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This website is currently under construction. Future Content is planned to be added.

ABOUT AUTHOR

Hi, I'm Sbiis Saibian. I am an amateur mathematician, but I also like creative pursuits. A few years back in 2004 I became interested in the possibility of variations of Tetris, and thus I became familiar with the polyominoes. I decided to make a website devoted to them especially as game pieces.

What are Polyominoes ?

Almost everyone know what "Tetris" is. Tetris is a game where you have to stack up "tetris pieces" together in order to produce rows. Each Tetris piece is made up of 4 unit squares joined together as one figure. The word "tetris" is a combination of the word "tetra-" meaning 4, and "tennis".

Polyominoes are a generalization of this idea. Take n unit squares and connect them together by sides and you have an nth order polyomino. For any n, there is always a finite number of distinct nth order polyominoes.

About This Site

This website is here to identify, organize and provide "petnames" for the simplest kinds of polyominoes that can be created, and considering how to use them to construct Tetris-variant games. Mathematicians sometimes take up the subject of "polyominoes" because of their interest in the "tiling problem". That is, given any plane figure, determine whether it can be used to tile the plane. A vast majority of the "simple" polyominoes can be tiled in such a way.

Our interest here is however not tiling, but classifying the known Polyominoes and considering their merits as "game pieces". Mathematicians have yet to devise an algorithm to count up all nth order polyominoes more efficiently than a brute force search, or to generate all polyomino forms. As far as I know, the Polyominoes have only been enumerated up to the 24th order so far!

This website classifies Polyominoes by their "order" ( the number of squares used in their construction ). In general, as the order increases so does the variety and complexity of these shapes.

Who is this site for?

The main appeal here is to hobbyists and Tetris fanatics. Here is a website devoted to the pieces, the games, and even the "theory".

So let's get started ...

THE POLYOMINO SEARCH

In Theory it would be the goal to classify EVERY polyomino. However this is not possible as there is an infinite number of them. It is therefore our goal to classify as many of the "simplest" polyominoes as possible.

The following links bring you the various pages which discuss various orders of polyominoes, beginning from lowest order and continuing to higher and higher orders. Order is always the number of squares used in the construction which is always a counting number ( ie. 1,2,3, ... ).

Note: The count of "distinct" nth order polyominoes is based on the criteria by which we define distinct. In this case distinctness can be defined as follows ...

Two Polyominoes of the same order are distinct if and only if one can not become the other using only the operations of rotation and translation.

I count reflections as distinct provided they can not become each other merely through rotation and translation. Most people who study the tiling problem however treat reflections as the same tile regardless. Keep in mind that these people will count the number of nth order polyominoes differently.

( I recently discovered that the set of Polyominoes I use are often referred to as the 1-sided polyominoes or "chiral" polyominoes, where as the set used by those studying tesselations are called 2-sided. From this point onward I will only be referring to Chiral polyominoes )

The reason for my particular format is based on the way Tetris is set up. Tetris treats the pieces as distinct based on rotation and translation alone.

Here are the links ...

Note: In each "article" I consider the full set of chiral polyominoes of the given order. The "Polyomino Petnames" found throughout the articles are of my own devising for the purposes of identifying them as "game pieces".

Monominoes : Here we discuss 1st order polyominoes, or Monominoes, and the dull game of "Monis"

Dominoes : These are 2nd order polyominoes. They are used to play "Domis"

Trominoes : 3rd order polyominoes. These are the first to exhibit some variety. We can use them in the game of "Tris"

Tetrominoes : 4th order polyominoes. These are the Tetris pieces that have become so prevalent thanks to the game of Tetris. On this page we identify all of the pieces, some interesting properties of them, and lastly we try to look into what makes Tetris so great.

Pentominoes : (Coming Soon) Here I will introduce all of the Pentominoes and cosider the challenges of the 5th order game, Pentris.

ADDITIONAL INFORMATION

Additional Information : Here I give some final thoughts, discuss other ideas and plans for the site and also provide some links to other sites on Polyominoes

FEEDBACK

Do you have a comment you'd like to make? A suggestion, an idea, or have you spotted an error somewhere you'd like corrected? Is there an idea you'd like to discuss? E-mail me at ...

sbiissaibian at aol.com

Thank you for visiting this site.

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Copyright Sbiis Saibian 2008