Penrose Tilings

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Penrose Tilings
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Infinite Polite Speeches, Königs Theorem,
Penrose Tilings and Aperiodicity
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Königs Island

• On Königs Island people say only two words
  Ba and Bu.

• Citizens dont care what you talk about, as long
  as you say it politely.

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The Morse-Thue Rulesfor 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 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.
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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.)
Königs Theorem
It is possible to speak forever without offending
anyone
Or
There is an infinite polite speech.
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Proving Königs 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.
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Morse-Thue Sequence
0, 1, 1, 0,
1, 0, 0, 1,
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Morse-Thue Sequence
0, 1, 1, 0,
0, 1, 1, 0, 1, 0, 0, 1,
1, 0, 0, 1,
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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).
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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.
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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
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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!

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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, . . .
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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
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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.

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Oops?
Its 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.
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Penrose Kites and Darts
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Kites and Darts Tile the Plane
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Penrose Tiling is Aperiodic
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Penrose Tiling is Aperiodic