During a talk, I mention the Demonstration Tiles with Finite Convex Spectra. One particular tile was of interest, it’s basically a half square and equilateral triangle. It can be used for many interesting tilings. The tile can be seen in Math Magic.
Does the tile have a name? It does! According to OEIS A342285, it’s seen in Goldberg, M. (1955). “Central Tessellations,” Scripta Mathematica, 21, pp. 253-260. See Fig.7b.
goldberg=-1,-2-
3
,-1,-
3
,{0,0},1,-
3
;​​goldberg2=1,-2-
3
,{0,-2},{0,0},
3
,-1;​​goldberg3=1,-2-
3
,1,-
3
,{0,0},-1,-
3
;​​goldberg4=1,2+
3
,{0,2},{0,0},
3
,1;
In[]:=
Graphics[{EdgeForm[Black],Blue,Polygon[goldberg]},ImageSize->Small]
Out[]=
Here’s a quick program for lots of rotations and translations.
In[]:=
TranslateRotate[shape_,vec_,num_]:=​​Module{base},​​base=shape+Table[vec,Length[shape]];TableRotationMatrix
2nπ
num
.#1&/@base,{n,0,num-1};
With that, we can get the dodecagon in 48 pieces.
Graphics[EdgeForm[Black],Lighter[Blue],(Polygon/@TranslateRotate[goldberg,#,6])&/@{0,0},0,2+2
3
,1,-2-
3
,​​Lighter[Purple],(Polygon/@TranslateRotate[goldberg2,#,6])&/@1,-
3
,-2-
3
,3+2
3
,-4-2
3
,0,​​Green,(Polygon/@TranslateRotate[goldberg3,#,6])&/@-1,-2-
3
,​​Cyan,(Polygon/@TranslateRotate[goldberg4,#,6])&/@-4-2
3
,0]
Out[]=

CITE THIS NOTEBOOK

The Goldberg tile​
by Ed Pegg​
Wolfram Community, STAFF PICKS, April 20 2023
​https://community.wolfram.com/groups/-/m/t/2902586