Introduction

A polyomino is a figure made up of squares joined by their edges:

The smallest polyomino is a monomino, consisting of only a single square, followed by the domino with two squares. Tetrominoes, polyominoes with four squares, are featured in the popular video game Tetris. Pentominoes, with five squares, are the subject of many puzzles and games, such as the problem of fitting the 12 pentominoes in a rectangular grid.

A polyomino is fixed if different rotations and reflections are considered separate. It is one-sided if rotations are considered the same figure but not reflections. It is a free polyomino if reflections and rotations are considered the same. Unless otherwise stated, we will consider a polyomino to be a free polyomino.

This site provides interactive explorations of several interesting properties of polyominoes:

• catalog: a list of polyominoes up to octominoes with a summary of their properties.
• symmetry: an explanation of the different types of symmetries a polyomino can have.
• classes: a categorization of polyominoes based on convexity and directedness.
• packing: fitting a set of polyominoes tightly in some container.
• tiling: whether a single polyomino can completely fill the plane.
• genealogy: a "family tree" showing how polyominoes are built from simpler polyominoes.