In[]:=
ArrayTrim[matrix_]:=Transpose[DeleteCases[​​Transpose[DeleteCases[matrix,{0..}]],{0..}]]
In[]:=
IterateAddSquare[matrix_]:=Module[{newMatrix,pos},​​newMatrix=ArrayPad[ArrayTrim[matrix],1];pos=Position[​​ArrayPad[ArrayTrim[matrix],1],1];​​Map[ArrayPad[First[Sort[​​ResourceFunction["ArrayRotations"][ArrayTrim[#]]]],1]&,​​Catenate[Outer[If[newMatrix[[Sequence@@(#1+#2)]]==0,​​ReplacePart[newMatrix,#1+#2->1],Nothing]&,​​pos,ReIm[Exp[IPi/2#]]&/@Range[4],1]]]]
In[]:=
PolyominoArrays[UpTo[weight_?Positive]]:=Map[ArrayTrim,​​NestList[Union[Catenate[IterateAddSquare/@#]​​]&,{ArrayPad[{{1}},1]},weight-1],{2}];
In[]:=
PolyominoArrays[weight_?Positive]:=Map[ArrayTrim,​​Nest[Union[Catenate[IterateAddSquare/@#]​​]&,{ArrayPad[{{1}},1]},weight-1]];
In[]:=
PolyominoArrays[3]
Out[]=
{{{1,1,1}},{{0,1},{1,1}}}
In[]:=
ArrayPlot/@PolyominoArrays[3]
Out[]=

,

In[]:=
ArrayPlot/@PolyominoArrays[5]
Out[]=

,
,
,
,
,
,
,
,
,
,
,

In[]:=
ResourceFunction["PolyominoPlot"]/@PolyominoArrays[5]
Out[]=

,
,
,
,
,
,
,
,
,
,
,

In[]:=
InputForm

Out[]//InputForm=
Graphics[{{GrayLevel[1], EdgeForm[GrayLevel[0]], {},
{GrayLevel[0], Polygon[{{1/2, 1/2}, {1/2, 3/2}, {3/2, 3/2}, {5/2, 3/2}, {5/2, 5/2}, {5/2, 7/2}, {7/2, 7/2}, {7/2, 5/2}, {7/2, 3/2}, {7/2, 1/2}, {5/2, 1/2},
{3/2, 1/2}} -> {}]}}, {GrayLevel[1], EdgeForm[GrayLevel[0]], {},
{GrayLevel[1], Polygon[{{1/2, 3/2}, {1/2, 5/2}, {1/2, 7/2}, {3/2, 7/2}, {5/2, 7/2}, {5/2, 5/2}, {5/2, 3/2}, {3/2, 3/2}} -> {}]}}}]
In[]:=
[[1]]
Out[]=

,EdgeForm
,{},
,Polygon
Number of points: 12
Embedding dimension: 2
,
,EdgeForm
,{},
,Polygon
Number of points: 8
Embedding dimension: 2

In[]:=
Cases[#[[1]],{_,_,_,_,{x_}}:>x,Infinity]&/@(ResourceFunction["PolyominoPlot"]/@PolyominoArrays[5])
Out[]=
Polygon
Number of points: 12
Embedding dimension: 2
,Polygon
Number of points: 10
Embedding dimension: 2
,Polygon
Number of points: 4
Embedding dimension: 2
,Polygon
Number of points: 12
Embedding dimension: 2
,Polygon
Number of points: 4
Embedding dimension: 2
,Polygon
Number of points: 12
Embedding dimension: 2
,Polygon
Number of points: 8
Embedding dimension: 2
,Polygon
Number of points: 12
Embedding dimension: 2
,Polygon
Number of points: 12
Embedding dimension: 2
,Polygon
Number of points: 12
Embedding dimension: 2
,Polygon
Number of points: 8
Embedding dimension: 2
,Polygon
Number of points: 12
Embedding dimension: 2
,Polygon
Number of points: 12
Embedding dimension: 2
,Polygon
Number of points: 12
Embedding dimension: 2
,Polygon
Number of points: 12
Embedding dimension: 2
,Polygon
Number of points: 12
Embedding dimension: 2

​​test1=Module{unit,init,res,steps=15,ct=0},SeedRandom[235234];​​unit=Polygon
Number of points: 12
Embedding dimension: 2
;​​init=AddCandidates[​​InitializePolygonAggregation[{unit},{Range[Length@@unit]},unit]​​];​​Monitor[​​res=NestWhileList[(ct++;RandomAddPolygon[#])&,​​init,!FailureQ[#]&,1,steps],ct];​​out=res;​​res=If[FailureQ[Last[res]],$Failed,​​FixGraphicsScale[10]@RandomAggregationGraphics[#]&/@​​res]
In[]:=
Polygon
Number of points: 12
Embedding dimension: 2
//InputForm
Out[]//InputForm=
Polygon[{{3/2, 1/2}, {3/2, 3/2}, {1/2, 3/2}, {1/2, 5/2}, {3/2, 5/2}, {5/2, 5/2}, {5/2, 7/2}, {7/2, 7/2}, {7/2, 5/2}, {7/2, 3/2}, {5/2, 3/2}, {5/2, 1/2}} -> {}]

Regular Polygons

Multiway Polypolyominoes

Penrose tilings