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Algebraic Substitution Tilings

Source Notebook

Substitution tiling systems based on algebraic barycentric coordinates

Details

A large collection of famous substitution tiling systems.

Examples

Basic Examples (1) 

Data for the Pinwheel tiling:

In[1]:=
ResourceData[\!\(\* TagBox["\"\<Algebraic Substitution Tilings\>\"", #& , BoxID -> "ResourceTag-Algebraic Substitution Tilings-Input", AutoDelete->True]\)]["Pinwheel"]
Out[1]=

Scope & Additional Elements (2) 

Currently available substitution tilings:

In[2]:=
keys = Keys[ResourceData[\!\(\* TagBox["\"\<Algebraic Substitution Tilings\>\"", #& , BoxID -> "ResourceTag-Algebraic Substitution Tilings-Input", AutoDelete->True]\)]]
Out[2]=

The roots, based on SqrtSpace, with 1.47, 1.32 and 1.84 being the supergolden ratio, plastic constant and tribonacci constant:

In[3]:=
Tally[ResourceData[\!\(\* TagBox["\"\<Algebraic Substitution Tilings\>\"", #& , BoxID -> "ResourceTag-Algebraic Substitution Tilings-Input", AutoDelete->True]\)][#]["Root"] & /@ keys]
Out[3]=

The psi quad tiling, with the root being psi, ψ, the supergolden ratio:

In[4]:=
ps = ResourceData[\!\(\* TagBox["\"\<Algebraic Substitution Tilings\>\"", #& , BoxID -> "ResourceTag-Algebraic Substitution Tilings-Input", AutoDelete->True]\)]["PsiQuad"]
Out[4]=

The points of the psi quad tiling after algebraic conversion with SqrtSpace:

In[5]:=
pts = ResourceFunction["SqrtSpace"][ps["Root"], #] & /@ ps["AlgebraicPointList"]
Out[5]=

Show the points:

In[6]:=
Graphics[{EdgeForm[Black], White, Polygon[pts[[#]]] & /@ ps["PolygonReplacementRules"][[1, 2]], Green, Opacity[.2], Polygon[pts[[{1, 2, 3, 4}]]], Opacity[1], Black, Table[Style[Text[n, pts[[n]]], 20], {n, 1, 6}]}, ImageSize -> Small]
Out[6]=

Visualizations (2) 

Data for the Kites and Darts tiling system:

In[7]:=
kd = ResourceData[\!\(\* TagBox["\"\<Algebraic Substitution Tilings\>\"", #& , BoxID -> "ResourceTag-Algebraic Substitution Tilings-Input", AutoDelete->True]\)]["KitesandDarts"]
Out[7]=

Using AlgebraicSubstitutionTiling, show the substitution rule and three steps:

In[8]:=
Row[Table[ ResourceFunction["AlgebraicSubstitutionTiling"][{kd["Root"], kd["AlgebraicPointList"], kd["PolygonReplacementRules"], kd["PolygonType"]}, k], {k, 0, 3}]]
Out[8]=

Wolfram Research, "Algebraic Substitution Tilings" from the Wolfram Data Repository (2022)  

Data Resource History

  • Date Created:

Source Metadata

  • Title: Substitution Tilings
  • Creator: Ed Pegg Jr
  • Publisher: Wolfram Demonstrations Project
  • Date: April 2019
  • Citation:

See Also

Data Downloads

Publisher Information

  • Prepared for the Wolfram Data Repository By: Ed Pegg Jr
  • Publisher of Record: Wolfram Research