The 17 Kenyon tiles are here: http://www.math.brown.edu/~rkenyon/gallery/all17.pdf
My tiling has a self-similar overlap, but it does rtile.
I call this approach the Minimal Pisot model of badly aproximated rational polynomials.
With some little derivation and programing efforts I got the following polynomials.
{-1 - 1/x^(14/9) + x^(4/9), -1 - 1/x^(3/2) + Sqrt[x], -1 - 1/x^(10/7) + x^(
4/7), -1 - 1/x^(7/5) + x^(3/5), -1 - 1/x^(4/3) + x^(2/3), -1 - 1/x^(5/4) +
x^(3/4), -1 - 1/x^(6/5) + x^(4/5), -1 - 1/x^(8/7) + x^(6/7), -1 - 1/x^(
10/9) + x^(8/9), -1 - x^(6/7) + x^(20/7), -1 - x + x^3, -1 - x^(6/5) + x^(
16/5), -1 - x^(4/3) + x^(10/3), -1 - x^(3/2) + x^(7/2), -1 - x^(8/5) + x^(
18/5)}
I, then, used the Akiyama Curley tile type program to sift through the polynomials.
This polynomial bar two was near the last experiment.
This experimental approach is an Edison light bulb approach.
Since it took me a year or so to find this with trying every approach I could think of,
I can’t claim much except I kept at it feeling there was a tiling there somewhere.
I actually think there are others.
(*Mathematica program*)
Clear[f, dlst, pt, cr, ptlst, M, r, p, rotate, r0]
(*IFS 2343 tilingby Roger L.Bagula 20 Jan 2018©*)

allColors = ColorData["Legacy"][[3, 1]];
firstCols = {"LightBlue", "Blue", "Cyan", "White", "Yellow", "Red", "White",
"DeepNaplesYellow", "Tomato", "Pink", "LightPink", "White", "Purple",
"DarkOrchid", "Magenta", "DodgerBlue", "GoldOchre", "LightPink", "Magenta",
"Green", "DarkOrchid", "LightSalmon", "LightPink", "Sienna", "Green",
"Mint", "DarkSlateGray", "ManganeseBlue", "SlateGray", "DarkOrange",
"MistyRose", "DeepNaplesYellow", "GoldOchre", "SapGreen", "Yellow",
"Yellow", "Tomato", "DeepNaplesYellow", "DodgerBlue", "Cyan", "Red",
"Blue", "DeepNaplesYellow", "Green", "Magenta", "DarkOrchid",
"LightSalmon", "LightPink", "Sienna", "Green", "Mint", "DarkSlateGray",
"ManganeseBlue", "SlateGray", "DarkOrange", "MistyRose",
"DeepNaplesYellow", "GoldOchre", "SapGreen", "Yellow", "LimeGreen"};
cols = ColorData["Legacy", #] & /@
Join[firstCols, Complement[allColors, firstCols]];

(*IFS definition in Mathematica*)
Clear[x, y]
NSolve[-1 - 1/x^(4/3) + x^(2/3) == 0, x]
r = x /. NSolve[-1 - 1/x^(4/3) + x^(2/3) == 0, x][[3]]

(*definition of complex power constants*)
c = Re[r]
s = Im[r]
c1 = Re[r^(2/3)]
s1 = Im[r^(2/3)]
c2 = Re[r^(4/3)]
s2 = Im[r^(4/3)]

(*IFS program type*)

f[1, {x_, y_}] = {x*c2 - s2*y + c2, s2*x + c2*y + s2};
f[2, {x_, y_}] = {x*c1 - s1*y + c1, s1*x + c1*y + s1};

pt = {0.5, 0.5};
it = 3000000;

dlst = Table[Which[(r = RandomReal[]) <= 1/3 + 0.02, 1, True, 2], {it}];
cr[n_] := cr[n] = cols[[n]];
cr2[n_] := cr2[n] = cols[[n + 4]];
cr3[n_] := cr3[n] = cols[[n + 8]];
cr4[n_] := cr4[n] = cols[[n + 12]];
aa = Table[pt = f[dlst[[j]], pt], {j, Length[dlst]}];
ptlst = Point[Developer`ToPackedArray[aa],
VertexColors -> Developer`ToPackedArray[cr2 /@ dlst]];
g2 = Graphics[{PointSize[.001], ptlst}, AspectRatio -> Automatic,
ImageSize -> 1500, Background -> Black, PlotRange -> {{-1, 1}, {-1, 1.}} 1.5] (* complement set*)
bb = Table[pt = f[dlst[[j]], pt], {j, Length[dlst]}];
cc = Table[bb[[i]]/(bb[[i]].bb[[i]]), {i, Length[bb]}];
ptlst1 = Point[Developer`ToPackedArray[cc],
VertexColors -> Developer`ToPackedArray[cr /@ dlst]];
gb = Graphics[{PointSize[.001], ptlst1}, AspectRatio -> Automatic,
ImageSize -> 1000, PlotRange -> {{-4, 4}, {-4, 4}}*5/4, Background -> Black]

Show[{g2, gb}, PlotRange -> {{-4, 4}, {-4, 4}} 3]
(* sphere conformal map*)

bb0 = Table[{2*cc[[i, 1]], 2*cc[[i, 2]],
1 - cc[[i]].cc[[i]]}/(1 + cc[[i]].cc[[i]]), {i, Length[cc]}];
ptlsb = Point[Developer`ToPackedArray[bb0],
VertexColors -> Developer`ToPackedArray[cr3 /@ dlst]];
cc0 = Table[{2*cc[[i, 1]],
2*cc[[i, 2]], -(1 - cc[[i]].cc[[i]])}/(1 + cc[[i]].cc[[i]]), {i,
Length[cc]}];
ptlsc = Point[Developer`ToPackedArray[cc0],
VertexColors -> Developer`ToPackedArray[cr2 /@ dlst]];
g22a = Graphics3D[{PointSize[.001], ptlsb, ptlsc}, AspectRatio -> Automatic,
PlotRange -> {{-1.01, 1.01}, {-1.01, 1.01}, {-1.01, 1.01}},
ImageSize -> {1000, 1000}, Background -> Black, ViewPoint -> {-5, 5, 2},
Boxed -> False]
(* end*) https://lh3.googleusercontent.com/dsPV4EAXCDdEcwJt5n-zXnfXZuCynYAaJBm0ECSNB6RQxeiwGYKpqfGCZNBt9ZqOieOzAF756jZusmDYMle4lkj6ltdCOI1gzIioEw=s220

https://lh3.googleusercontent.com/jCSbpJitg3jQNtyghBYYFLjGIX5u84gIqkEs_HZfg2esZ0hUCK0Wlw1TWs62SruNkb3fpFyKXGSUmBeEz8K35TqpseH2WTtHj9RzGg=w170-h220

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Oh , Damn, it's #9...

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 20/01/2018 21:43, Roger Bagula wrote:

The 17 Kenyon tiles are here: http://www.math.brown.edu/~rkenyon/gallery/all17.pdf
My tiling has a self-similar overlap, but it does rtile.
I call this approach the Minimal Pisot model of badly aproximated rational polynomials.
With some little derivation and programing efforts I got the following polynomials.
{-1 - 1/x^(14/9) + x^(4/9), -1 - 1/x^(3/2) + Sqrt[x], -1 - 1/x^(10/7) + x^(
4/7), -1 - 1/x^(7/5) + x^(3/5), -1 - 1/x^(4/3) + x^(2/3), -1 - 1/x^(5/4) +
x^(3/4), -1 - 1/x^(6/5) + x^(4/5), -1 - 1/x^(8/7) + x^(6/7), -1 - 1/x^(
10/9) + x^(8/9), -1 - x^(6/7) + x^(20/7), -1 - x + x^3, -1 - x^(6/5) + x^(
16/5), -1 - x^(4/3) + x^(10/3), -1 - x^(3/2) + x^(7/2), -1 - x^(8/5) + x^(
18/5)}
I, then, used the Akiyama Curley tile type program to sift through the polynomials.
This polynomial bar two was near the last experiment.
This experimental approach is an Edison light bulb approach.
Since it took me a year or so to find this with trying every approach I could think of,
I can’t claim much except I kept at it feeling there was a tiling there somewhere.
I actually think there are others.
(*Mathematica program*)
Clear[f, dlst, pt, cr, ptlst, M, r, p, rotate, r0]
(*IFS 2343 tilingby Roger L.Bagula 20 Jan 2018©*)

allColors = ColorData["Legacy"][[3, 1]];
firstCols = {"LightBlue", "Blue", "Cyan", "White", "Yellow", "Red", "White",
"DeepNaplesYellow", "Tomato", "Pink", "LightPink", "White", "Purple",
"DarkOrchid", "Magenta", "DodgerBlue", "GoldOchre", "LightPink", "Magenta",
"Green", "DarkOrchid", "LightSalmon", "LightPink", "Sienna", "Green",
"Mint", "DarkSlateGray", "ManganeseBlue", "SlateGray", "DarkOrange",
"MistyRose", "DeepNaplesYellow", "GoldOchre", "SapGreen", "Yellow",
"Yellow", "Tomato", "DeepNaplesYellow", "DodgerBlue", "Cyan", "Red",
"Blue", "DeepNaplesYellow", "Green", "Magenta", "DarkOrchid",
"LightSalmon", "LightPink", "Sienna", "Green", "Mint", "DarkSlateGray",
"ManganeseBlue", "SlateGray", "DarkOrange", "MistyRose",
"DeepNaplesYellow", "GoldOchre", "SapGreen", "Yellow", "LimeGreen"};
cols = ColorData["Legacy", #] & /@
Join[firstCols, Complement[allColors, firstCols]];

(*IFS definition in Mathematica*)
Clear[x, y]
NSolve[-1 - 1/x^(4/3) + x^(2/3) == 0, x]
r = x /. NSolve[-1 - 1/x^(4/3) + x^(2/3) == 0, x][[3]]

(*definition of complex power constants*)
c = Re[r]
s = Im[r]
c1 = Re[r^(2/3)]
s1 = Im[r^(2/3)]
c2 = Re[r^(4/3)]
s2 = Im[r^(4/3)]

(*IFS program type*)

f[1, {x_, y_}] = {x*c2 - s2*y + c2, s2*x + c2*y + s2};
f[2, {x_, y_}] = {x*c1 - s1*y + c1, s1*x + c1*y + s1};

pt = {0.5, 0.5};
it = 3000000;

dlst = Table[Which[(r = RandomReal[]) <= 1/3 + 0.02, 1, True, 2], {it}]; cr[n_] := cr[n] = cols[[n]];
cr2[n_] := cr2[n] = cols[[n + 4]];
cr3[n_] := cr3[n] = cols[[n + 8]];
cr4[n_] := cr4[n] = cols[[n + 12]];
aa = Table[pt = f[dlst[[j]], pt], {j, Length[dlst]}];
ptlst = Point[Developer`ToPackedArray[aa],
VertexColors -> Developer`ToPackedArray[cr2 /@ dlst]];
g2 = Graphics[{PointSize[.001], ptlst}, AspectRatio -> Automatic,
ImageSize -> 1500, Background -> Black, PlotRange -> {{-1, 1}, {-1, 1.}} 1.5]
(* complement set*)
bb = Table[pt = f[dlst[[j]], pt], {j, Length[dlst]}];
cc = Table[bb[[i]]/(bb[[i]].bb[[i]]), {i, Length[bb]}];
ptlst1 = Point[Developer`ToPackedArray[cc],
VertexColors -> Developer`ToPackedArray[cr /@ dlst]];
gb = Graphics[{PointSize[.001], ptlst1}, AspectRatio -> Automatic,
ImageSize -> 1000, PlotRange -> {{-4, 4}, {-4, 4}}*5/4, Background -> Black]

Show[{g2, gb}, PlotRange -> {{-4, 4}, {-4, 4}} 3]
(* sphere conformal map*)

bb0 = Table[{2*cc[[i, 1]], 2*cc[[i, 2]],
1 - cc[[i]].cc[[i]]}/(1 + cc[[i]].cc[[i]]), {i, Length[cc]}];
ptlsb = Point[Developer`ToPackedArray[bb0],
VertexColors -> Developer`ToPackedArray[cr3 /@ dlst]];
cc0 = Table[{2*cc[[i, 1]],
2*cc[[i, 2]], -(1 - cc[[i]].cc[[i]])}/(1 + cc[[i]].cc[[i]]), {i,
Length[cc]}];
ptlsc = Point[Developer`ToPackedArray[cc0],
VertexColors -> Developer`ToPackedArray[cr2 /@ dlst]];
g22a = Graphics3D[{PointSize[.001], ptlsb, ptlsc}, AspectRatio -> Automatic,
PlotRange -> {{-1.01, 1.01}, {-1.01, 1.01}, {-1.01, 1.01}},
ImageSize -> {1000, 1000}, Background -> Black, ViewPoint -> {-5, 5, 2},
Boxed -> False]
(* end*) https://lh3.googleusercontent.com/dsPV4EAXCDdEcwJt5n-zXnfXZuCynYAaJBm0ECSNB6RQxeiwGYKpqfGCZNBt9ZqOieOzAF756jZusmDYMle4lkj6ltdCOI1gzIioEw=s220

https://lh3.googleusercontent.com/jCSbpJitg3jQNtyghBYYFLjGIX5u84gIqkEs_HZfg2esZ0hUCK0Wlw1TWs62SruNkb3fpFyKXGSUmBeEz8K35TqpseH2WTtHj9RzGg=w170-h220

As you subsequently note is the 9th tile in Kenyon's set.

What you've effectively done is take this tile with dissection equation
x + x^3 = 1, and replaced the x^3 element by an x^2 element, with an
overlap of x^5.

I haven't attempted to do anything in this area myself, but I've noted a strategy at

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

The modified dissection equation

x + x^2 - x^5 = 1

is the same as

(x - 1)(x + 1)(x + x^3 - 1)

so has the same Perron number as x + x^3 = 1

The rules of the game if you're attempting to add to Kenyon's set is
that that the tile should be able to be dissected into directly (no reflections) self-similar non-overlapping parts. You can relax the rules
in various ways, but then you're not addressing Kenyon's conjecture that
there are only 17 self-similar order 2 tiles.

You can relax the rules to allow reflections, which gives several more
order 2 rep-tiles. (I have a conjecture that reflections don't work for
cubic Perron numbers because the unit cells don't have reflectional
symmetry.) Counting gets messy because there's a couple of continuously deformable tiles (which to confuse the matter further include two of the
Kenyon 17 among the sets).

You can relax the rules to allow graph directed IFSs rather than
stateless IFSs. I've played around with graph directed IFSs before, but
not directed at this particular problem. I recall Dmitri (ifstile.com) producing some order 2 tiles based on graph directed IFSs, including a different isoceles triangle to the one in Kenyon's set. This may be
equivalent to or somewhat more restrictive that relaxing the rules to
allow overlaps.

You also can relax the rules to allow the disssection to be self-affine.
This converts Kenyon's 17 tiles into 17 continously deformable sets, but
there might be additional tiles. (I know that there are an order 3
self-affine triangles which are not equivalent to the order 3
self-similar triangles, though in all cases the sets include all
possible trianges - it's the dissections that are different. The order 3 self-similar triangle is the half-sunburst; the other self-affine ones
are the order 3 starburst and order 3 cloudburst.)

--
SRH

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

My badly approximated rational approach was successful in one thing:
it was a new way to get the same old minimal Pisot cubic tiles, LOL.
I haven't given up.
Thanks you for your suggestions.
I've had success with Kleinian group Thurston-Cannon maps using a two matrix Moran approach: it turns out that the Twin dragon type tiling corresponds to a
4_1 Figure 8 knot as a quadratic space filling set. The Weeks hyperbolic 3 manifold has been identified as being associated with the Minimal Pisot cubic as well. It appears that finding Thurston -Cannon sphere filling sets with two generators is in
parallel to the Kenyon 17 problem. Riemannian geometry is universal surfaces as well as spaces.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

I had forgotten about this two transform overlap square space fill whose second transform is a chair:
Although the 4 set is a space filling set there is a internal pattern
that forms ( filling more slowly). The Sierpinski overlap sets on two transforms
use the rotational symmetry effect to cover the sets.

(* Mathematica*)
Clear[cr, cols, cr2, cr3, cr4, firstCols]
allColors = ColorData["Legacy"][[3, 1]];
firstCols = {"White", "DodgerBlue", "AliceBlue", "LightBlue" , "Cyan",
"ManganeseBlue" , "Blue", "Magenta", "Purple", "DarkOrchid", "White",
"AliceBlue", "LightBlue" , "Cyan", "ManganeseBlue", "DodgerBlue" , "Blue",
"Magenta", "Purple", "DarkOrchid", "DeepNaplesYellow", "Gold", "Banana",
"Yellow", "Pink", "Tomato", "Red", "DarkOrange", "Orange",
"DeepNaplesYellow", "Gold", "Banana", "Yellow", "LightYellow", "Orange",
"Pink", "LightPink", "Yellow", "LightYellow", "LightPink", "White",
"DeepNaplesYellow", "Orange", "DarkOrange", "Tomato", "Red", "Tomato",
"Pink", "LightPink", "DeepNaplesYellow", "Orange", "DarkOrange", "Tomato",
"White", "Pink", "Banana", "LightBlue", "DodgerBlue", "Cyan", "White",
"Purple", "DarkOrchid", "Magenta", "ManganeseBlue", "DeepNaplesYellow",
"Orange", "DarkOrange", "Tomato", "GoldOchre", "LightPink", "Magenta",
"Green", "DarkOrchid", "LightSalmon", "LightPink", "Sienna", "Green",
"Mint", "DarkSlateGray", "ManganeseBlue", "SlateGray", "DarkOrange",
"MistyRose", "DeepNaplesYellow", "GoldOchre", "SapGreen", "Yellow",
"Yellow", "Tomato", "DeepNaplesYellow", "DodgerBlue", "Cyan", "Red",
"Blue", "DeepNaplesYellow", "Green", "Magenta", "DarkOrchid",
"LightSalmon", "LightPink", "Sienna", "Green", "Mint", "DarkSlateGray",
"ManganeseBlue", "SlateGray", "DarkOrange", "MistyRose",
"DeepNaplesYellow", "GoldOchre", "SapGreen", "Yellow", "LimeGreen"};
cols = ColorData["Legacy", #] & /@
Join[firstCols, Complement[allColors, firstCols]];
rotate[theta_] := {{Cos[theta], -Sin[theta]}, {Sin[theta], Cos[theta]}}; Clear[f, dlst, pt, cr, ptlst, r, m]
m = 4;
dlst = Table[ Random[Integer, {1, 2}], {n, 2500000}];
r[3] = 2;
r[4] = 2;
r[5] = (Sqrt[5] + 1)/2 + 1;
r[6] = 3;
r[7] = 2 + Sqrt[1.5];
r[8] = 2 + Sqrt[2];
M = Table[
If[n == 1, {{1, 0}, {0, 1}}/
r[m], {{Cos[2*Pi/m], -Sin[2*Pi/m]}, {Sin[2*Pi/m], Cos[2*Pi/m]}}], {n, 1,
8}]
in = Table[If[n == 1, {0, 0}, {-1/2, Sqrt[3]/2}], {n, 1, 8}]
f[j_, {x_, y_}] := M[[j]]. {x, y} + in[[j]]
pt = {0.0, 0.0};
cr[n_] := cr[n] = cols[[n]];
cr2[n_] := cr2[n] = cols[[n + 4]];

ptlst = Point[
Developer`ToPackedArray[Table[pt = f[dlst[[j]], pt], {j, Length[dlst]}]],
VertexColors -> Developer`ToPackedArray[cr /@ dlst]];
g2 = Graphics[{PointSize[.001], ptlst}, AspectRatio -> Automatic,
PlotRange -> All, ImageSize -> {1000, 1000}, Background -> Black]' Export["Sierpinski4overlap.jpg", g2]
(* end*) https://i.pinimg.com/564x/33/7b/9e/337b9ec9a79642432cfccc39f47d638f.jpg

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)