[PPT] - Embedding computations in tilings (Part 1: fixed point tilings) PowerPoint Presentation

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Embedding computations in tilings (Part 1: fixed point tilings) Andrei Romashchenko 31 May 2016

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What is a tile?

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What is a tile? In this mini-course: Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·}

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What is a tile? In this mini-course: Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·} Wang Tile: a unit square with colored sides.

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What is a tile? In this mini-course: Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·} Wang Tile: a unit square with colored sides. i.e, an element of C 4, e.g.,

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What is a tile? In this mini-course: Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·} Wang Tile: a unit square with colored sides. i.e, an element of C 4, e.g., Tile set: a set τ ⊂ C 4

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What is a tile? In this mini-course: Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·} Wang Tile: a unit square with colored sides. i.e, an element of C 4, e.g., Tile set: a set τ ⊂ C 4 Tiling: a mapping f : Z2 → τ that respects the matching rules

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Tiling: a mapping f : Z2 → τ such that f (i, j).right = f (i + 1, j).left, e.g., + f (i, j).top = f (i, j + 1).bottom, e.g., +

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Tiling: a mapping f : Z2 → τ such that f (i, j).right = f (i + 1, j).left, e.g., + f (i, j).top = f (i, j + 1).bottom, e.g., +

Example. A finite pattern from a valid tiling:

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τ-tiling is a mapping f : Z2 → τ that respects the local rules.

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τ-tiling is a mapping f : Z2 → τ that respects the local rules. T ∈ Z2 is a period if f (x + T) = f (x) for all x.

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τ-tiling is a mapping f : Z2 → τ that respects the local rules. T ∈ Z2 is a period if f (x + T) = f (x) for all x.

Theorem. There exists a tile set τ such that

(i) τ-tilings exist, and

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τ-tiling is a mapping f : Z2 → τ that respects the local rules. T ∈ Z2 is a period if f (x + T) = f (x) for all x.

Theorem. There exists a tile set τ such that

(i) τ-tilings exist, and (ii) all τ-tilings are aperiodic.

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A construction of an aperiodic tile set:

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A construction of an aperiodic tile set:

◮ define self-similar tile sets

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A construction of an aperiodic tile set:

◮ define self-similar tile sets ◮ observe that every self-similar tile set is aperiodic

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A construction of an aperiodic tile set:

◮ define self-similar tile sets ◮ observe that every self-similar tile set is aperiodic ◮ construct some self-similar tile set

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Macro-tile:

Macro-color Macro-color Macro-color Macro-color N N

an N × N square made of matching τ-tiles

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Fix a tile set τ and an integer N > 1.

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Fix a tile set τ and an integer N > 1. Definition 1. A τ-macro-tile: an N × N square made of matching τ-tiles.

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Fix a tile set τ and an integer N > 1. Definition 1. A τ-macro-tile: an N × N square made of matching τ-tiles. Definition 2. A tile set ρ is simulated by τ: there exists a family of τ-macro-tiles R such that

◮ R is isomorphic to ρ, and ◮ every τ-tiling can be uniquely split by an N × N grid into

macro-tiles from R.

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Example. A tile set ρ: Trivial tile set (only one color) ✭✐ ✰ ✶❀ ❥✮ ✭✐❀ ❥✮ ✭✐❀ ❥✮ ✭✐❀ ❥ ✰ ✶✮

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Example. A tile set ρ: Trivial tile set (only one color) A tile set τ: A tile set that simulates a trivial tile set ρ ✭✐ ✰ ✶❀ ❥✮ ✭✐❀ ❥✮ ✭✐❀ ❥✮ ✭✐❀ ❥ ✰ ✶✮

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Example. A tile set ρ: Trivial tile set (only one color) A tile set τ: A tile set that simulates a trivial tile set ρ ✭✐ ✰ ✶❀ ❥✮ ✭✐❀ ❥✮ ✭✐❀ ❥✮ ✭✐❀ ❥ ✰ ✶✮

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✭✵❀ ✵✮ ✭✵❀ ✵✮ ✭✵❀ ◆ − ✶✮ ✭✵❀ ✵✮ ✭◆ − ✶❀ ✵✮ ✭✵❀ ✵✮ ✭✵❀ ◆ − ✶✮ ✭◆ − ✶❀ ✵✮ ◆ 9 / 18

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Self-similar tile set: a tile set that simulates a set of macrotiles isomorphic to itself.

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Self-similar tile set: a tile set that simulates a set of macrotiles isomorphic to itself.

Proposition. Self-similar tile set is aperiodic.

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Self-similar tile set: a tile set that simulates a set of macrotiles isomorphic to itself.

Proposition. Self-similar tile set is aperiodic.

Sketch of the proof:

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Self-similar tile set: a tile set that simulates a set of macrotiles isomorphic to itself.

Proposition. Self-similar tile set is aperiodic.

Sketch of the proof:

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Self-similar tile set: a tile set that simulates a set of macrotiles isomorphic to itself.

Proposition. Self-similar tile set is aperiodic.

Sketch of the proof:

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Simulating a given tile set ρ by macro-tiles.

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Simulating a given tile set ρ by macro-tiles. Representation of the tile set ρ:

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Simulating a given tile set ρ by macro-tiles. Representation of the tile set ρ:

colors of a tile set ρ

= ⇒ k-bits strings

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Simulating a given tile set ρ by macro-tiles. Representation of the tile set ρ:

colors of a tile set ρ

= ⇒ k-bits strings

a tile set ρ

= ⇒ a predicate P(x1, x2, x3, x4)

n 4-tuples of colors

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Simulating a given tile set ρ by macro-tiles. Representation of the tile set ρ:

colors of a tile set ρ

= ⇒ k-bits strings

a tile set ρ

= ⇒ a predicate P(x1, x2, x3, x4)

n 4-tuples of colors
a TM that accepts
nly 4-tuples of colors

for the ρ-tiles

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The scheme of implementation:

Turing machine

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