Penrose TilingsPenrose Tilings

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

König’s IslandKönig’s Island

•On König’s Island people say only two words: “Ba” On König’s Island people say only two words: “Ba” and “Bu”.and “Bu”.

•Citizens don’t care what you talk Citizens don’t care what you talk aboutabout, as long as , as long as you say it politely. you say it politely.

Ba, BuBa, Bu, Bu

The Morse-Thue Rulesfor Polite Speech

1. The number of “bu’s” and “ba’s” in a polite speech can 1. The number of “bu’s” and “ba’s” in a polite speech can differ by no more than one.differ by no more than one.

2. In a polite speech, the 2. In a polite speech, the 2n2nthth word must be the opposite of word must be the opposite of the the nnthth word. word.

Examples of polite speeches:Examples of polite speeches:

bu.bu.

bu, ba, bu.bu, ba, bu.

bu, ba, ba, bu, bu.bu, ba, ba, bu, bu.

bu, ba, ba, bu, ba, bu, ba.bu, ba, ba, bu, ba, bu, ba.

bu, ba, ba, bu, ba, bu, bu, ba, bu.bu, ba, ba, bu, ba, bu, bu, ba, bu.

bu, ba, ba, bu, ba, bu, bu, ba, ba , bu, babu, ba, ba, bu, ba, bu, bu, ba, ba , bu, ba

Facts About Polite SpeechFacts About Polite SpeechThere are polite speeches of arbitrary length.There are polite speeches of arbitrary length.

((If you know how long you have to speak, you can fill the If you know how long you have to speak, you can fill the time politely, no matter how long it is.)time politely, no matter how long it is.)

Every initial segment of a polite speech is a polite speech.Every initial segment of a polite speech is a polite speech.

(Once you stick your foot in your mouth you can’t talk your (Once you stick your foot in your mouth you can’t talk your way out of it.)way out of it.)

König’s TheoremIt is possible to speak forever without offending anyoneIt is possible to speak forever without offending anyone

There is an infinite polite speech.There is an infinite polite speech.

OrOr

Proving König’s TheoremStep 1Step 1: Note that there must be infinitely many polite speeches. : Note that there must be infinitely many polite speeches.

Step 2Step 2: There must either be infinitely many polite speeches : There must either be infinitely many polite speeches beginning with “bu” or infinitely many beginning with “ba.” beginning with “bu” or infinitely many beginning with “ba.” Suppose it is “ba.”Suppose it is “ba.”

Step 3Step 3: There must either be infinitely many polite speeches : There must either be infinitely many polite speeches beginning with “ba, bu” or infinitely many beginning with beginning with “ba, bu” or infinitely many beginning with “ba, ba.”“ba, ba.”

Continuing inductivelyContinuing inductively we can construct an infinite polite we can construct an infinite polite speech.speech.

Morse-Thue Sequence

1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0,

Morse-Thue Sequence

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,

Morse-Thue Sequence

1, 0, 0, 1, 1, 0, 0, 1,

0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0 . . 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0 . .

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”1 = “bu”

0 = “ba”0 = “ba”

The Morse-Thue sequence is an infinite polite speech (under the Morse-Thue rules).

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 …

1

10 1 01 0

0 0 1 0 1 1 0 0 1 1 0

Morse-Thue sequence is self-similar under this block-renaming rule.

1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 …

Block renaming is a local rule on M-T.

How do we divide

…0 0 1 0 1 1… into blocks?

Only possible way: …0 0 1 0 1 1…

Start in the middle.

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!

Run that by me again…?

1, 0, 0, 1, . . .1, 0, 0, 1, . . .

½ P ½ P

P PP

• 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!

Subtleties1 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…

1 0 1 0 1 0

… 0 1 0 1 0 1 0 …

… 0 1 0 1 0 1 0 … … 0 1 0 1 0 1 0 …

Why Is “Locality” Important?1, 0, 0, 1, . . .1, 0, 0, 1, . . .

½ 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.

½ P

P PP

Oops? It’s OK, block renaming is a local rule!

1, 0, 0, 1, …1, 0, 0, 1, … ?, ?, 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 DartsPenrose Kites and Darts

Kites and Darts Tile the PlaneKites and Darts Tile the Plane

Penrose Tiling is AperiodicPenrose Tiling is Aperiodic

25 tiles

16 tiles

Penrose Tiling is AperiodicPenrose Tiling is Aperiodic