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.


L tetromino

Area: 4.

Perimeter: 10.

Size: 2x3.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 2.

Square order: 4.

Odd order: ∞.

Prime rectangles: 2.

Some facts:

Smallest rectangle tilings

Smallest rectangle (2x4):

Smallest square (4x4):

No odd rectangles exist.

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
N>0
1
0
2
0
0
3
0
0
0
4
0
2
0
10
5
0
0
0
0
0
6
0
0
0
42
0
0
7
0
0
0
0
0
0
0
8
0
4
4
182
436
4340
16708
141970
9
0
0
0
0
0
0
0
611468
0
10
0
0
0
790
0
0
0
4417084
0
0
11
0
0
0
0
0
0
0
≥1
0
0
0
12
0
8
0
3432
0
517450
0
≥1
0
≥1
0
≥1
13
0
0
0
0
0
0
0
≥1
0
0
0
0
0
14
0
0
0
14914
0
0
0
≥1
0
0
0
≥1
0
0
15
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
16
0
16
24
64814
443144
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
17
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
18
0
0
0
281680
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
0
0
19
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
0
20
0
32
0
1224182
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
21
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
0
0
8k
22
0
0
0
5320310
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
4k
23
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
0
0
8k
24
0
64
144
23122148
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
N>0
x
4k
8k
2k
8k
4k
8k
all
8k
4k
8k
2k
8k
4k
8k
all
8k
4k
8k
2k

Smallest prime reptiles

Smallest prime reptile (4Lx2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
L tetromino
1
4
14
59009
17593273
≥267754502500

Smallest tori tilings

Smallest torus (2x4):

Smallest square torus (4x4):

Tori tilings' solutions count (including translations)

w \ h
1
2
3
4
5
1
0
2
0
0
3
0
0
0
4
0
32
0
992
5
0
0
0
0
0
6
0
0
0
8096
0
7
0
0
0
0
0
8
0
256
864
97920
275840
9
0
0
0
0
0
10
0
0
0
≥500000
0
11
0
0
0
0
0
12
0
2048
0
≥500000
0

Smallest Baiocchi figures

Smallest Baiocchi figure (area 8):

Smallest Baiocchi figure without holes (area 16):

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 $A(w; h)$

$N(n; m) = T(n; m) = 0, \qquad 8\nmid nm \tag{1}$

Assume L tetromino tiles $n\times m$ rectangles for $nm\not\equiv 0\pmod{8}$.

Place numbers in rectangles' cells according to function $F(x,y)\equiv4x+1\pmod{8}$, where $x$ and $y$ are cells' coordinates (zero-based). On the one hand, L tetromino, no matter how placed, covers sum congruent to $0\pmod{8}$. Then sum covered by all tetrominoes is also congruent to $0\pmod{8}$. On the other hand, rectangle covers sum congruent to $\sum_{x=0}^{n-1}\sum_{y=0}^{m-1}\left(4x+1\right)$, which is not congruent to $0\pmod{8}$ for $nm\not\equiv 0\pmod{8}$. Contradiction, as tetromino tiles this rectangle and thus sum covered by all tetrominoes should be equal to sum covered by rectangle. Thus only assumption we made is false - L tetromino doesn't tile $n\times m$ rectangles for $nm\not\equiv 0\pmod{8}$. Q.E.D.

$T(n; m) = 1, \qquad 8\mid nm \tag{2}$

Attributions

  1. Odd rectangle non-existance first proved here: http://www.cflmath.com/~reid/Polyomino/l4_rect.html

See Also

I tetrominoO tetromino