Beginning from the version 1.2, IFStile program can analitically find exact Hausdorff dimension of the tile's boundary (as the ratio of logarithms of algebraic numbers).

You can select any IFS and use "Info to console" menu item to print information.

Some examples:

Rauzy tile boundary
-----------------------
Base: x^3+x^2+x-1
used roots:
-0.7718445063+1.115142508*i
-0.7718445063-1.115142508*i
p = |prod(roots)| ~= 1.839286755
Graph: x^4-2x-1
x ~= 1.395336994
dim=2*log(x)/log(p) ~= 1.093364164


Levy Curve boundary
-----------------------
Base: x^2-2x+2
used roots:
1+1*i
1-1*i
p = |prod(roots)| ~= 2
Graph: x^9-3x^8+3x^7-3x^6+2x^5+4x^4-8x^3+8x^2-16x+8
x ~= 1.954776401
dim=2*log(x)/log(p) ~= 1.934007186


Scorpion tile boundary
-----------------------
Base: x^2-2x+2
used roots:
1+1*i
1-1*i
p = |prod(roots)| ~= 2
Graph: x^14-x^13-x^12-x^11-x^10+x^9-x^8+3x^7-8x^6+10x^4-6x^3+8x^2-4x-8
x ~= 1.908050865
dim=2*log(x)/log(p) ~= 1.864199263


Pentadendrite "boundary"
-----------------------
Base: x^4-9x^3+31x^2-49x+31
used roots:
2.809016994+0.5877852523*i
2.809016994-0.5877852523*i
p = |prod(roots)| ~= 8.236067977
Graph: x-3
x ~= 3
dim=2*log(x)/log(p) ~= 1.042068089


Tame Twin Dragon boundary
-----------------------
Base: x^2-x+2
used roots:
0.5+1.322875656*i
0.5-1.322875656*i
p = |prod(roots)| ~= 2
Graph: x^3-x-2
x ~= 1.521379707
dim=2*log(x)/log(p) ~= 1.210760533


Golden Bee boundary
-----------------------
Base: x^4+x^2-1
used roots:
+1.27201965*i
-1.27201965*i
p = |prod(roots)| ~= 1.618033989
Graph: x^4-x^2-1
x ~= 1.27201965
dim=2*log(x)/log(p) ~= 1

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Hi!

Is this the algorithm you are using

http://people.clas.ufl.edu/avince/files/HausdorffDim.pdf

or is there another?

Some generalization, that uses
directed graph IFS representation of the boundary.

Am I right in expected countablegons such as the triangular and
rectangular spirals and the stepped countablegons to have a boundary dimensions of 1?

Yes, you are right, for more examples, see: https://ifstile.com/view/Polynomial_(-2)

You can save boundary of the rectangular spiral to file and check, that all strongly connected components of the graph are simple segments + some cantor sets with dimension<1

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Дмитрий Мехонцев and Steward Robert Hinsley,

In chemistry ( and even plane tiling) groups are used to
categorize the symmetry of the geometry involved.
Is there any way to calculate a spectrum associated with
a tiling? Do tiles with lower dimensional borders, have
lower energy spectra? This question is "close" to a fractal
drum type question of are their sound alike fractal tiles?

Roger Bagula

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On 07/08/2016 17:37, Дмитрий Мехонцев wrote:

Beginning from the version 1.2, IFStile program can analitically find exact Hausdorff dimension of the tile's boundary (as the ratio of logarithms of algebraic numbers).

You can select any IFS and use "Info to console" menu item to print information.

Is this the algorithm you are using

http://people.clas.ufl.edu/avince/files/HausdorffDim.pdf

or is there another?

Am I right in expected countablegons such as the triangular and
rectangular spirals and the stepped countablegons to have a boundary
dimensions of 1?

--
SRH

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On 09/08/2016 21:44, Roger Bagula wrote:

Дмитрий Мехонцев and Steward Robert Hinsley,

In chemistry ( and even plane tiling) groups are used to
categorize the symmetry of the geometry involved.
Is there any way to calculate a spectrum associated with
a tiling? Do tiles with lower dimensional borders, have
lower energy spectra? This question is "close" to a fractal
drum type question of are their sound alike fractal tiles?

Roger Bagula

For a given Perron number there are tiles with boundaries of different dimensions. For example for the Perron number 1 + i the dimension of the boundary varies from 1 (right isoceles triangle) to close to 2 (Levy
curve). You could call that a "tile boundary dimension spectrum" of the
Perron number.

There are multiple tiles with not only the same group but also (when appropriately scaled) the same tiling vectors. For example there are
probably somewhere in excess of 100,000 grouped element derivatives of
the flowsnake (there are in excess of 9,000 Pacmen), all of which have a
unit cell which is a flowsnake (with 2, 3 or 6 copies in the unit cell).
Even if you restrict yourself to those with 2 copies in the unit cell
there is a wide variation in the dimension of the boundary. The lowest dimension boundary might that of Ventrella's palidromic 7-dragon; the
highest might be one of the Pacmen. You could call that "tile boundary dimension spectrum" of the unit cell.

[A mildly interesting question - do all Pacmen with the same number of
copies in the unit cell hae the same boundary dimension? They all rather similar, but that might just mean a limited range in boundary
dimensions, rather than the same boundary dimension.]

You could equally well consider the values of the boundary dimensions of
all tiles with the same space group, or all tiles with the same point group.

But you have to bear in mind that a spectrum is the most general sense
is either a list of values (with intensities) or a function. Outside the various used in physical science mathematics has a number of difference meanings for spectrum.

https://en.wikipedia.org/wiki/Spectrum_(disambiguation)#Mathematics

But turning to the spectrum (resonant frequencies) of a fractal drum,
this has been investigated (Google spectrum fractal drum), with the Koch snowflake in particular being studied. The question you may wish to ask
is what results about the spectra of fractal drums can be obtained analytically, in which case my answer is that I have no idea, but I'm
not optimistic. For the particular case of whether different fractal
drums sound alike, one can consider the generalisation of flowsnakes to fractals composed of nested hexagons of equal sized elements; as the
number of nested hexagons tends to infinity the boundary tends to a
hexagon, which the boundary dimension (I conjecture) monotonically
decreasing. My intuition would be that at least above some minimal
number of nested hexagons the spectrum monotonically approaches that of
a hexagon, and for any degree of difference between spectra there is a
number of nested hexagons n such the the difference between the spectra
of flowsnake-n and flowsnake-n+1 is smaller.

I'd like to say that the number of fractal tiles of order n increases
rapidly with n, and draw conclusions of the decreasing difference
between adjacent values of various numbers that can be associated with
the tiles, but there's the slight problem that the number of order 4
tiles seems to be already countably infinite, so we have find more
useful measure than raw number of tiles.

Given that phonons are quantised there is presumably a mapping between a frequency spectrum and an energy spectrum, but it's probably better to
refer to a frequency spectrum as this is more directly accessible.

--
SRH

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Stewart Robert Hinsley
You certainly get the idea ( probably better than me as usual) I've been doing 3d
IFS experiments in making hyperbolic 3 manifolds of of the
first Akiyama tiling( 1,5} as the same polynomial
base:x^3-x-1
has been linked to the Weeks minimal hyperbolic 3 manifold. So the frequency spectrum of the 2d tile would be parallel to Dr Weeks' idea of the vacuum. Such an hyperbolic 3 manifold would tile space and have a characteristic set of frequencies.
In other words the Perron number tiling idea doesn't just work in 2d , but 3d as well. And the fractal drum frequency results have physical implications about the shape of the universe. These types of tiling manifolds may also explain part of the dark
mass / dark energy.
My idea has been that there is a third sort of particle
that with leptons and bosons make up a hyperbolic 3 manifold.
This kind of particle can be developed statistically as I have derived. Since bosons were named for Dr. Bose,
I'd have to name the new particle type weeksons, I guess.
Here is my post of the derivation of a third particle type distribution law: (---------------------------------------)
I thought of doing this before but never got the derivation right.
I did the derivation last night by hand using the Wall derivation “pattern”.
Today I checked the result using Mathematica.
This derivation show that the anharmonic group/ Sym_3: on{-1,1}n instead of {0,1};
{1/(1+x),1/(1-x),(1+x)/(1-x),(+x,1-x,(1-x)/(1+x)}
for x=-a-b*e[i]: energy
gives a mapping of thermodynamic types.
(* Mathematica*)
(*argument for derivation from :*)
(* Chemical Thermodynamics \
Frederick T. Wall*)
\
(*https://www.amazon.com/Chemical-Thermodynamics-Course-Study-chemistry/dp/\ 0716701731/ref=sr_1_1?ie=UTF8&qid=1470597514&sr=8-1&keywords=Chemical+\ Thermodynamics+Frederick+T.+Wall*)
(*page 229 Boltzman distribution: \
N[i]=g[i]*Exp[-a-b*e[i]]*)
(*page 297 Boson distribution: \
N[i]=g[i]/(Exp[-a-b*e[i]]-1)*)
(*page 305 Fermi Dirac Lepton distribution: \
N[i]=g[i]/(Exp[-a-b*e[i]]+1)*)
Clear[n, m]
(* log of Sterling approximation*)
f[n_] = n*Log[n] - n;
(* Lepton-Fermion: binomial*)
a = n!/((n - m)!*m!);
(* Boson combinations*)
b = (n + m)!/(n!*m!);
(* product of the two types gives a third statistical \ combination:(m+n)!/((m!)^2 (-m+n)!)*)
W = a*b;
((m + n)!/((m!)^2 (-m + n)!))
(* log of activity W in Sterling approximations*)

g = f[m + n] - 2*f[m] - f[n - m];
(* differential variation of a Log[W] action*)
d = D[g, {n, 1}];
(* solution of distribution law*)
Solve[Exp[en] - d == 0, n]
(* derived form*)
N[i] = g[i]*(Exp[-a - b*e[i]] + 1)/(-Exp[-a - b*e[i]] + 1)
(* end*)

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