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 - 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.

w \ h
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
0
2
0
1
3
0
0
0
4
0
1
0
1
5
0
0
0
0
0
6
0
1
0
1
0
1
7
0
0
0
0
0
0
0
8
0
≥1
0
≥1
0
≥1
?
≥1
9
0
0
0
0
0
0
0
?
0
10
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
11
0
0
0
0
0
0
0
?
0
0
0
12
0
≥1
0
≥1
0
≥1
?
≥1
?
≥1
?
≥1
13
0
0
0
0
0
0
0
?
0
0
0
?
0
14
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
15
0
0
0
0
0
0
0
?
0
0
0
?
0
0
0
16
0
≥1
0
≥1
0
≥1
?
≥1
?
≥1
?
≥1
?
≥1
?
≥1
17
0
0
0
0
0
0
0
?
0
0
0
?
0
0
0
?
0
18
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
19
0
0
0
0
0
0
0
?
0
0
0
?
0
0
0
?
0
0
0
20
0
≥1
0
≥1
0
≥1
?
≥1
?
≥1
?
≥1
?
≥1
?
≥1
?
≥1
?
≥1
N>0
x
2k
x
2k
x
2k
?
?
?
?
?
?
?
?
?
?
?
?
?
?

Smallest prime reptiles¶

Smallest prime reptile (4Ox2):

polyomino \ n²
O tetromino
1
1
1
1
1
1

Smallest tori tilings¶

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

w \ h
1
2
3
4
5
6
7
8
1
0
2
0
4
3
0
0
0
4
0
8
0
12
5
0
0
0
0
0
6
0
16
0
20
0
28
7
0
0
0
0
0
0
0
8
0
32
0
36
0
44
0
60

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 $A(w; h)$

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

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

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

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