I started working on writing up my 2002 studies on metallic and cubic
tiles, and in the process found several hundred more tiles associated
with the 6th cubic Pisot number. I've not finished - I've got caught in
the rabbit hole of the infinite number of different tilings based on the symmetric 6th cubic Pisot tile, but I might as well let people look at
the tiles.


http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/6thcubic/6thcubic.php

--
SRH

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Stewart Robert Hinsley
Happy New Year!
I'm glad to see your fractal tile site back up on the web and you actively finding new original tiles.

The number of tiles does seem a lot, but you are doing higher order versions based on composite polynomials.

I've always thought that the minimal Pisots form a kind of orthogonal field like the Cyclotomic field in number theory with a golden mean limiting real scalar.
The x^5 minimal Pisot suggests that there may be a limited number of them like the Heegner number based hyperbolic tori. Also a link to the hyperbolic knot minimal polynomials might exist, but I don't know it.

This tiling work is good work: thanks for posting it.
Roger Baguls

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On 30/12/2015 16:01, Roger Bagula wrote:

Stewart Robert Hinsley
Happy New Year!
I'm glad to see your fractal tile site back up on the web and you actively finding new original tiles.

The number of tiles does seem a lot, but you are doing higher order versions based on composite polynomials.

There seem to be fewer order 3 tiles than for the "4th cubic"
(tribonacci number) by a far margin. I don't know for certain that the
number of order 5 and order 7 tiles is high because of the existence of
a symmetric order 3 tile, rather than making it easier to find them.

Heuristic driven surveys for order 3 tiles take about a day of computer
time. Heuristic driven surveys for order 4 and higher aren't
practicable, so what's found is what's generatable by mechanical
derivation with the only question being whether the generated fractal is connected or not.

Anyway, I think I've now worked out how to tile all the tiles - the last
batch of 105 turned out to neatly fall out with 4 or 6 copies making up
a unit cell coextensive with the parent symmetric tile - the PC is
chuntering away plotting the weight function to confirm unit cell, as
some of the unit cells are too messy (Levy curve territory) for clear
visual confirmation.

I've written up the tilings for the other 2 order 3 tiles.

http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/6thcubic/complex.php http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/6thcubic/external.php

I've always thought that the minimal Pisots form a kind of orthogonal field like the Cyclotomic field in number theory with a golden mean limiting real scalar.
The x^5 minimal Pisot suggests that there may be a limited number of them like the Heegner number based hyperbolic tori. Also a link to the hyperbolic knot minimal polynomials might exist, but I don't know it.

This tiling work is good work: thanks for posting it.
Roger Baguls

--
SRH

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On Sunday, January 3, 2016 at 3:03:47 PM UTC-8, Stewart Robert Hinsley wrote:

On 30/12/2015 16:01, Roger Bagula wrote:

Stewart Robert Hinsley
Happy New Year!
I'm glad to see your fractal tile site back up on the web and you actively finding new original tiles.

The number of tiles does seem a lot, but you are doing higher order versions based on composite polynomials.

There seem to be fewer order 3 tiles than for the "4th cubic"
(tribonacci number) by a far margin. I don't know for certain that the
number of order 5 and order 7 tiles is high because of the existence of
a symmetric order 3 tile, rather than making it easier to find them.

Heuristic driven surveys for order 3 tiles take about a day of computer
time. Heuristic driven surveys for order 4 and higher aren't
practicable, so what's found is what's generatable by mechanical
derivation with the only question being whether the generated fractal is connected or not.

Anyway, I think I've now worked out how to tile all the tiles - the last batch of 105 turned out to neatly fall out with 4 or 6 copies making up
a unit cell coextensive with the parent symmetric tile - the PC is
chuntering away plotting the weight function to confirm unit cell, as
some of the unit cells are too messy (Levy curve territory) for clear
visual confirmation.

I've written up the tilings for the other 2 order 3 tiles.

http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/6thcubic/complex.php
http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/6thcubic/external.php

I've always thought that the minimal Pisots form a kind of orthogonal field like the Cyclotomic field in number theory with a golden mean limiting real scalar.
The x^5 minimal Pisot suggests that there may be a limited number of them like the Heegner number based hyperbolic tori. Also a link to the hyperbolic knot minimal polynomials might exist, but I don't know it.

This tiling work is good work: thanks for posting it.
Roger Baguls

--
SRH

At about x^8 the minimal Pisots seem to limit at the GoldenRatio: I'm wondering if minimal Pisots exist above
the 8th power with real roots/ratios about 1.61.
They probably don't but the scanning I've done in no way rules it out.
As the quintic minimal Pisot made a new rule polynomial pattern that the sexitic and heptic and octic seemed to follow, maybe 9 and 10 are some new pattern we haven't yet found?

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I don't know if this sequence gives the absolute minimum minimal Pisots or not, but it appears close:

I reconstructed the Pisot sequence based on the quintic polynomial(x^5):
solved as absolute value of roots:
Table[Table[
Abs[x] /. NSolve[1 - x^2 + (x^3 + x^4 - x^5)*x^n == 0, x][[i]], {i,
5 + n}], {n, 0, 10}]

{{0.891548, 0.891548, 0.933645, 0.933645, 1.44327}, {0.845219,
0.962786, 0.962786, 0.921954, 0.921954, 1.50159}, {0.901, 0.901,
0.977383, 0.977383, 0.913516, 0.913516, 1.54522}, {0.863901,
0.961331, 0.961331, 0.981389, 0.981389, 0.909067, 0.909067,
1.57368}, {0.90816, 0.90816, 0.978951, 0.978951, 0.982621, 0.982621,
0.907465, 0.907465, 1.59118}, {0.877993, 0.959757, 0.959757,
0.984975, 0.984975, 0.982822, 0.982822, 0.907601, 0.907601,
1.60176}, {0.914119, 0.914119, 0.978288, 0.978288, 0.987583,
0.987583, 0.98256, 0.98256, 0.908735, 0.908735, 1.60813}, {0.889045,
0.958832, 0.958832, 0.985355, 0.985355, 0.988864, 0.988864,
0.982075, 0.982075, 0.91042, 0.91042, 1.61198}, {0.919235, 0.919235,
0.977348, 0.977348, 0.988644, 0.988644, 0.989522, 0.989522,
0.981482, 0.981482, 0.912391, 0.912391, 1.61433}, {0.958452,
0.958452, 0.897979, 0.98503, 0.98503, 0.9904, 0.9904, 0.989846,
0.989846, 0.980848, 0.980848, 0.914491, 0.914491,
1.61576}, {0.923699, 0.923699, 0.976496, 0.976496, 0.988776,
0.988776, 0.991422, 0.991422, 0.989976, 0.989976, 0.980209,
0.980209, 0.916629, 0.916629, 1.61663}}
The real ratio/root sequence is:
In[588]:= Table[
Abs[x] /.
NSolve[1 - x^2 + (x^3 + x^4 - x^5)*x^n == 0, x][[n + 5]], {n, 0, 10}]

{1.44327, 1.50159, 1.54522, 1.57368, 1.59118, 1.60176, 1.60813, 1.61198, 1.61433, 1.61576, 1.61663}
When divided by the GoldenRatio:
'{0.891989, 0.928037, 0.954996, 0.972587, 0.983406, 0.98994, 0.993878, 0.996261, 0.997709, 0.998592, 0.999134}
show the limiting effect.

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On 30/12/2015 16:01, Roger Bagula wrote:

Stewart Robert Hinsley
Happy New Year!
I'm glad to see your fractal tile site back up on the web and you actively finding new original tiles.

The number of tiles does seem a lot, but you are doing higher order versions based on composite polynomials.

I'm looking into golden tiles [perron number ((1+sqrt(5))/2)e^(ipi/2)],
and they appear to be a richer set. The first symmetric dissection of
the golden rectangle is order 7, so there are fewer order 5 tiles, but
the golden rectangle being d2-symmetric rather than c2-symmetric means
that there are more demi-tiles (dissections are c2- or d1-symmetric, so
there are no quarter tiles) - I haven't counted up yet, but it looks as
if the total of order 2 to 7 tiles might run up into the 400 or 500 region.

The trihextals [perron number (sqrt(3))e^(ipi/6)] are richer yet. They
have several c2-symmetric order 3-tiles, so there are a great number of
order 5 and order 7 demi-tiles.

I've always thought that the minimal Pisots form a kind of orthogonal field like the Cyclotomic field in number theory with a golden mean limiting real scalar.
The x^5 minimal Pisot suggests that there may be a limited number of them like the Heegner number based hyperbolic tori. Also a link to the hyperbolic knot minimal polynomials might exist, but I don't know it.

This tiling work is good work: thanks for posting it.
Roger Baguls

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Forward from Steward Robert Hinsley:
I've written up a summary about what I know about tiling and Perron numbers

http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Perron.php

I've also redone a web page from my old site, adding the next four tiles in the series in the process.

http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/nxx2x3/nxx2x3.php

And generated some 23 new tiles for the 8th unit cubic Pisot and 12 new tiles for the 12th unit cubic Pisot.

http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/8thcubic/8thcubic.php http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/12thcubic/12thcubic.php

You can pass this on to Dieter Steeman and the rest of the folks at TrueTile.

--
SRH

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On 20/07/2016 19:16, Roger Bagula wrote:

Forward from Steward Robert Hinsley:
I've written up a summary about what I know about tiling and Perron numbers

http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Perron.php

I've also redone a web page from my old site, adding the next four tiles in the series in the process.

http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/nxx2x3/nxx2x3.php

And generated some 23 new tiles for the 8th unit cubic Pisot and 12 new tiles for the 12th unit cubic Pisot.

http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/8thcubic/8thcubic.php
http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Cubic/12thcubic/12thcubic.php

You can pass this on to Dieter Steeman and the rest of the folks at TrueTile.

I've linked all the fractal stuff on my new web site together

http://www.stewart.hinsley.me.uk/Fractals/IFS/Tiles/Tiles.php

and added some material on other types of IFS fractal tiles.

--
alias Ernest Major

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