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Penrose Tilings

Penrose Tilings. Infinite Polite Speeches, K ö nig’s Theorem, Penrose Tilings and Aperiodicity. Ba, Bu. Ba, Bu, Bu. König’s Island. On K ö nig’s Island people say only two words: “Ba” and “Bu”. Citizens don’t care what you talk about , as long as you say it politely.

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Penrose Tilings

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  1. Penrose Tilings

  2. Infinite Polite Speeches, König’s Theorem, Penrose Tilings and Aperiodicity

  3. Ba, Bu Ba, Bu, Bu König’s Island • On König’s Island people say only two words: “Ba” and “Bu”. • Citizens don’t care what you talk about, as long as you say it politely.

  4. The Morse-Thue Rulesfor Polite Speech 1. The number of “bu’s” and “ba’s” in a polite speech can differ by no more than one. 2. In a polite speech, the 2nth word must be the opposite of the nth word. Examples of polite speeches: bu. bu, ba, ba, bu, ba, bu, bu, ba, bu. bu, ba, ba, bu, ba, bu, bu, ba, ba , bu, ba bu, ba, bu. bu, ba, ba, bu, bu. bu, ba, ba, bu, ba, bu, ba.

  5. Facts About Polite Speech There are polite speeches of arbitrary length. (If you know how long you have to speak, you can fill the time politely, no matter how long it is.) Every initial segment of a polite speech is a polite speech. (Once you stick your foot in your mouth you can’t talk your way out of it.) König’s Theorem It is possible to speak forever without offending anyone Or There is an infinite polite speech.

  6. Proving König’s Theorem Step 1: Note that there must be infinitely many polite speeches. Step 2: There must either be infinitely many polite speeches beginning with “bu” or infinitely many beginning with “ba.” Suppose it is “ba.” Step 3: There must either be infinitely many polite speeches beginning with “ba, bu” or infinitely many beginning with “ba, ba.” Continuing inductivelywe can construct an infinite polite speech.

  7. Morse-Thue Sequence 0, 1, 1, 0, 1, 0, 0, 1,

  8. Morse-Thue Sequence 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1,

  9. 1 = “bu” 0 = “ba” Morse-Thue Sequence 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0 . . The Morse-Thue sequence is an infinite polite speech (under the Morse-Thue rules).

  10. 01 0 10 1 1 0 0 1 1 0 0 0 0 1 1 1 Self-similarity in M-T 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 … Morse-Thue sequence is self-similar under this block-renaming rule.

  11. Block renaming is a local rule on M-T. 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 … Start in the middle. • How do we divide • …0 0 1 0 1 1… into blocks? • Only possible way: …0 0 1 0 1 1…

  12. M-T is aperiodic 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 … • Suppose M-T is periodic with (shortest) period P. • The block-renamed sequence would have to repeat after exactly P/2 terms. • But block-renamed sequence is M-T!

  13. P P P ½ P ½ P Run that by me again…? • Suppose M-T is periodic with (shortest) period P. • The block-renamed sequence would have to repeat after exactly P/2 terms. • But block-renamed sequence is M-T! 1, 0, 0, 1, . . .

  14. 1 0 1 0 1 0 Subtleties 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 … It seems the argument we just gave might prove that 1,0,1,0, . . . is aperiodic! (Huh?) Unlike the previous, this block-renaming rule is not local… … 0 1 0 1 0 1 0 … … 0 1 0 1 0 1 0 … … 0 1 0 1 0 1 0 …

  15. P P P ½ P ½ P Why Is “Locality” Important? 1, 0, 0, 1, . . . • Assumptions we made: • P is even • Break occurs between blocks---we can neatly “shrink” each individual block inito a block of half the size.

  16. 1, 0, 0, 1, … Oops? It’s OK, block renaming is a local rule! ?, ?, 1, 0, 0, 1, . . Since block renaming is local, the string at beginning of the second block must be divided up in precisely the same way as the string in the first block.

  17. Penrose Kites and Darts

  18. Kites and Darts Tile the Plane

  19. Penrose Tiling is Aperiodic

  20. 25 tiles

  21. 16 tiles

  22. Penrose Tiling is Aperiodic

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