penrose tilings. infinite polite speeches, könig’s theorem, penrose tilings and aperiodicity

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

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Penrose Tilings Infinite Polite Speeches, Knigs Theorem, Penrose Tilings and Aperiodicity Knigs Island On Knigs Island people say only two words: Ba and Bu.On Knigs Island people say only two words: Ba and Bu. Citizens dont care what you talk about, as long as you say it politely.Citizens dont care what you talk about, as long as you say it politely. Ba, Bu Ba, Bu, Bu The Morse-Thue Rules for Polite Speech 1. The number of bus and bas in a polite speech can differ by no more than one. 2. In a polite speech, the 2n th word must be the opposite of the n th word. Examples of polite speeches: bu. bu, ba, bu. bu, ba, ba, bu, bu. bu, ba, ba, bu, ba, bu, ba. bu, ba, ba, bu, ba, bu, bu, ba, bu. bu, ba, ba, bu, ba, bu, bu, ba, ba, bu, ba 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 cant talk your way out of it.) Knigs Theorem It is possible to speak forever without offending anyone There is an infinite polite speech. Or Proving Knigs 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 inductively we can construct an infinite polite speech. Morse-Thue Sequence 1, 0, 0, 1, 0, 1, 1, 0, Morse-Thue Sequence 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, Morse-Thue Sequence 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0.. 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1 = bu 0 = ba The Morse-Thue sequence is an infinite polite speech (under the Morse-Thue rules). Self-similarity in M-T Morse-Thue sequence is self-similar under this block- renaming rule. Block renaming is a local rule on M-T. How do we divide into blocks? Only possible way: Start in the middle. M-T is aperiodic 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! Run that by me again? 1, 0, 0, 1,... P PPP 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! Subtleties 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 Why Is Locality Important? 1, 0, 0, 1,... P Assumptions we made: P is even Break occurs between blocks---we can neatly shrink each individual block inito a block of half the size. P PPP Oops? Its OK, block renaming is a local rule! 1, 0, 0, 1, ?, ?, 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. Penrose Kites and Darts Kites and Darts Tile the Plane Penrose Tiling is Aperiodic 25 tiles 16 tiles Penrose Tiling is Aperiodic