Tilings Encyclopedia | M. Prunescu

Denote the elements of the field $F_{4}$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_{2}: w^{2} + w + 1 = 0$. Mothman is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = x^{2} + (w + 1) y^{2} + z^{2}$. This recurrent double sequence can be also obtained using a system of substitutions of type 2 -> 4 with 15 rules, as it follows.

Vampire

Denote the elements of the field $F_4$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_2: w^2 + w + 1 = 0$. Vampire is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = w x + (w + 1) x^2 + (w + 1) y^2 + w z + (w + 1) z^2$.

Treasure

Denote the elements of the field $F_4$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_2: w^2 + w + 1 = 0$. Treasure is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = w x + y + w z + 1$. This recurrent double sequence can be also obtained using a system of substitutions of type 2 -> 4 with 12 rules, as it follows.

Single Bat

Denote the elements of the field $F_4$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_2: w^2 + w + 1 = 0$. Single Bat is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = x + (w + 1) x^2 + w y^2 + z + (w + 1) z^2$.

Pairs of Squares

Denote the elements of the field $F_4$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_2: w^2 + w + 1 = 0$. Pairs of Squares is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = (w + 1) x + (w + 1) y^2 + (w + 1) z$.

Open Peano

Denote the elements of the field $F_4$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_2: w2 + w + 1 = 0$. Open Peano is a recurrent double sequence defined by $a(i, 0) = a(0, j) = w + 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = w x^2 + w y + w z^2$.

Infinity

Denote the elements of the field $F_{4}$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_{2}: w^{2} + w + 1 = 0$. Infinity is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = x^{2} + (w + 1) y^{2} + z$. This recurrent double sequence can be also obtained using a system of substitutions of type 2 -> 4 with 5 rules.

Dragonul

Denote the elements of the field $F_{4}$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_{2}: w^{2} + w + 1 = 0$. Dragonul is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z ) = x^{2} + w x + (w + 1) y^{2} + w y + w z$.

Diamond

Denote the elements of the field $F_{4}$ by $\{0, 1, w, w + 1\}$, where w satisfies the following equation with coefficients in $F_{2}: w^{2} + w + 1 = 0$. Diamond is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = w x^{2} + y + (w + 1) y^{2} + w z^{2}$.

Bat in Cone

Denote the elements of the field F4 by {0, 1, w, w + 1}, where w satisfies the following equation with coefficients in F2: w2 + w + 1 = 0. Bat in Cone is a recurrent double sequence defined by a(i, 0) = a(0, j) = 1 and a(i, j) = f( a(i, j-1), a(i-1, j-1), a(i-1, j) ), where f(x, y, z ) = x + x2 + w y + z + z2.

M. Prunescu

Discovered Tilings