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.


Monominoes

Area: 1.

Perimeter: 4.

Size: 1x1.

Is rectangular: yes.

Is convex: yes.

Holes: 0.

Order: 1.

Square order: 1.

Odd order: 1.

Prime rectangles: 1.

Some facts:

Smallest rectangle tilings

Smallest rectangle and smallest square and smallest odd rectangle (1x1):

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
111P
211C11C
311C11C11C
411C11C11C11C
511C11C11C11C11C
611C11C11C11C11C11C
N>0allallallallallall

Smallest prime reptiles

Smallest prime reptile (1Ox2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
O monomino11P1P1C1P1C

Smallest tori tilings

Smallest torus and smallest square torus and smallest odd torus (1x1):

Tori tilings' solutions count (including translations)

w \ h123456
111
21111
3111111
411111111
51111111111
6111111111111

Smallest Baiocchi figures

Smallest Baiocchi figure (area 1):

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 $A(w; h)$

$N(w; h) = 1 \tag{1}$

$T(w; h) = 1 \tag{2}$

$A(w; h) = 1 \tag{3}$

$G(T; x; y) = \frac{xy}{(1-x)(1-y)} \tag{4}$

$G(A; x; y) = \frac{xy}{(1-x)(1-y)} \tag{5}$

See Also

I tetromino, L tetromino, O tetromino and T tetrominoP1 11-omino