This morning, Vesa Timonen sent me a new shape found by a searching program of his that looked like a new irreptile based on the Plastic constant. It didn’t take long to verify it.
I talk about the Plastic constant extensively in my post Shattering the Plane.
We can draw it with AnglePath:
path=RootReduce[With[{r=Root[-1-#1+#1^3&,1,0]},​​AnglePath[{{r^6,0°},{r^7,120°},{r^4,120°},{r^6,-120°},{r^8,120°},{r^7,120°},​​{r^4,60°},{r^2,120°},{r^0,-120°},{r^3,120°},{r^5,120°}}]]];​​Graphics[{Line[path],Style[Text[ResourceFunction["IntegerChop"][Quiet@N[Log[Root[-1-#1+#1^3&,1,0],EuclideanDistance@@#]]],Mean[#],BackgroundWhite],16]&/@Partition[path,2,1]}]
Out[]=
At the Math Magic Irreptiles page and at the Wikipedia Rep-tile page, a similar solution by Karl Scherer can be seen.
In[]:=
path=AnglePath[{{4,0°},{3,120°},{2,120°},{1,-120°},{2,120°},{3,-120°},​​{6,-120°},{1,120°},{2,-120°},{4,-120°},{6,120°},{8,120°},{4,120°},{8,-120°},{12,120°},{6,120°}}];​​Graphics[{Line[path]}]
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It’s also similar to the Ammann Chair, a shape independently found by Karl Scherer. It’s based on the Golden Ratio:
In[]:=
path=RootReduce[With[{r=RootReduce[Sqrt[GoldenRatio]]},​​AnglePath[{{r^3,90°},{r^4,90°},{r^5,90°},{r^2,90°},{r^1,90°},{r^0,-90°},​​{r^2,0°},{r^5,-90°},{r^6,-90°},{r^3,-90°}}]]];​​Graphics[{Line[path],Style[Text[ResourceFunction["IntegerChop"][Quiet@N[Log[Sqrt[GoldenRatio],EuclideanDistance@@#]]],Mean[#],BackgroundWhite],16]&/@Partition[path,2,1]}]
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Karl Scherer wrote a large Demonstration for Irreptiles:
I also include several irreptiles in my resource function AlgebraicSubstitutionTiling:
In[]:=
Column[{Text@#,ResourceFunction["AlgebraicSubstitutionTiling"][#,1,{"N"}]},Alignment->Center]&/@{"Tritan3","Kite","Equithirds","AmmannChair","WaltonChair","TribonacciChordQuadrangle","PsiChordQuadrangle","RhoChordQuadrangle","SqrtTwo","SqrtPhi","SqrtPsi","SqrtRho","SqrtChi"}
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
Tritan3
,
Kite
,
Equithirds
,
AmmannChair
,
WaltonChair
,
TribonacciChordQuadrangle
,
PsiChordQuadrangle
,
RhoChordQuadrangle
,
SqrtTwo
,
SqrtPhi
,
SqrtPsi
,
SqrtRho
,
SqrtChi

I’ll need to add this new discovery in a few places. Note: A solution in the Plastic constant often implies a related solution with the supergolden ratio.

CITE THIS NOTEBOOK

New Vesa Timonen irreptile: pentagon cut into its own smaller copies & plastic constant powers sides​
by Ed Pegg​
Wolfram Community, STAFF PICKS, June 27, 2024
​https://community.wolfram.com/groups/-/m/t/3200723