POLYOMINO TILINGS

Polyomino Tilings

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.


O tetromino

Area: 4.

Perimeter: 8.

Size: 2x2.

Is rectangular: yes.

Is convex: yes.

Holes: 0.

Order: 1.

Square order: 1.

Odd order: 1.

Prime rectangles: 1.

Smallest rectangle tilings

Smallest rectangle and smallest square and smallest odd rectangle (2x2):

Rectangle tilings' solutions count (including symmetric)

Blue number (P) - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).

Green number (W) - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).

Red number (C) - 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.

w \ h123456
10
2011P
3000
4011C011C
500000
6011C011C011C
N>0x2kx2kx2k

Smallest prime reptiles

Smallest prime reptile (4Ox2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
O tetromino11P1P1C1P1C

Smallest tori tilings

Smallest torus and smallest square torus and smallest odd torus (2x2):

Tori tilings' solutions count (including translations)

w \ h12345678
100
20044
3000000
40088001212
50000000000
6001616002020002828
700000000000000
8003232003636004444006060

Smallest Baiocchi figures

Smallest Baiocchi figure (area 4):

Formulas

$N(w; h)$ - 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 $A(w; h)$

$N(n; 2m + 1) = T(n; 2m + 1) = 0 \tag{1}$

$N(2n; 2m) = T(2n; 2m) = A(2n; 2m) = 1 \tag{2}$

$N(w; h) = T(w; h) = A(w; h) = \frac{1}{4} \left(1+(-1)^w\right) \left(1+(-1)^h\right) \tag{3}$

$G(T; x; y) = G(A; x; y) = \frac{x^2 y^2}{\left(1-x^2\right)\left(1-y^2\right)} \tag{4}$

See Also

L tetrominoT tetromino