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.

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 and smallest square and smallest odd rectangle (2x2):

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 \ h | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

1 | 0 | |||||

2 | 0 | 11P | ||||

3 | 0 | 0 | 0 | |||

4 | 0 | 11C | 0 | 11C | ||

5 | 0 | 0 | 0 | 0 | 0 | |

6 | 0 | 11C | 0 | 11C | 0 | 11C |

N>0 | x | 2k | x | 2k | x | 2k |

Smallest prime reptile (4Ox2):

polyomino \ n² | 1² | 2² | 3² | 4² | 5² | 6² |
---|---|---|---|---|---|---|

O tetromino | 1 | 1P | 1P | 1C | 1P | 1C |

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

w \ h | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

1 | 00 | |||||||

2 | 00 | 44 | ||||||

3 | 00 | 00 | 00 | |||||

4 | 00 | 88 | 00 | 1212 | ||||

5 | 00 | 00 | 00 | 00 | 00 | |||

6 | 00 | 1616 | 00 | 2020 | 00 | 2828 | ||

7 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | |

8 | 00 | 3232 | 00 | 3636 | 00 | 4444 | 00 | 6060 |

Smallest Baiocchi figure (area 4):

$N(w; h)$ - 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 $A(w; h)$

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

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

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

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