Perron Number Tiling Systems
Author
Roger Bagula
Title
Perron Number Tiling Systems
Description
Four Programs for calculating Dr. Richard Kenyon's method for plane tilings from Perron numbers by substitutions. The construction of self-similar tilings , Geom. and Func. Analysis 6,(1996):417-488. Thurston showed that the expansion constant of a self-similar tiling of the plane must be a complex Perron number (algebraic integer strictly larger in modulus than its Galois conjugates except for its complex conjugate). Here we prove that, conversely, for every complex Perron number there exists a self-similar tiling. We also classify the expansion constants for self-similar tilings which have a rotational symmetry of order n.
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http://www.notebookarchive.org/2018-10-10ql6gy/
DOI
https://notebookarchive.org/2018-10-10ql6gy
Date Added
2018-10-02
Date Last Modified
2018-10-02
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This notebook has not been updated since 2006. 
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A Translation of
"The Construction of Self-Similar Tiles",
Richard Kenyon, into Mathematica
Submitted to Math Source by Roger L. Bagula 30 April 2005
A Translation of
"The Construction of Self-Similar Tiles",
Richard Kenyon, into Mathematica
Submitted to Math Source by Roger L. Bagula 30 April 2005
"The Construction of Self-Similar Tiles",
Richard Kenyon, into Mathematica
Submitted to Math Source by Roger L. Bagula 30 April 2005
http://www.math.ubc.ca/~kenyon/papers/index.html
The construction of self-similar tilings , Geom. and Func. Analysis 6,(1996):417-488. Thurston showed that the expansion constant of a self-similar tiling of the plane must be a complex Perron number (algebraic integer strictly larger in modulus than its Galois conjugates except for its complex conjugate). Here we prove that, conversely, for every complex Perron number there exists a self-similar tiling. We also classify the expansion constants for self-similar tilings which have a rotational symmetry of order n.
Section 6: page 15-18
http://www.math.ubc.ca/~kenyon/papers/index.html
The construction of self-similar tilings , Geom. and Func. Analysis 6,(1996):417-488. Thurston showed that the expansion constant of a self-similar tiling of the plane must be a complex Perron number (algebraic integer strictly larger in modulus than its Galois conjugates except for its complex conjugate). Here we prove that, conversely, for every complex Perron number there exists a self-similar tiling. We also classify the expansion constants for self-similar tilings which have a rotational symmetry of order n.
Section 6: page 15-18
The construction of self-similar tilings , Geom. and Func. Analysis 6,(1996):417-488. Thurston showed that the expansion constant of a self-similar tiling of the plane must be a complex Perron number (algebraic integer strictly larger in modulus than its Galois conjugates except for its complex conjugate). Here we prove that, conversely, for every complex Perron number there exists a self-similar tiling. We also classify the expansion constants for self-similar tilings which have a rotational symmetry of order n.
Section 6: page 15-18
Kenyon Polynomial type: λ^n-p λ^(n-1)+q λ+r
First new Polynomial subtype :λ^n-p λ^(n-2)+q λ+r
Second new Polynomial subtype :λ^n-p λ^(n-4)+q λ+r
Second new Polynomial subtype :λ^n-p λ^(n-4)+q λ+r
Fractal set Found inside Kenyon tiles by Null Substitutions
Fractal set Found inside Kenyon tiles by Null Substitutions
Comments on the Theory behind the new Polynomials and the fractal subsets by Roger L. Bagula
The way I got the ideas for new higher substitutions and polynomials is hard to put together.
I had done work on Pisot Theta[0], Theta[1] and Theta[2] trying to get IFS tiles.
S.R. Hinsley claimed to have a Kenyon tile IFS. I spent several months trying to duplicate that.
I finally went to the original article and tried from there. Up until lately I've had little luck with substitution fractals and tiles.
I had done work on Pisot Theta[0], Theta[1] and Theta[2] trying to get IFS tiles.
S.R. Hinsley claimed to have a Kenyon tile IFS. I spent several months trying to duplicate that.
I finally went to the original article and tried from there. Up until lately I've had little luck with substitution fractals and tiles.
In probably totally unrelated work on minimal surfaces ,
CMC ( constant mean curvature surfaces),
and solitons, I went into the theory of elliptical equations pretty deeply.
If we make the Kenyon polynomial into:
(dx/dt)2=x^n-p*x^(n-1)+q*x+r
Then these algebraic numbers are associated with the cycles of the
lattice that the elliptic equation establishes on the plane.
That's why the degenerates are lines or quadrilaterals, I suppose.
It stills seems strange that this type of cycling behavior should give minimal substitutions
and tiles. I had thought of using Weierstrass elliptical equations in
Weierstrass fractal functions before as Besicovitch-Ursell like functions,
but I doubt that would give tiles. The two numbers {g2,g3}
map to the complex Perron numbers, I think , for n=3.
It gets harder for higher n, but seems to be related to a lattice structure of:
{a,b,c}-> -{a,b,c}
The Complex Perron number
z0=x0+I*y0
acts as the self-similar scaling/ affine scaling of the tiles.
In IFS terms that comes out like
S=scale number=x0/(x02+y02)-I*y0/(x02+y02)
z'=S*z+a(i)
a(i) are the digit set of the tile. I used that in the Reptiles I constructed.
This result isn't a "total" answer, but it makes the tiles make more sense
and my polynomial which is mostly hyper elliptic with substitution symmetry:
{a,b,c,d,e}-->- {a,b,c,d,e}
Which is the next Prime Galois field level.
The Reptiles are the quadratic two symbol version of this:
{a,b}->- {a,b}
So we have levels: 2,3,5
The next polynomial of Kenyon in Galois field terms would/should be:
{a,b,c,d,e,f,g}-> - {a,b,c,d,e,f,g}
w2=x^n-p*x^(n-4)+q*x+r
At higher levels it appears that negative q at least will give tiles ( I got one last night
from my elliptical invariant analysis of polynomial).
The higher level seem to roughly correspond to Theta(0),Theta(1) and Theta(2)
levels of minimal symmetry.
CMC ( constant mean curvature surfaces),
and solitons, I went into the theory of elliptical equations pretty deeply.
If we make the Kenyon polynomial into:
(dx/dt)2=x^n-p*x^(n-1)+q*x+r
Then these algebraic numbers are associated with the cycles of the
lattice that the elliptic equation establishes on the plane.
That's why the degenerates are lines or quadrilaterals, I suppose.
It stills seems strange that this type of cycling behavior should give minimal substitutions
and tiles. I had thought of using Weierstrass elliptical equations in
Weierstrass fractal functions before as Besicovitch-Ursell like functions,
but I doubt that would give tiles. The two numbers {g2,g3}
map to the complex Perron numbers, I think , for n=3.
It gets harder for higher n, but seems to be related to a lattice structure of:
{a,b,c}-> -{a,b,c}
The Complex Perron number
z0=x0+I*y0
acts as the self-similar scaling/ affine scaling of the tiles.
In IFS terms that comes out like
S=scale number=x0/(x02+y02)-I*y0/(x02+y02)
z'=S*z+a(i)
a(i) are the digit set of the tile. I used that in the Reptiles I constructed.
This result isn't a "total" answer, but it makes the tiles make more sense
and my polynomial which is mostly hyper elliptic with substitution symmetry:
{a,b,c,d,e}-->- {a,b,c,d,e}
Which is the next Prime Galois field level.
The Reptiles are the quadratic two symbol version of this:
{a,b}->- {a,b}
So we have levels: 2,3,5
The next polynomial of Kenyon in Galois field terms would/should be:
{a,b,c,d,e,f,g}-> - {a,b,c,d,e,f,g}
w2=x^n-p*x^(n-4)+q*x+r
At higher levels it appears that negative q at least will give tiles ( I got one last night
from my elliptical invariant analysis of polynomial).
The higher level seem to roughly correspond to Theta(0),Theta(1) and Theta(2)
levels of minimal symmetry.
On the Fractal subsets found for Kenyon tiles using Mauldin-Williams theory and null substitutions:
{1,λ, λ^2}->{1,λ,0},{1, 0, λ^2},{0,λ, λ^2}
Essentially this result is like replacing one symbol with a {} null substitution.
Any substitution can be represented as a state machine digraph of the Mauldin-Williams type.
Removing one vertex seems to result in tiles producing fractals.
{1,λ, λ^2}->{1,λ,0},{1, 0, λ^2},{0,λ, λ^2}
Essentially this result is like replacing one symbol with a {} null substitution.
Any substitution can be represented as a state machine digraph of the Mauldin-Williams type.
Removing one vertex seems to result in tiles producing fractals.
Some References:
Related paper link to Perron Numbers:
http://www.google.com/url?sa=U&start=1&q=http://www.mrlonline.org/mrl/2004-011-003/2004-011-003-001.pdf&e=10187
Or
http://www.mrlonline.org/mrl/2004-011-003/2004-011-003-001.pdf
On Mauldin -Williamd Digraphs:
"On the Hausdorff Dimension of Some Graphs," (with S.C.Williams),Transactions American Mathematical Society 298 (2) (1986),793-803.
http://www.math.unt.edu/~mauldin/papers/no60.pdf
"Hausdorff Dimension in Graph Directed Constructions," (with S.C.Williams),Transactions American Mathematical Society,309(1988),811-829.
http://www.math.unt.edu/~mauldin/papers/no67.pdf
Related paper link to Perron Numbers:
http://www.google.com/url?sa=U&start=1&q=http://www.mrlonline.org/mrl/2004-011-003/2004-011-003-001.pdf&e=10187
Or
http://www.mrlonline.org/mrl/2004-011-003/2004-011-003-001.pdf
On Mauldin -Williamd Digraphs:
"On the Hausdorff Dimension of Some Graphs," (with S.C.Williams),Transactions American Mathematical Society 298 (2) (1986),793-803.
http://www.math.unt.edu/~mauldin/papers/no60.pdf
"Hausdorff Dimension in Graph Directed Constructions," (with S.C.Williams),Transactions American Mathematical Society,309(1988),811-829.
http://www.math.unt.edu/~mauldin/papers/no67.pdf
Cite this as: Roger Bagula, "Perron Number Tiling Systems" from the Notebook Archive (2006), https://notebookarchive.org/2018-10-10ql6gy
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