# 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:

• It is the smallest polyomino.

## Smallest rectangle tilings¶

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

## 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.

## Smallest prime reptiles¶

Smallest prime reptile (1Ox2):

polyomino \ n²
O monomino
1
1
1
1
1
1

## Smallest tori tilings¶

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

w \ h
1
2
3
4
5
6
1
1
2
1
1
3
1
1
1
4
1
1
1
1
5
1
1
1
1
1
6
1
1
1
1
1
1

## 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 $A(w; h)$

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

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

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

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

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