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 “I have discovered a truly remarkable proof which this margin is too small to contain”

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?

Follow A: T 1 A = B, T 4 B = C, T 1 C = D so T 1 T 4 T 1 A = D Other possibilities: T 4 T 1 T 1 A = D T 2 T 1 T 4 T 1 T 4 A = D, and many more

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 transformationTo 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!A pattern generator is like a MRCM!

Pattern Generator Start with the original tile M and a list of transformations {T i } 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)

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

p1oP-gramTranslations only p22222P-gram4 180 o pm**Rect2 par reflections pmm*2222Recth,v reflections pgxxRectGlide reflection pgg22xRectGlide o pmg22*Rect180 o,refl | g-refl cmx*RhomRefl, g-refl cmm2*22Rhom180 o,refl + g-refl

p4442Square4 90 o p4m*442Square4 90 o on g-refl, refl p4g4*2Square4 90 o ~on g-refl, refl p3333Eq tri3 120 o p3m1*333Eq tri3 120 o on g-refl, refl p31m3*3Eq tri3 120 o ~on g-refl, refl p6632Eq tri6 60 o p6m*632Eq tri6 60 o, g-refl, refl

Examples Web page:

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 todayDon’t forget your project description is due today