An Introduction to Self-Similar and Combinatorial Tiling Part II
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Transcript of An Introduction to Self-Similar and Combinatorial Tiling Part II
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An Introduction to Self-Similar and Combinatorial
Tiling Part IIMoisés RiveraVassar College
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Self-Similar vs. Combinatorial Tiling
• Substitution/Inflate and Subdivide Rule
Self-Similar Tiling
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Self-Similar vs. Combinatorial Tiling
• Substitution/Inflate and Subdivide Rule• No geometric resemblance to itself • Substitution of non-constant length
Combinatorial Tiling Self-Similar Tiling
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One-Dimensional Symbolic Substitution
• A is a finite set called an alphabet whose elements are letters.
• A* is the set of all words with elements from A.
• A symbolic substitution is any map σ : A → A*.
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One-Dimensional Symbolic Substitution
• A is a finite set called an alphabet whose elements are letters.
• A* is the set of all words with elements from A.
• A symbolic substitution is any map σ : A → A*.
Let A = {a,b} and let σ(a) = ab and σ(b) = a.
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One-Dimensional Symbolic Substitution
• A is a finite set called an alphabet whose elements are letters.
• A* is the set of all words with elements from A.
• A symbolic substitution is any map σ : A → A*.
Let A = {a,b} and let σ(a) = ab and σ(b) = a.
If we begin with a we get:
a → ab → ab a → ab a ab → ab a ab ab a → ab a ab ab a ab a ab →···
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One-Dimensional Symbolic Substitution
• A is a finite set called an alphabet whose elements are letters.
• A* is the set of all words with elements from A.
• A symbolic substitution is any map σ : A → A*.
Let A = {a,b} and let σ(a) = ab and σ(b) = a.
If we begin with a we get:
a → ab → ab a → ab a ab → ab a ab ab a → ab a ab ab a ab a ab →···
• The block lengths are Fibonacci numbers 1, 2, 3, 5, 8, 13, …
• substitution of non-constant length or combinatorial substitution
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Substitution Matrix
• The substitution matrix M is the n×n matrix with entries given by
mij = the number of tiles of type i in the substitution of the tile of type j
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Substitution Matrix
• The substitution matrix M is the n×n matrix with entries given by
mij = the number of tiles of type i in the substitution of the tile of type j
σ(a) = ab σ(b) = a
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Substitution Matrix
• The substitution matrix M is the n×n matrix with entries given by
mij = the number of tiles of type i in the substitution of the tile of type j
σ(a) = ab σ(b) = a
• substitution matrix for one-dimensional Fibonacci substitution:
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Eigenvectors and Eigenvalues
• Eigenvector - a non-zero vector v that, when multiplied by square matrix A yields the original vector multiplied by a single number λ
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Eigenvectors and Eigenvalues
• Eigenvector - a non-zero vector v that, when multiplied by square matrix A yields the original vector multiplied by a single number λ
• λ is the eigenvalue of A corresponding to v.
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Eigenvectors and Eigenvalues
• Eigenvector - a non-zero vector v that, when multiplied by square matrix A yields the original vector multiplied by a single number λ
• λ is the eigenvalue of A corresponding to v.
• This equation has non-trivial solutions if and only if
•Solve for λ to find eigenvalues.
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Expansion Constant• Eigenvalues are the roots of the characteristic polynomial
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Expansion Constant• Eigenvalues are the roots of the characteristic polynomial
• Perron Eigenvalue - largest positive real valued eigenvalue that is larger in modulus than the other eigenvalues of the matrix
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Expansion Constant• Eigenvalues are the roots of the characteristic polynomial
• Perron Eigenvalue - largest positive real valued eigenvalue that is larger in modulus than the other eigenvalues of the matrix
• Perron Eigenvalue of Fibonacci substitution matrix:
the golden mean
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The Fibonacci Direct Product Substitution
The direct product of the one-dimensional Fibonacci substitution with itself.
{a,b}x{a,b}
where (a,a) = 1, (a,b) = 2, (b,a) = 3, (b,b) = 4.
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The Fibonacci Direct Product Substitution
The direct product of the one-dimensional Fibonacci substitution with itself.
{a,b}x{a,b}
where (a,a) = 1, (a,b) = 2, (b,a) = 3, (b,b) = 4.
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The Fibonacci Direct Product Substitution
The direct product of the one-dimensional Fibonacci substitution with itself.
{a,b}x{a,b}
where (a,a) = 1, (a,b) = 2, (b,a) = 3, (b,b) = 4.
• Not self-similar• Not an inflate and subdivide rule
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Several Iterations of Tile Types
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Several Iterations of Tile Types
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Substitution Matrix
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Substitution Matrix
The substitution matrix for the Fibonacci Direct Product is
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Expansion Constant• Eigenvalues are the roots of the characteristic polynomial
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• Perron Eigenvalue of Fibonacci Direct Product substitution matrix:
the golden mean squared
Expansion Constant• Eigenvalues are the roots of the characteristic polynomial
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Replace-and-rescale Method
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Replace-and-rescale Method
Rescale volumes by the Perron Eigenvalue raised to the nth power: 1/γ2n where n corresponds to the nth-level of our block.
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Replace-and-rescale Method
Rescale volumes by the Perron Eigenvalue raised to the nth power: 1/γ2n where n corresponds to the nth-level of our block.
This results in a level-0 tile with different lengths that the original. Repeat for other tile types. The substitution rule is now self-similar.
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Self-Similar Fibonacci Direct Product
Combinatorial Substitution Rule
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Self-Similar Fibonacci Direct Product
Combinatorial Substitution Rule
Self-Similar Inflate and Subdivide Rule
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Combinatorial Tiling
Replace-and-rescale MethodCompare level-5 tiles of the Fibonacci Direct Product (left)
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Combinatorial Tiling Self-Similar Tiling
Replace-and-rescale MethodCompare level-5 tiles of the Fibonacci Direct Product (left) and the self-similar tiling (right).
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Replace-and-rescale Method
• Not known whether replace-and-rescale method always works
• Replace-and-rescale method works in all known examples
Self-Similar Fibonacci DPV
Self-Similar non-Pisot DPV
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References
• N.P. Frank, A primer of substitution tilings of the Euclidean plane, Expositiones Mathematicae, 26 (2008) 4, 295-386
Further Readings• R. Kenyon, B. Solomyak, On the Characterization of
Expansion Maps for Self-Affine Tiling, Discrete Comput Geom, 43 (2010), 577-593