POLYOMINO TILINGS

Polyomino Tilings

Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.

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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:

• It is smallest rectifiable polyomino which does not tile any rectangle with odd number of pieces.

Smallest rectangle tilings¶

Smallest rectangle (2x4):

Smallest square (4x4):

No odd rectangles exist.

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 \ h12345678N>0
10
200
3000
4022P01010C
500000
60004242C00
70000000
8044C44P182182C436436C43404340C1670816708C141970141970C
90000000611468611468C8k
10000790790C00044170844417084C4k
110000000≥1≥1C8k
12088C034323432C0517450517450C0≥1≥1C2k
130000000≥1≥1C8k
140001491414914C000≥1≥1C4k
150000000≥1≥1C8k
1601616C2424C6481464814C443144443144C≥1≥1C≥1≥1C≥1≥1Call
170000000≥1≥1C8k
18000281680281680C000≥1≥1C4k
190000000≥1≥1C8k
2003232C012241821224182C0≥1≥1C0≥1≥1C2k
210000000≥1≥1C8k
2200053203105320310C000≥1≥1C4k
230000000≥1≥1C8k
2406464C144144C2312214823122148C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
N>0x4k8k2k8k4k8kall

Smallest prime reptiles¶

Smallest prime reptile (4Lx2):

Reptile tilings' solutions count (including symmetric)¶

polyomino \ n²
L tetromino14P14P59009C17593273P≥267754502500C

Smallest tori tilings¶

Smallest torus (2x4):

Smallest square torus (4x4):

Tori tilings' solutions count (including translations)¶

w \ h12345
100
20000
3000000
400323200992992
50000000000
60000008096809600
70000000000
8002562568648649792097920275840275840
90000000000
10000000≥500000≥50000000
110000000000
12002048204800≥500000≥50000000

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.

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