Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.
Then click "Show" button.
You may also see list of all polyomino sets for which data is available here.
Area: 8.
Size: 3x4.
Holes: 0.
Order: 2.
Square order: 2.
Prime rectangles: ≥ 1.
Smallest rectangle and smallest square (4x4):
Blue number - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).
Green number - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).
Purple number - prime rectangle (unknown if weakly or strongly prime).
Red number - composite rectangle (which can be divided into two rectangles tileable by this set).
Gray number - it is unknown whether rectangle is prime or composite.
Question mark (?) - solution count is unknown.
Click on underlined numbers to view picture with one solution.
Smallest prime reptile (8Gx4):
$N(w; h)$ - number of ways to tile $w\times h$ rectangle (including symmetric solutions)
$T(w; h) = \begin{cases} 1, & N(w; h) \geq 1 \\ 0, & \text{else} \end{cases}$ - tileability function, $1$ if tiles rectangle, $0$ otherwise
$A(w; h) = \left(N(w; h)\right)^{\frac{1}{wh}}$ - average number of ways to tile cell in $w\times h$ rectangle (including symmetric solutions)
$G(T; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}T(w; h)x^wy^h$ - bivariate generating function of $T(w; h)$
$G(A; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}A(w; h)x^wy^h$ - bivariate generating function of $A(w; h)$
$N(w; h) = T(w; h) = A(w; h) = 0, \qquad w \not\equiv 0\pmod{4} \tag{1}$
$T(4n; 4m) = 1, \qquad n,m \geq 1 \tag{2}$
$N(4n; 4m) = 4^{nm}, \qquad n,m \geq 1 \tag{3}$
$G(T; x; y) = \frac{x^4y^4}{(1-x^4)(1-y^4)} \tag{4}$
$A(4n; 4m) = \sqrt[8]{2}, \qquad n,m \geq 1 \tag{5}$
$G(A; x; y) = \sqrt[8]{2}G(T; x; y) = \frac{\sqrt[8]{2}\,x^4y^4}{(1-x^4)(1-y^4)} \tag{6}$
