sciences

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  1. Governor’s School for the Sciences Mathematics Day 12

  2. MOTD: Pierre Fermat • 1601 to 1665 (France) • Lawyer and Judge • Worked in number theory • Most famous for ‘Fermat’s Last Theorem’: xn + yn = zn only has integer solutions for n=2 • “I have discovered a truly remarkable proof which this margin is too small to contain”

  3. Tilings (Regular Patterns) • Given a tile and a collection of transformations, is it legal? i.e. does it produce a regular pattern • First try at an answer: Use the tile and transformations to construct some of the pattern; no conflicts means it may be legal • How can we construct a pattern?

  4. Follow A: T1A = B, T4B = C, T1C = D so T1T4T1A = D • Other possibilities: T4T1T1A = D T2T1T4T1T4A = D, and many more

  5. What did we learn? • There are many different ways to get from point to point • To be a tiling, all ways must result in the same transformation • To build a pattern you need to apply all combinations of the transformations • A pattern generator is like a MRCM!

  6. Pattern Generator • Start with the original tile M and a list of transformations {Ti} • Apply all the transformations to M, saving all the images (and M) • Repeat, applying all the transformations to the new set of tiles, adding the new images to the set of tiles • After N repetitions, every combination of N transformations will have been applied to the original tile M • (Like an MRCM, except save all the images)

  7. Labelling the 17 Patterns • Various ways; depends on background from crystallography or geometry • The basic idea is to encode the various transformations and possibly tile type • Table summarizes the results

  8. Examples • Web page: http://www2.spsu.edu/math/tile

  9. Lab Time • Explore program Kali • Try to determine all 12 patterns generated by a square tile using a modified MRCM program • Don’t forget your project description is due today

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