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Governor’s School for the Sciences

Dive into the mathematical journey initiated by Pierre Fermat, the renowned lawyer and judge from France, whose work in number theory culminated in the famous Fermat's Last Theorem. This session explores the legal aspects of tiling patterns—uses transformations of a tile to create regular patterns. We discuss methods to generate patterns by applying combinations of transformations, analyzing results, and understanding the necessity for consistency in transformations for tiling legality. Learn about practical applications and tools like the Kali program for constructing and categorizing patterns.

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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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