<|"Pinwheel" -> <|"Root" -> 1, "AlgebraicPointList" -> {{0, 0}, {1, 2}, {5, 0}, {1, 0}, {1, 1}, {3, 1}, {3, 0}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{1, 4, 2}, {4, 5, 6}, {6, 7, 4}, {2, 5, 6}, {6, 7, 3}}}, "PolygonType" -> {{1, 1, 1, 1, 1}}, "Name" -> "pinwheel tiling", "Discoverers" -> {"John Conway", "Charles Radin"}|>, "Domino" -> <|"Root" -> 1, "AlgebraicPointList" -> {{-1, 0}, {-1, 2}, {3, 2}, {3, 0}, {0, 0}, {0, 1}, {2, 1}, {2, 0}, {0, 2}, {2, 2}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{5, 6, 7, 8}, {6, 9, 10, 7}, {5, 1, 2, 9}, {4, 8, 10, 3}}}, "PolygonType" -> {{1, 1, 1, 1}}, "Name" -> "domino tiling", "Discoverers" -> {"Folklore"}|>, "Tromino" -> <|"Root" -> 1, "AlgebraicPointList" -> {{0, -2}, {4, -2}, {4, 2}, {2, 2}, {2, 0}, {0, 0}, {2, -2}, {2, -1}, {1, -1}, {1, 0}, {4, 0}, {3, 0}, {3, -1}, {3, 1}, {2, 1}}, "PolygonReplacementRules" -> {{1, 2, 3, 4, 5, 6} -> {{6, 1, 7, 8, 9, 10}, {7, 2, 11, 12, 13, 8}, {9, 13, 14, 15, 5, 10}, {11, 3, 4, 15, 14, 12}}}, "PolygonType" -> {{1, 1, 1, 1}}, "Name" -> "tromino tiling", "Discoverers" -> {"Folklore"}|>, "LTetromino" -> <|"Root" -> 1, "AlgebraicPointList" -> {{-2, 0}, {-2, 4}, {4, 4}, {4, 2}, {0, 2}, {0, 0}, {-2, 3}, {-1, 3}, {-1, 1}, {0, 1}, {0, 4}, {3, 4}, {3, 3}, {1, 3}, {1, 2}}, "PolygonReplacementRules" -> {{1, 2, 3, 4, 5, 6} -> {{6, 1, 7, 8, 9, 10}, {2, 11, 10, 9, 8, 7}, {5, 11, 12, 13, 14, 15}, {3, 4, 15, 14, 13, 12}}}, "PolygonType" -> {{1, 1, 1, 1}}, "Name" -> "L tetromino tiling", "Discoverers" -> {"Folklore"}|>, "PPentomino" -> <|"Root" -> 1, "AlgebraicPointList" -> {{0, -2}, {6, -2}, {6, 0}, {4, 0}, {4, 2}, {0, 2}, {3, -2}, {3, -1}, {2, -1}, {2, 0}, {0, 0}, {4, -1}, {3, 2}, {3, 1}, {2, 1}}, "PolygonReplacementRules" -> {{1, 2, 3, 4, 5, 6} -> {{1, 7, 8, 9, 10, 11}, {12, 5, 13, 14, 15, 9}, {2, 7, 8, 12, 4, 3}, {6, 13, 14, 15, 10, 11}}}, "PolygonType" -> {{1, 1, 1, 1}}, "Name" -> "P pentomino tiling", "Discoverers" -> {"Folklore"}|>, "Tritan3" -> <|"Root" -> 1, "AlgebraicPointList" -> {{-6, -6}, {-6, 0}, {0, 0}, {6, -6}, {-6, -3}, {-3, -3}, {0, -6}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{3, 6, 7, 4}, {1, 5, 6, 7}, {2, 5, 6, 3}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}}, "PolygonType" -> {{1, 2, 2}, {1}}, "Name" -> "semi-detached house tiling", "Discoverers" -> {"Dirk Frettlöh"}|>, "Tritan4" -> <|"Root" -> 1, "AlgebraicPointList" -> {{-4, 0}, {0, 0}, {0, 2}, {-2, 2}, {-2, 0}, {-2, 1}, {-3, 1}, {-1, 1}, {-1, 2}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{1, 5, 6, 7}, {7, 8, 9, 4}, {2, 5, 6, 8}, {2, 3, 9, 8}}}, "PolygonType" -> {{1, 1, 1, 1}}, "Name" -> "tritan rep-tile tiling", "Discoverers" -> {"Folklore"}|>, "TriangleDuo" -> <|"Root" -> 2, "AlgebraicPointList" -> {{{0}, {0}}, {{16}, {0}}, {{4}, {28}}, {{9}, {7}}, {{36}, {28}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{2, 4, 1}, {3, 4, 1}}, {5, 3, 1} -> {{4, 3, 5}, {3, 4, 1}}}, "PolygonType" -> {{1, 2}, {1, 2}}, "Name" -> "triangle duo tiling", "Discoverers" -> {"Ludwig Danzer", "Chaim Goodman-Strauss"}|>, "WaltonChair" -> <|"Root" -> 2, "AlgebraicPointList" -> {{{8}, {16}}, {{8}, {36}}, {{0}, {36}}, {{0}, {0}}, {{32}, {0}}, {{32}, {16}}, {{2}, {16}}, {{2}, {9}}, {{2}, {4}}, {{8}, {4}}, {{8}, {0}}, {{0}, {9}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4, 5, 6} -> {{10, 9, 7, 6, 5, 11}, {9, 8, 12, 4, 11, 10}, {7, 8, 12, 3, 2, 1}}, {1, 2, 3, 4, 5, 6} -> {{1, 2, 3, 4, 5, 6}}}, "PolygonType" -> {{1, 2, 2}, {1}}, "Name" -> "Walton chair tiling", "Discoverers" -> {"Dale Walton"}|>, "TwoTriangle" -> <|"Root" -> 2, "AlgebraicPointList" -> {{{0}, {0}}, {{64}, {0}}, {{4}, {28}}, {{16}, {0}}, {{9}, {7}}, {{49}, {7}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{4, 3, 1}, {6, 4, 2}, {4, 6, 5}}, {1, 2, 3} -> {{1, 2, 3}}}, "PolygonType" -> {{1, 2, 2}, {1}}, "Name" -> "two triangle tiling", "Discoverers" -> {"Ed Pegg Jr"}|>, "Drafter3" -> <|"Root" -> 3, "AlgebraicPointList" -> {{{4}, {0}}, {{Rational[9, 4]}, {Rational[3, 4]}}, {{0}, {0}}, {{1}, {0}}, {{1}, {Rational[1, 3]}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{5, 2, 1}, {5, 4, 1}, {5, 4, 3}}}, "PolygonType" -> {{1, 1, 1}}, "Name" -> "30-60-90 3 tiling", "Discoverers" -> {"Folklore"}|>, "Drafter4" -> <|"Root" -> 3, "AlgebraicPointList" -> {{{4}, {0}}, {{Rational[9, 4]}, {Rational[3, 4]}}, {{0}, {0}}, {{1}, {0}}, {{Rational[9, 4]}, {0}}, {{Rational[9, 16]}, {Rational[3, 16]}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{1, 5, 2}, {4, 5, 2}, {4, 6, 2}, {4, 6, 3}}}, "PolygonType" -> {{1, 1, 1, 1}}, "Name" -> "30-60-90 4 tiling", "Discoverers" -> {"Folklore"}|>, "HalfHex" -> <|"Root" -> 3, "AlgebraicPointList" -> {{{16}, {0}}, {{9}, {3}}, {{1}, {3}}, {{0}, {0}}, {{1}, {0}}, {{Rational[9, 4]}, {Rational[3, 4]}}, {{Rational[25, 4]}, {Rational[3, 4]}}, {{9}, {0}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{2, 7, 6, 3}, {2, 7, 8, 1}, {3, 6, 5, 4}, {5, 6, 7, 8}}}, "PolygonType" -> {{1, 1, 1, 1}}, "Name" -> "half-hex rep-tile tiling", "Discoverers" -> {"Folklore"}|>, "QuarterHex" -> <|"Root" -> 3, "AlgebraicPointList" -> {{{4}, {12}}, {{16}, {12}}, {{16}, {0}}, {{0}, {0}}, {{4}, {0}}, {{1}, {3}}, {{9}, {3}}, {{16}, {3}}, {{Rational[25, 4]}, {Rational[3, 4]}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{7, 8, 2, 1}, {7, 8, 3, 5}, {5, 9, 6, 4}, {7, 9, 6, 1}}}, "PolygonType" -> {{1, 1, 1, 1}}, "Name" -> "quarter-hex rep-tile tiling", "Discoverers" -> {"Folklore"}|>, "Sphinx" -> <|"Root" -> 3, "AlgebraicPointList" -> {{{9}, {0}}, {{-9}, {0}}, {{-1}, {12}}, {{0}, {3}}, {{4}, {3}}, {{0}, {0}}, {{-1}, {3}}, {{Rational[-9, 4]}, {Rational[3, 4]}}, {{Rational[-25, 4]}, {Rational[3, 4]}}, {{Rational[1, 4]}, {Rational[3, 4]}}, {{Rational[9, 4]}, {Rational[3, 4]}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4, 5} -> {{2, 6, 7, 8, 9}, {5, 7, 6, 10, 11}, {6, 1, 5, 11, 10}, {9, 3, 4, 7, 8}}}, "PolygonType" -> {{1, 1, 1, 1}}, "Name" -> "sphinx tiling", "Discoverers" -> {"Folklore"}|>, "Kite" -> <|"Root" -> 3, "AlgebraicPointList" -> {{{Rational[-1, 4]}, {Rational[-3, 4]}}, {{1}, {-3}}, {{Rational[25, 4]}, {Rational[-3, 4]}}, {{1}, {3}}, {{0}, {0}}, {{1}, {0}}, {{4}, {0}}, {{1}, {Rational[1, 3]}}, {{Rational[1, 4]}, {Rational[3, 4]}}, {{Rational[9, 4]}, {Rational[3, 4]}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{1, 5, 6, 2}, {6, 7, 3, 2}, {6, 8, 9, 5}, {6, 8, 10, 7}, {9, 8, 10, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}}, "PolygonType" -> {{1, 1, 2, 2, 2}, {1}}, "Name" -> "kite tiling", "Discoverers" -> {"Folklore"}|>, "Equithirds" -> <|"Root" -> 3, "AlgebraicPointList" -> {{{27}, {0}}, {{0}, {9}}, {{-27}, {0}}, {{3}, {0}}, {{0}, {1}}, {{-3}, {0}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{1, 4, 2}, {2, 6, 3}, {4, 5, 6}, {6, 5, 2}, {2, 5, 4}}, {1, 2, 3} -> {{1, 2, 3}}}, "PolygonType" -> {{1, 1, 2, 2, 2}, {1}}, "Name" -> "equithirds tiling", "Discoverers" -> {"Bill Kalahurka", "Ludwig Danzer"}|>, "Dog" -> <|"Root" -> 3, "AlgebraicPointList" -> {{{0}, {-12}}, {{-9}, {-12}}, {{-9}, {-3}}, {{0}, {0}}, {{Rational[9, 4]}, {Rational[-3, 4]}}, {{0}, {-3}}, {{-1}, {-3}}, {{-4}, {-12}}, {{Rational[-9, 4]}, {Rational[-75, 4]}}, {{Rational[-9, 4]}, {Rational[-3, 4]}}, {{1}, {-3}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4, 5} -> {{1, 6, 7, 8, 9}, {3, 2, 8, 7, 10}, {4, 10, 7, 11, 5}}}, "PolygonType" -> {{1, 1, 1}}, "Name" -> "Dog fractal", "Discoverers" -> {"Yoshiaki Araki\:200e"}|>, "Limhex" -> <|"Root" -> 3, "AlgebraicPointList" -> {{{Rational[9, 4]}, {3}}, {{1}, {Rational[3, 4]}}, {{Rational[9, 4]}, {0}}, {{1}, {Rational[-3, 4]}}, {{0}, {Rational[-3, 4]}}, {{Rational[-1, 4]}, {0}}, {{Rational[-9, 4]}, {0}}, {{-4}, {Rational[3, 4]}}, {{Rational[-9, 4]}, {3}}, {{-1}, {Rational[27, 4]}}, {{0}, {Rational[27, 4]}}, {{Rational[1, 4]}, {3}}, {{0}, {3}}, {{Rational[-1, 4]}, {3}}, {{Rational[-9, 16]}, {Rational[27, 16]}}, {{Rational[-25, 16]}, {Rational[27, 16]}}, {{Rational[-25, 16]}, {Rational[75, 16]}}, {{Rational[-9, 4]}, {Rational[27, 4]}}, {{Rational[-25, 16]}, {Rational[147, 16]}}, {{Rational[-9, 16]}, {Rational[147, 16]}}, {{Rational[-1, 16]}, {Rational[147, 16]}}, {{Rational[-1, 16]}, {Rational[75, 16]}}, {{-1}, {Rational[3, 4]}}, {{Rational[-9, 16]}, {Rational[3, 16]}}, {{-1}, {0}}, {{Rational[-49, 16]}, {Rational[3, 16]}}, {{Rational[-81, 16]}, {Rational[3, 16]}}, {{Rational[-25, 4]}, {Rational[3, 4]}}, {{Rational[-81, 16]}, {Rational[27, 16]}}, {{-4}, {3}}, {{Rational[9, 16]}, {Rational[27, 16]}}, {{Rational[1, 4]}, {Rational[3, 4]}}, {{Rational[9, 16]}, {Rational[3, 16]}}, {{Rational[1, 4]}, {0}}, {{0}, {0}}, {{Rational[-1, 16]}, {Rational[3, 16]}}, {{Rational[1, 16]}, {Rational[27, 16]}}, {{Rational[25, 16]}, {Rational[3, 16]}}, {{Rational[25, 16]}, {Rational[-3, 16]}}, {{Rational[9, 4]}, {Rational[-3, 4]}}, {{Rational[25, 16]}, {Rational[-27, 16]}}, {{Rational[9, 16]}, {Rational[-27, 16]}}, {{Rational[1, 16]}, {Rational[-27, 16]}}, {{Rational[1, 16]}, {Rational[-3, 16]}}, {{Rational[-9, 16]}, {Rational[-27, 16]}}, {{-1}, {Rational[-3, 4]}}, {{Rational[-9, 4]}, {Rational[-3, 4]}}, {{Rational[-49, 16]}, {Rational[-3, 16]}}, {{Rational[-1, 4]}, {Rational[-3, 4]}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} -> {{13, 14, 15, 16, 9, 17, 18, 19, 20, 21, 11, 22}, {15, 23, 24, 25, 7, 26, 27, 28, 29, 30, 9, 16}, {31, 32, 33, 34, 35, 36, 24, 23, 15, 14, 13, 37}, {35, 34, 33, 38, 3, 39, 40, 41, 42, 43, 5, 44}, {45, 46, 47, 48, 7, 25, 24, 36, 35, 44, 5, 49}}}, "PolygonType" -> {{1, 1, 1, 1, 1}}, "Name" -> "limhex tiling", "Discoverers" -> {"Joshua Socolar"}|>, "AmmannA4" -> <|"Root" -> Sqrt[2], "AlgebraicPointList" -> {{{Rational[-1, 2], Rational[-1, 4]}, {Rational[-5, 2], Rational[-7, 4]}}, {{1, Rational[1, 2]}, {Rational[-5, 2], Rational[-7, 4]}}, {{1, Rational[1, 2]}, {-1, Rational[-1, 2]}}, {{Rational[5, 2], Rational[7, 4]}, {-1, Rational[-1, 2]}}, {{Rational[5, 2], Rational[7, 4]}, {Rational[1, 2], Rational[1, 4]}}, {{0, 0}, {Rational[1, 2], Rational[1, 4]}}, {{0, 0}, {0, 0}}, {{Rational[-1, 2], Rational[-1, 4]}, {0, 0}}, {{0, 0}, {Rational[-5, 2], Rational[-7, 4]}}, {{0, 0}, {-2, -1}}, {{Rational[1, 2], Rational[-1, 4]}, {-2, -1}}, {{Rational[1, 2], Rational[-1, 4]}, {Rational[-1, 2], Rational[-1, 4]}}, {{-1, Rational[1, 2]}, {Rational[-1, 2], Rational[-1, 4]}}, {{-1, Rational[1, 2]}, {-1, Rational[-1, 2]}}, {{Rational[-1, 2], Rational[-1, 4]}, {-1, Rational[-1, 2]}}, {{1, Rational[1, 2]}, {Rational[-1, 2], Rational[1, 4]}}, {{Rational[1, 2], Rational[1, 4]}, {Rational[-1, 2], Rational[1, 4]}}, {{Rational[1, 2], Rational[1, 4]}, {0, 0}}, {{0, 0}, {Rational[-1, 2], Rational[-1, 4]}}, {{Rational[1, 2], Rational[-1, 4]}, {-1, Rational[-1, 2]}}, {{2, 1}, {Rational[-1, 2], Rational[1, 4]}}, {{2, 1}, {0, 0}}, {{Rational[5, 2], Rational[7, 4]}, {0, 0}}, {{1, Rational[1, 2]}, {Rational[1, 2], Rational[1, 4]}}, {{1, Rational[1, 2]}, {1, Rational[-1, 2]}}, {{Rational[1, 2], Rational[1, 4]}, {1, Rational[-1, 2]}}, {{Rational[1, 2], Rational[-1, 4]}, {Rational[1, 2], Rational[1, 4]}}, {{Rational[1, 2], Rational[-1, 4]}, {0, 0}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4, 5, 6, 7, 8} -> {{1, 9, 10, 11, 12, 13, 14, 15}, {3, 16, 17, 18, 7, 19, 12, 20}, {17, 21, 22, 23, 5, 24, 25, 26}, {3, 2, 9, 10, 11, 20}, {7, 8, 15, 14, 13, 19}, {7, 6, 24, 25, 26, 18}, {3, 4, 23, 22, 21, 16}}, {1, 2, 24, 27, 28, 8} -> {{1, 9, 10, 11, 12, 13, 14, 15}, {3, 16, 17, 18, 7, 19, 12, 20}, {3, 2, 9, 10, 11, 20}, {7, 8, 15, 14, 13, 19}, {27, 24, 16, 17, 18, 28}}}, "PolygonType" -> {{1, 1, 1, 2, 2, 2, 2}, {1, 1, 2, 2, 2}}, "Name" -> "Ammann A4 tiling", "Discoverers" -> {"Robert Ammann"}|>, "Labyrinth" -> <|"Root" -> Sqrt[2], "AlgebraicPointList" -> {{{-1, 0}, {-1, 0}}, {{1, 0}, {-1, 0}}, {{1, 0}, {1, 0}}, {{-1, 0}, {1, 0}}, {{-3, 2}, {-3, 2}}, {{3, -2}, {-3, 2}}, {{3, -2}, {3, -2}}, {{-3, 2}, {3, -2}}, {{-17, 12}, {-1, 0}}, {{17, -12}, {-1, 0}}, {{1, 0}, {-17, 12}}, {{1, 0}, {17, -12}}, {{17, -12}, {1, 0}}, {{-17, 12}, {1, 0}}, {{-1, 0}, {17, -12}}, {{-1, 0}, {-17, 12}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{5, 6, 7, 8}, {1, 9, 5, 16}, {2, 11, 6, 10}, {3, 13, 7, 12}, {4, 15, 8, 14}, {5, 9, 10, 6}, {6, 11, 12, 7}, {7, 13, 14, 8}, {8, 15, 16, 5}}, {1, 2, 7, 4} -> {{5, 6, 7, 8}, {1, 9, 5, 16}, {2, 11, 6, 10}, {4, 15, 8, 14}, {5, 9, 10, 6}, {8, 15, 16, 5}}, {1, 8, 7, 2} -> {{5, 6, 7, 8}, {1, 9, 5, 16}, {2, 11, 6, 10}, {5, 9, 10, 6}}}, "PolygonType" -> {{1, 2, 2, 2, 2, 3, 3, 3, 3}, {1, 2, 2, 2, 3, 3}, {1, 2, 2, 3}}, "Name" -> "labyrinth tiling", "Discoverers" -> {"Folklore"}|>, "AmmannChair" -> <|"Root" -> GoldenRatio, "AlgebraicPointList" -> {{{0, 0}, {-1, -1}}, {{3, 5}, {-1, -1}}, {{3, 5}, {1, 0}}, {{1, 2}, {1, 0}}, {{1, 2}, {2, 3}}, {{0, 0}, {2, 3}}, {{0, 1}, {1, 0}}, {{0, 1}, {0, 0}}, {{0, 0}, {0, 0}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4, 5, 6} -> {{2, 3, 7, 8, 9, 1}, {6, 5, 4, 7, 8, 9}}, {1, 2, 3, 4, 5, 6} -> {{1, 2, 3, 4, 5, 6}}}, "PolygonType" -> {{1, 2}, {1}}, "Name" -> "Ammann chair tiling", "Discoverers" -> {"Robert Ammann", "Karl Scherer"}|>, "AmmannPhiChair" -> <|"Root" -> GoldenRatio, "AlgebraicPointList" -> {{{0, 0}, {0, 0}}, {{0, 0}, {2, 3}}, {{2, 3}, {2, 3}}, {{2, 3}, {1, 1}}, {{5, 8}, {1, 1}}, {{5, 8}, {0, 0}}, {{1, 1}, {0, 0}}, {{1, 1}, {1, 0}}, {{1, 0}, {1, 0}}, {{1, 0}, {1, 1}}, {{0, 0}, {1, 1}}, {{2, 3}, {1, 0}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4, 5, 6} -> {{6, 5, 4, 12, 8, 7}, {3, 12, 9, 10, 11, 2}, {1, 11, 10, 9, 8, 7}}, {2, 1, 7, 8, 12, 3} -> {{3, 12, 9, 10, 11, 2}, {1, 11, 10, 9, 8, 7}}}, "PolygonType" -> {{1, 1, 2}, {1, 2}}, "Name" -> "Ammann phi chair tiling", "Discoverers" -> {"Robert Ammann"}|>, "KitesandDarts" -> <|"Root" -> GoldenRatio, "AlgebraicPointList" -> {{{0, 0}, {0, 0}}, {{Rational[1, 4], 0}, {Rational[7, 4], -1}}, {{2, -1}, {0, 0}}, {{Rational[1, 4], 0}, {Rational[-7, 4], 1}}, {{5, -3}, {0, 0}}, {{Rational[5, 4], Rational[-3, 4]}, {Rational[47, 4], Rational[-29, 4]}}, {{Rational[5, 4], Rational[-3, 4]}, {Rational[-47, 4], Rational[29, 4]}}, {{Rational[13, 4], -2}, {Rational[-7, 4], 1}}, {{Rational[5, 2], Rational[-5, 4]}, {Rational[9, 2], Rational[-11, 4]}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{2, 3, 5, 6}, {4, 7, 5, 3}, {1, 5, 7, 8}}, {4, 1, 5, 9} -> {{4, 7, 5, 3}, {1, 5, 7, 8}}}, "PolygonType" -> {{1, 1, 2}, {1, 2}}, "Name" -> "kites and darts tiling", "Discoverers" -> {"Roger Penrose"}|>, "RobinsonTriangle" -> <|"Root" -> GoldenRatio, "AlgebraicPointList" -> {{{-1, 0}, {0, 0}}, {{Rational[1, 2], Rational[-1, 4]}, {Rational[1, 2], Rational[1, 4]}}, {{2, -1}, {0, 0}}, {{Rational[-1, 4], 0}, {Rational[7, 4], -1}}, {{0, 0}, {0, 0}}, {{Rational[-1, 2], Rational[-3, 4]}, {Rational[1, 2], Rational[1, 4]}}, {{Rational[-1, 4], Rational[-1, 4]}, {Rational[3, 4], Rational[-1, 4]}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{2, 4, 5}, {2, 3, 5}, {4, 5, 1}}, {1, 6, 5} -> {{5, 7, 1}, {7, 1, 6}}}, "PolygonType" -> {{1, 1, 2}, {1, 2}}, "Name" -> "Robinson triangle tiling", "Discoverers" -> {"Raphael Robinson"}|>, "PenroseRhomb" -> <|"Root" -> GoldenRatio, "AlgebraicPointList" -> {{{1, 1}, {0, 0}}, {{Rational[1, 2], Rational[-1, 4]}, {Rational[-1, 2], Rational[-1, 4]}}, {{-1, 0}, {0, 0}}, {{Rational[1, 2], Rational[-1, 4]}, {Rational[1, 2], Rational[1, 4]}}, {{Rational[-5, 2], Rational[5, 4]}, {Rational[1, 2], Rational[1, 4]}}, {{0, 0}, {0, 0}}, {{Rational[1, 4], Rational[1, 4]}, {Rational[-3, 4], Rational[1, 4]}}, {{Rational[1, 4], Rational[1, 4]}, {Rational[3, 4], Rational[-1, 4]}}, {{Rational[5, 4], 0}, {Rational[3, 4], 1}}, {{Rational[-1, 4], 0}, {Rational[3, 4], 1}}, {{Rational[-1, 4], 0}, {Rational[-3, 4], -1}}, {{Rational[-1, 2], Rational[-3, 4]}, {Rational[1, 2], Rational[1, 4]}}, {{Rational[-5, 4], Rational[3, 4]}, {Rational[3, 4], Rational[-1, 4]}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{3, 5, 4, 6}, {6, 7, 1, 8}, {4, 9, 8, 6}}, {3, 10, 6, 11} -> {{3, 12, 10, 13}, {6, 8, 13, 3}}}, "PolygonType" -> {{1, 1, 2}, {1, 2}}, "Name" -> "Penrose rhomb tiling", "Discoverers" -> {"Roger Penrose"}|>, "Binary" -> <|"Root" -> GoldenRatio, "AlgebraicPointList" -> {{{5, 0}, {0, 0}}, {{Rational[5, 4], 0}, {Rational[-3, 4], -1}}, {{0, 0}, {0, 0}}, {{Rational[5, 4], 0}, {Rational[3, 4], 1}}, {{Rational[1, 4], Rational[1, 4]}, {Rational[3, 4], Rational[-1, 4]}}, {{1, 1}, {0, 0}}, {{Rational[1, 4], Rational[1, 4]}, {Rational[-3, 4], Rational[1, 4]}}, {{Rational[5, 2], Rational[3, 4]}, {Rational[1, 2], Rational[1, 4]}}, {{Rational[5, 2], Rational[3, 4]}, {Rational[-1, 2], Rational[-1, 4]}}, {{5, 5}, {0, 0}}, {{Rational[5, 4], Rational[5, 4]}, {Rational[-3, 4], Rational[1, 4]}}, {{Rational[5, 4], Rational[5, 4]}, {Rational[3, 4], Rational[-1, 4]}}, {{2, 3}, {0, 0}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{5, 6, 7, 3}, {8, 6, 5, 4}, {7, 6, 9, 2}, {8, 1, 9, 6}}, {10, 11, 3, 12} -> {{5, 6, 7, 3}, {5, 12, 13, 6}, {13, 11, 7, 6}}}, "PolygonType" -> {{1, 1, 1, 2}, {1, 2, 2}}, "Name" -> "binary tiling", "Discoverers" -> {"Claude Godrèche", "Frédéric Lançon"}|>, "BirdsandBees" -> <|"Root" -> GoldenRatio, "AlgebraicPointList" -> {{{Rational[1, 2], Rational[-1, 4]}, {Rational[-1, 2], Rational[-1, 4]}}, {{Rational[1, 4], Rational[1, 4]}, {Rational[-7, 4], Rational[-11, 4]}}, {{Rational[1, 4], Rational[1, 4]}, {Rational[-3, 4], Rational[1, 4]}}, {{Rational[5, 4], 2}, {Rational[-3, 4], -1}}, {{1, 0}, {0, 0}}, {{2, 3}, {0, 0}}, {{Rational[1, 4], Rational[1, 4]}, {Rational[3, 4], Rational[-1, 4]}}, {{Rational[5, 4], 2}, {Rational[3, 4], 1}}, {{Rational[1, 2], Rational[-1, 4]}, {Rational[1, 2], Rational[1, 4]}}, {{Rational[1, 4], Rational[1, 4]}, {Rational[7, 4], Rational[11, 4]}}, {{Rational[-1, 2], Rational[1, 4]}, {Rational[1, 2], Rational[1, 4]}}, {{Rational[-1, 4], Rational[-1, 4]}, {Rational[7, 4], Rational[11, 4]}}, {{Rational[-1, 4], Rational[-1, 4]}, {Rational[3, 4], Rational[-1, 4]}}, {{Rational[-5, 4], -2}, {Rational[3, 4], 1}}, {{-1, 0}, {0, 0}}, {{-2, -3}, {0, 0}}, {{Rational[-1, 4], Rational[-1, 4]}, {Rational[-3, 4], Rational[1, 4]}}, {{Rational[-5, 4], -2}, {Rational[-3, 4], -1}}, {{Rational[-1, 2], Rational[1, 4]}, {Rational[-1, 2], Rational[-1, 4]}}, {{Rational[-1, 4], Rational[-1, 4]}, {Rational[-7, 4], Rational[-11, 4]}}}, "PolygonReplacementRules" -> {{20, 14, 8, 2} -> {{1, 2, 4, 3}, {3, 4, 6, 5}, {5, 6, 8, 7}, {13, 14, 16, 15}, {15, 16, 18, 17}, {17, 18, 20, 19}, {19, 20, 2, 1}, {19, 13, 7, 1}}, {20, 14, 8, 2} -> {{7, 8, 10, 9}, {9, 10, 12, 11}, {11, 12, 14, 13}, {19, 20, 2, 1}, {19, 13, 7, 1}}}, "PolygonType" -> {{1, 2, 1, 1, 2, 1, 2, 1}, {1, 2, 1, 1, 2}}, "Name" -> "birds and bees", "Discoverers" -> {"Joshua Socolar"}|>, "PsiQuad" -> <|"Root" -> Root[-1 - #1^2 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{1, 0, 0}, {0, 0, 0}}, {{Rational[-1, 4], Rational[1, 2], Rational[-1, 4]}, {Rational[-1, 4], Rational[-1, 2], Rational[3, 4]}}, {{Rational[-1, 2], Rational[-1, 4], Rational[1, 4]}, {Rational[3, 2], Rational[-3, 4], Rational[-1, 4]}}, {{Rational[5, 2], Rational[1, 4], Rational[-5, 4]}, {Rational[3, 2], Rational[-3, 4], Rational[-1, 4]}}, {{Rational[19, 4], Rational[-7, 4], -1}, {Rational[-15, 4], Rational[-1, 4], 2}}, {{Rational[1, 2], Rational[1, 4], Rational[-1, 4]}, {Rational[-3, 2], Rational[3, 4], Rational[1, 4]}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{2, 3, 4, 5}, {5, 4, 1, 6}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}}, "PolygonType" -> {{1, 2}, {3}, {1}}, "Name" -> "psi quad tiling", "Discoverers" -> {"Ed Pegg Jr"}|>, "RhoQuad" -> <|"Root" -> Root[-1 - #1 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{1, 0, 0}, {0, 0, 0}}, {{1, Rational[1, 4], Rational[-3, 4]}, {0, Rational[-3, 4], Rational[1, 4]}}, {{0, Rational[-1, 4], Rational[-1, 4]}, {0, Rational[-3, 4], Rational[1, 4]}}, {{0, 0, Rational[-1, 4]}, {-1, 0, Rational[3, 4]}}, {{0, 0, 0}, {0, 0, 0}}, {{1, -1, Rational[1, 4]}, {-1, 0, Rational[3, 4]}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{4, 5, 2, 3}, {2, 5, 6, 1}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}}, "PolygonType" -> {{2, 3}, {1}, {2}}, "Name" -> "rho quad tiling", "Discoverers" -> {"Ed Pegg Jr"}|>, "PsiTriple" -> <|"Root" -> Root[-1 - #1^2 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{0, 0, 0}, {0, 0, 0}}, {{1, 0, 0}, {0, 0, 0}}, {{Rational[-1, 4], Rational[1, 2], Rational[-1, 4]}, {Rational[-1, 4], Rational[-1, 2], Rational[3, 4]}}, {{Rational[-1, 2], Rational[-1, 4], Rational[1, 4]}, {Rational[3, 2], Rational[-3, 4], Rational[-1, 4]}}, {{Rational[-1, 2], Rational[-1, 4], Rational[-3, 4]}, {Rational[3, 2], Rational[-3, 4], Rational[-1, 4]}}, {{Rational[1, 4], Rational[-1, 4], 0}, {Rational[-1, 4], Rational[-3, 4], 0}}}, "PolygonReplacementRules" -> {{2, 3, 5} -> {{4, 1, 3}, {3, 1, 2}, {5, 4, 3}}, {2, 3, 5} -> {{2, 3, 5}}, {2, 3, 5} -> {{2, 3, 5}}, {2, 3, 5} -> {{2, 3, 5}}}, "PolygonType" -> {{4, 3, 2}, {1}, {2}, {3}}, "Name" -> "psi triple tiling", "Discoverers" -> {"Ed Pegg Jr"}|>, "TribonacciChordQuadrangle" -> <|"Root" -> Root[-1 - #1 - #1^2 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{0, 0, 0}, {0, 0, 0}}, {{0, 0, 0}, {1, 1, 1}}, {{3, Rational[9, 2], Rational[11, 2]}, {3, Rational[9, 2], Rational[11, 2]}}, {{7, 11, 13}, {0, 0, 0}}, {{1, 1, 1}, {0, 0, 0}}, {{1, 2, 3}, {0, 0, 0}}, {{0, Rational[-1, 2], Rational[1, 2]}, {1, Rational[3, 2], Rational[3, 2]}}, {{1, 2, 3}, {2, 3, 4}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{3, 8, 6, 4}, {6, 5, 7, 8}, {7, 2, 1, 5}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}}, "PolygonType" -> {{1, 2, 3}, {1}, {2}}, "Name" -> "tribonacci chord quadrangle 3-3", "Discoverers" -> {"Folklore", "Laurenz Andritz"}|>, "PsiChordQuadrangle" -> <|"Root" -> Root[-1 - #1^2 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{0, 0, 0}, {3, 2, 4}}, {{0, 0, 0}, {0, 0, 0}}, {{4, 3, 6}, {0, 0, 0}}, {{5, 4, 8}, {2, 0, 1}}, {{1, 2, 1}, {1, 1, 3}}, {{3, 1, 4}, {1, 1, 2}}, {{0, 0, 1}, {0, 0, 0}}, {{3, 1, 4}, {0, 0, 0}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{1, 5, 7, 2}, {7, 8, 6, 5}, {6, 4, 3, 8}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}}, "PolygonType" -> {{2, 3, 4}, {1}, {2}, {3}}, "Name" -> "psi chord quadrangle 4-3", "Discoverers" -> {"Laurenz Andritz"}|>, "PsiWedge" -> <|"Root" -> Root[-1 - #1^2 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{0, 0, 0}, {0, 0, 0}}, {{0, 0, 0}, {0, 3, -2}}, {{0, 0, 0}, {1, 0, 0}}, {{1, -2, 1}, {0, 0, 0}}, {{1, -2, 1}, {-1, 1, 0}}, {{1, -2, 1}, {1, 0, 0}}, {{0, 0, 1}, {0, 0, 0}}, {{4, -8, 4}, {-1, 1, 0}}, {{-1, 1, 0}, {1, 0, 0}}}, "PolygonReplacementRules" -> {{9, 3, 1, 7} -> {{8, 5, 4, 7}, {2, 1, 4, 5}, {5, 6, 3, 2}, {9, 6, 5, 8}}, {9, 3, 1, 7} -> {{9, 3, 1, 7}}, {9, 3, 1, 7} -> {{9, 3, 1, 7}}}, "PolygonType" -> {{1, 2, 2, 3}, {1}, {2}}, "Name" -> "psi wedge tiling", "Discoverers" -> {"Dieter Steemann"}|>, "PsiTrap" -> <|"Root" -> Root[-1 - #1^2 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{-1, 2, -1}, {-8, 8, -1}}, {{1, -2, 1}, {-8, 8, -1}}, {{1, 2, 1}, {-8, 8, -1}}, {{-1, 0, 0}, {0, 0, 0}}, {{1, 0, 0}, {0, 0, 0}}, {{9, 0, 0}, {0, 0, 0}}, {{7, -3, 0}, {24, -9, -6}}, {{-1, -1, -1}, {2, 1, -5}}, {{1, 1, 1}, {2, 1, -5}}, {{5, 1, 5}, {2, 1, -5}}}, "PolygonReplacementRules" -> {{10, 3, 1, 8} -> {{9, 5, 4, 8}, {2, 3, 6, 7}, {10, 9, 7, 6}, {4, 1, 2, 5}}, {10, 3, 1, 8} -> {{10, 3, 1, 8}}, {10, 3, 1, 8} -> {{10, 3, 1, 8}}}, "PolygonType" -> {{1, 2, 2, 3}, {1}, {2}}, "Name" -> "psi trap tiling", "Discoverers" -> {"Dale Walton"}|>, "TribonacciTrap" -> <|"Root" -> Root[-1 - #1 - #1^2 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{0, 0, 0}, {0, 0, 0}}, {{4, 6, 7}, {0, 0, 0}}, {{Rational[9, 4], Rational[7, 2], Rational[17, 4]}, {Rational[11, 4], Rational[7, 2], Rational[19, 4]}}, {{Rational[1, 4], Rational[1, 2], Rational[1, 4]}, {Rational[11, 4], Rational[7, 2], Rational[19, 4]}}, {{Rational[-1, 4], 0, Rational[1, 4]}, {Rational[5, 4], 2, Rational[3, 4]}}, {{Rational[1, 2], Rational[1, 2], 0}, {Rational[3, 2], Rational[3, 2], 4}}, {{Rational[1, 4], Rational[3, 2], Rational[1, 4]}, {Rational[11, 4], Rational[7, 2], Rational[19, 4]}}, {{Rational[1, 2], Rational[1, 2], 1}, {Rational[3, 2], Rational[3, 2], 4}}, {{Rational[3, 4], 1, Rational[5, 4]}, {Rational[5, 4], 2, Rational[3, 4]}}, {{Rational[5, 4], 2, 1}, {Rational[-1, 4], 0, 1}}, {{1, 2, 2}, {0, 0, 0}}, {{3, 4, Rational[21, 4]}, {1, 2, Rational[7, 4]}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{7, 10, 12, 3}, {2, 12, 10, 11}, {1, 11, 9, 5}, {5, 9, 8, 6}, {6, 8, 7, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}}, "PolygonType" -> {{1, 1, 1, 2, 3}, {1}, {2}}, "Name" -> "tribonacci trapezoid tiling", "Discoverers" -> {"Dale Walton"}|>, "RhoChordQuadrangle" -> <|"Root" -> Root[-1 - #1 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{0, 0, 0}, {2, 3, 2}}, {{0, 0, 0}, {0, 0, 0}}, {{3, 5, 4}, {0, 0, 0}}, {{Rational[93, 25], Rational[162, 25], Rational[121, 25]}, {Rational[17, 25], Rational[28, 25], Rational[24, 25]}}, {{Rational[17, 25], Rational[28, 25], Rational[24, 25]}, {Rational[28, 25], Rational[52, 25], Rational[41, 25]}}, {{2, 3, 2}, {1, 2, 1}}, {{0, 0, 1}, {0, 0, 0}}, {{2, 3, 2}, {0, 0, 0}}}, "PolygonReplacementRules" -> {{1, 2, 3, 4} -> {{1, 5, 7, 2}, {7, 8, 6, 5}, {6, 4, 3, 8}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}, {1, 2, 3, 4} -> {{1, 2, 3, 4}}}, "PolygonType" -> {{3, 4, 5}, {1}, {2}, {3}, {4}}, "Name" -> "rho chord quadrangle 5-3", "Discoverers" -> {"Laurenz Andritz"}|>, "SqrtTwo" -> <|"Root" -> 2, "AlgebraicPointList" -> {{{18}, {0}}, {{Rational[9, 8]}, {Rational[63, 8]}}, {{-18}, {0}}, {{-2}, {0}}, {{2}, {0}}, {{Rational[-9, 8]}, {Rational[7, 8]}}, {{Rational[1, 8]}, {Rational[7, 8]}}, {{Rational[49, 32]}, {Rational[63, 32]}}, {{Rational[81, 8]}, {Rational[7, 8]}}, {{Rational[-1, 2]}, {Rational[7, 2]}}, {{Rational[1, 2]}, {Rational[7, 2]}}, {{Rational[169, 32]}, {Rational[7, 32]}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{10, 4, 3}, {1, 9, 5}, {5, 7, 4}, {2, 8, 9}, {6, 10, 11}, {2, 11, 10}, {11, 7, 6}, {4, 6, 7}, {7, 8, 2}, {12, 5, 7}, {7, 8, 12}, {9, 12, 8}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}}, "PolygonType" -> {{2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4}, {1}, {2}, {3}}, "Name" -> "sqrt two tiling", "Discoverers" -> {"Ed Pegg Jr"}|>, "SqrtPhi" -> <|"Root" -> GoldenRatio, "AlgebraicPointList" -> {{{8, 12}, {0, 0}}, {{Rational[-1, 2], Rational[1, 4]}, {Rational[25, 2], Rational[81, 4]}}, {{-45, -72}, {0, 0}}, {{-1, -1}, {0, 0}}, {{0, 0}, {0, 0}}, {{1, 0}, {0, 0}}, {{Rational[-5, 2], Rational[-15, 4]}, {Rational[1, 2], Rational[5, 4]}}, {{Rational[1, 2], Rational[3, 4]}, {Rational[1, 2], Rational[5, 4]}}, {{Rational[9, 2], Rational[27, 4]}, {Rational[1, 2], Rational[5, 4]}}, {{Rational[-5, 4], -2}, {Rational[7, 4], 3}}, {{Rational[13, 4], 4}, {Rational[7, 4], 3}}, {{Rational[-29, 4], Rational[-45, 4]}, {Rational[19, 4], Rational[31, 4]}}, {{Rational[-13, 4], Rational[-21, 4]}, {Rational[19, 4], Rational[31, 4]}}, {{Rational[-1, 4], Rational[-1, 4]}, {Rational[19, 4], Rational[31, 4]}}, {{Rational[-17, 4], -6}, {Rational[7, 4], 3}}, {{-20, -32}, {0, 0}}, {{Rational[-73, 4], -28}, {Rational[7, 4], 3}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{4, 15, 16}, {2, 14, 12}, {2, 14, 11}, {13, 14, 5}, {17, 16, 3}, {16, 17, 15}, {11, 8, 14}, {5, 8, 14}, {1, 9, 6}, {15, 12, 17}, {12, 13, 7}, {5, 4, 10}, {6, 8, 9}, {7, 10, 13}, {9, 11, 8}, {10, 7, 4}, {5, 6, 8}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}}, "PolygonType" -> {{4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}}, "Name" -> "sqrt phi tiling", "Discoverers" -> {"Ed Pegg Jr"}|>, "SqrtPsi" -> <|"Root" -> Root[-1 - #1^2 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{4, 3, 5}, {0, 0, 0}}, {{Rational[1, 4], 0, Rational[1, 4]}, {Rational[7, 4], 1, Rational[11, 4]}}, {{-3, -2, -4}, {0, 0, 0}}, {{0, 0, 0}, {0, 0, 0}}, {{1, 1, 2}, {0, 0, 0}}, {{Rational[1, 4], Rational[1, 4], Rational[3, 4]}, {Rational[3, 4], Rational[-1, 4], Rational[1, 4]}}, {{0, Rational[5, 4], 1}, {1, Rational[3, 4], 1}}, {{Rational[1, 4], Rational[1, 4], Rational[-1, 4]}, {Rational[3, 4], Rational[-1, 4], Rational[1, 4]}}, {{Rational[-5, 4], Rational[-1, 4], Rational[-7, 4]}, {Rational[3, 4], Rational[-1, 4], Rational[1, 4]}}, {{0, -1, 0}, {0, 0, 0}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{2, 8, 9}, {1, 5, 7}, {4, 6, 2}, {5, 6, 4}, {10, 9, 3}, {9, 10, 8}, {7, 6, 5}, {6, 7, 2}, {8, 4, 10}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}}, "PolygonType" -> {{4, 5, 5, 6, 6, 6, 7, 7, 8}, {1}, {2}, {3}, {4}, {5}, {6}, {7}}, "Name" -> "sqrt psi tiling", "Discoverers" -> {"Ed Pegg Jr"}|>, "SqrtRho" -> <|"Root" -> Root[-1 - #1 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{1, 1, 0}, {0, 0, 0}}, {{Rational[-9, 4], Rational[-7, 2], Rational[-5, 2]}, {Rational[11, 4], Rational[9, 2], Rational[7, 2]}}, {{-16, -28, -20}, {0, 0, 0}}, {{-4, -7, -5}, {0, 0, 0}}, {{0, 0, 0}, {0, 0, 0}}, {{Rational[-25, 4], Rational[-25, 4], Rational[-25, 4]}, {Rational[3, 4], Rational[11, 4], Rational[-5, 4]}}, {{-8, Rational[-57, 4], -11}, {0, Rational[3, 4], 1}}, {{-2, Rational[-9, 4], -1}, {0, Rational[3, 4], 1}}, {{0, Rational[-1, 4], 0}, {0, Rational[3, 4], 1}}, {{Rational[-13, 4], Rational[-21, 4], -3}, {Rational[3, 4], Rational[7, 4], 1}}, {{-4, -5, -4}, {0, 3, 4}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{4, 8, 5}, {4, 7, 3}, {2, 8, 4}, {9, 8, 2}, {5, 9, 8}, {7, 6, 11}, {1, 5, 9}, {4, 6, 7}, {10, 6, 4}, {11, 10, 6}, {2, 11, 10}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}}, "PolygonType" -> {{5, 5, 6, 7, 8, 8, 9, 9, 10, 11, 12}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}, {11}}, "Name" -> "sqrt rho tiling", "Discoverers" -> {"Ed Pegg Jr"}|>, "SqrtChi" -> <|"Root" -> Root[-1 - #1 + #1^4 & , 2, 0], "AlgebraicPointList" -> {{{4, 7, 6, 5}, {0, 0, 0, 0}}, {{Rational[1, 4], 0, 0, 0}, {Rational[15, 4], 7, 6, 5}}, {{-6, -11, -9, -7}, {0, 0, 0, 0}}, {{-1, -1, -1, -2}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}}, {{3, 4, 2, 3}, {0, 0, 0, 0}}, {{Rational[-13, 4], Rational[-19, 4], Rational[-11, 4], Rational[-13, 4]}, {Rational[-1, 4], Rational[1, 4], Rational[5, 4], Rational[3, 4]}}, {{Rational[-11, 2], Rational[-19, 4], Rational[1, 4], Rational[1, 2]}, {Rational[-7, 2], Rational[-11, 4], Rational[13, 4], Rational[5, 2]}}, {{Rational[-1, 4], Rational[-1, 2], Rational[-1, 4], 0}, {Rational[-1, 4], Rational[1, 2], Rational[3, 4], 0}}, {{1, 0, Rational[1, 4], Rational[-3, 4]}, {0, 0, Rational[-1, 4], Rational[3, 4]}}, {{Rational[1, 4], Rational[3, 4], Rational[3, 4], Rational[1, 4]}, {Rational[-1, 4], Rational[1, 4], Rational[5, 4], Rational[3, 4]}}, {{Rational[9, 4], Rational[15, 4], Rational[11, 4], Rational[9, 4]}, {Rational[-1, 4], Rational[1, 4], Rational[5, 4], Rational[3, 4]}}, {{-1, Rational[-7, 4], Rational[-3, 2], Rational[-5, 4]}, {1, Rational[9, 4], Rational[3, 2], Rational[7, 4]}}, {{Rational[-1, 2], Rational[-5, 4], Rational[-1, 2], 1}, {Rational[1, 2], Rational[3, 4], Rational[3, 2], 2}}, {{Rational[-1, 2], Rational[-1, 2], Rational[1, 2], Rational[1, 4]}, {Rational[3, 2], Rational[7, 2], Rational[5, 2], Rational[3, 4]}}, {{Rational[1, 4], 0, Rational[1, 4], Rational[3, 4]}, {Rational[11, 4], 5, Rational[15, 4], Rational[13, 4]}}, {{Rational[1, 2], Rational[-1, 4], Rational[1, 4], Rational[-1, 4]}, {Rational[3, 2], Rational[13, 4], Rational[11, 4], Rational[9, 4]}}, {{Rational[-5, 4], Rational[-3, 2], Rational[3, 4], 1}, {Rational[7, 4], Rational[9, 2], Rational[19, 4], 5}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{4, 7, 3}, {5, 11, 6}, {12, 11, 16}, {15, 11, 5}, {10, 14, 8}, {9, 4, 7}, {6, 12, 11}, {13, 14, 18}, {16, 15, 11}, {17, 14, 10}, {5, 9, 4}, {8, 13, 14}, {1, 6, 12}, {2, 16, 15}, {18, 17, 14}, {7, 8, 13}, {2, 18, 17}, {5, 10, 9}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}}, "PolygonType" -> {{10, 10, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 17, 18, 19}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}, {11}, {12}, {13}, {14}, {15}, {16}, {17}, {18}}, "Name" -> "sqrt chi tiling", "Discoverers" -> {"Ed Pegg Jr"}|>, "Harvest" -> <|"Root" -> Root[-1 - #1 + #1^3 & , 1, 0], "AlgebraicPointList" -> {{{0, 0, 0}, {0, 0, 0}}, {{25, 50, 25}, {0, 0, 0}}, {{68, 112, 96}, {57, 113, 79}}, {{-47, 177, -9}, {-23, -32, 44}}, {{48, 32, -44}, {-23, -32, 44}}, {{-28, 48, 109}, {33, 22, 1}}, {{-188, 8, 164}, {188, 117, -139}}, {{-8, 28, 24}, {33, 22, 1}}, {{Rational[-1807, 49], Rational[-263, 49], Rational[2571, 49]}, {Rational[2407, 49], Rational[913, 49], Rational[-1496, 49]}}, {{Rational[-106668, 2401], Rational[26788, 2401], Rational[71404, 2401]}, {Rational[2407, 49], Rational[913, 49], Rational[-1496, 49]}}, {{-100, -75, 125}, {-23, -32, 44}}, {{-150, -125, 250}, {-23, -32, 44}}, {{Rational[-5475, 49], Rational[-1950, 49], Rational[9025, 49]}, {Rational[-1807, 49], Rational[-263, 49], Rational[2571, 49]}}, {{Rational[2652, 49], Rational[10468, 49], Rational[-3456, 49]}, {Rational[-1807, 49], Rational[-263, 49], Rational[2571, 49]}}}, "PolygonReplacementRules" -> {{5, 2, 3} -> {{7, 6, 3}, {4, 2, 5}, {9, 8, 6}, {13, 12, 11}, {8, 7, 6}, {10, 9, 8}, {14, 13, 12}, {12, 4, 14}, {11, 10, 9}, {6, 14, 13}, {5, 11, 10}}, {5, 2, 3} -> {{5, 2, 3}}, {5, 2, 3} -> {{5, 2, 3}}, {1, 2, 3} -> {{1, 5, 2}, {5, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}, {1, 2, 3} -> {{1, 2, 3}}}, "PolygonType" -> {{2, 3, 4, 4, 5, 6, 6, 6, 8, 8, 10}, {1}, {2}, {6, 3}, {4}, {5}, {6}, {7}, {8}, {9}}, "Name" -> "harvest substitution tiling", "Discoverers" -> {"Andrew Hudson"}|>, "QuarticPinwheel" -> <|"Root" -> Root[1 - #1 - 2*#1^2 - #1^3 + #1^4 & , 2, 0], "AlgebraicPointList" -> {{{0, 1, -1, 1}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 1, 0}}, {{1, -1, -2, -1}, {0, 0, 0, 0}}, {{-1, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}}, {{1, 0, 0, 0}, {0, 0, 0, 0}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{2, 4, 3}, {1, 6, 2}, {4, 5, 2}, {6, 5, 2}}, {1, 2, 3} -> {{1, 2, 3}}, {2, 5, 3} -> {{2, 4, 3}, {4, 5, 2}}, {2, 5, 3} -> {{2, 5, 3}}}, "PolygonType" -> {{1, 2, 3, 3}, {1}, {2, 4}, {3}}, "Name" -> "quartic pinwheel tiling", "Discoverers" -> {"Edmund Harriss"}|>, "WaltonQuintic" -> <|"Root" -> Root[-1 - #1^3 + #1^5 & , 1, 0], "AlgebraicPointList" -> {{{0, 0, 0, 0, 4}, {0, 0, 0, 0, 0}}, {{2, -1, 0, 2, -2}, {2, 3, 0, 6, 2}}, {{-4, -4, 0, -8, -4}, {0, 0, 0, 0, 0}}, {{-1, 3, 3, -2, 0}, {1, -3, 1, 2, 0}}, {{0, -4, 0, 0, -4}, {0, 0, 0, 0, 0}}, {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}, {{0, -16, -4, 0, 0}, {0, 0, 0, 0, 0}}, {{5, -7, 1, -2, -4}, {1, -3, 1, 2, 0}}, {{1, -3, 1, 2, -4}, {1, -3, 1, 2, 0}}, {{1, 1, 1, -2, 0}, {1, -3, 1, 2, 0}}, {{3, 3, -1, -2, 0}, {1, -3, 1, 2, 0}}}, "PolygonReplacementRules" -> {{1, 2, 3} -> {{5, 6, 2}, {1, 6, 2}, {3, 5, 2}}, {1, 6, 2} -> {{1, 4, 6}, {11, 10, 6}, {4, 10, 2}, {4, 11, 6}}, {3, 5, 2} -> {{7, 5, 8}, {9, 8, 5}, {8, 9, 2}, {3, 7, 8}}, {1, 2, 3} -> {{1, 2, 3}}, {3, 5, 2} -> {{3, 5, 2}}, {3, 5, 2} -> {{3, 5, 2}}, {3, 5, 2} -> {{3, 5, 2}}, {1, 2, 3} -> {{1, 2, 3}}}, "PolygonType" -> {{1, 2, 3}, {4, 8, 2, 7}, {8, 8, 3, 7}, {1}, {3}, {5}, {6}, {4}}, "Name" -> "Walton quintic tiling", "Discoverers" -> {"Dale Walton"}|>|>