additionalinformation
Additional Information
So we have considered all of the Polynomials up to order 4. So far we have the sequence of values ...
Poly(1) = 1
Poly(2) = 1
Poly(3) = 2
Poly(4) = 7
There is a significant amount of information out there on the polynominoes. Alot of this focuses on ways of packing together sets of them to form rectangles, or other shapes and designs. People have also counted the number of polynominoes all the way up to order 24, but the number of pieces starts to increases rather rapidly ( the growth rate is roughly exponential in nature ). These searches are performed by computer algorithms.
There is also several variants on polynominoes, such as using equilateral triangles or hexagons. It is also possible to consider higher dimensional polynominoes, such as Polycubes, and even Polytesseracts. Some rather usual variant's include half pieces !
http://en.wikipedia.org/wiki/Tetris : This is a rather extensive wikipedia article on the game of Tetris. It discusses the origins of the game, it's history, gameplay, a comparison of versions and features, and virtually anything you could want to know about the game.
http://en.wikipedia.org/wiki/Polyomino : A wikipedia article on Polynominoes. There is some interesting information here.
http://www.research.att.com/~njas/sequences/A000988 : The function Poly(n) is cataloged in the encyclopedia of integer sequences as function A000988(n). This link takes you directly to the citation. As you can see, the values grow rapidly. Poly(9) is already greater than a thousand. So far Poly(n) has only been computed up to n <=24.
http://web.mit.edu/puzzle/www/06/puzzles/epcot_center/pentris/ : This is an unusual link. You can play a rather frustrating version of Pentris, that is actually part of a unusual word puzzle. Be WARNED there is no way to rotate the pieces. This is part of what makes the puzzle difficult. Thankfully the solution is provided, and a casual read of this will be all the explanation you need. At least this will give you some idea of the difficulties involved in a game of Pentris, though you would at least be able to rotate in the "actual" game.
http://www.youtube.com/watch?v=8FB72STw7ak&NR=1 : Here is an awesome Youtube video of Game color Tetris !
FUTURE SITE PLANS
The next thing I want to do is create an article on the Pentominoes and the game of Pentris. There are 18 chiral polynominoes ( chiral meaning 1-sided in this case ). I will list them in my order and explain their properties and give them petnames.
SEQUENCE ALGORITHM
I've mentioned that the order in which I have choosen to present the Polynominoes is based on my own algorithm, but I never actually explained how it works. So I will now give you an explanation.
Firstly begin with the Monomino, or Nod. For every Nod in a Polynomino there are 4 sides on which to attach another nod. We try combinations of attaching Nods to the 4 sides in order, from top and moving around clockwise. So first we check up, then right, down, and finally left.
In the case of the Monomino, we first attach a Nod on top, but this automatically rules out the other 3 possibilities do to rotation.
Thus we form the only Domino. It is important to number the Nods in sequence. So the Domino has the numbers 1 and 2 inscribed on it's squares. 1 on bottom and 2 on top ( These numbers are an important part of the way we decide how to sequence the polynominoes).
The Next step is we attach the Nod in sequence, beginning with the highest number square, going around clockwise, and then continue to the next highest square, and so on, until all squares have been exhausted.
In the case of forming Trominoes, we first place a Nod on top of the Doon, to form Tropel. Next we turn clockwise around the "2-square", and place a Nod to the right of the 2-square of the Doon, thus forming Treeb. All other "latches" are canceled to do rotation.
Now we continue in this manner to give sequence number to all polynominoes. We always perform the construction with the previous order polynominoes in their proper order.
Thus to get the Tetrominoes in their proper sequence, we begin with Tropel, and when we have exhausted this, we move on to latching Nod's to Treeb.
The Tetril is formed by latching Nod on top of Tropel's 3-cell. Next we get the Jed from latching the Nod to the right of the 3-cell. It is not possible to attach a nod below the 3-cell as this space is already occupied by the cell marked 2. Moving to the last side, we form the Led by latching the nod to the left of the 3-cell. Finally we move on to latches of the 2-cell. We can't latch above, so we go to the right, which forms the Tad. At this point all of the possibilities for Tropel have been exhausted so we move on to Treeb. With Treeb we begin with by placing a Nod on top of the 3-cell (which is the one facing right in the Treeb). This forms the Zaw. Next we move to the right of the 3-cell but this would form a led so we skip this. Next would be just below the 3-cell, this forms a 2x2 or the Basib. We move on to the 2-cell but we find that we can only crate Tad's in this way. So we finally move to the 1-cell. Only when we attach a Nod to the left of the 1-cell to we get our one last final piece, the Saw.
So we get 4 tetrominoes from Tropel and 3 from Treeb for a grand total of 7.
We can continue this for Pentominoes and beyond. The key is to make sure that all of our polynominoes have their cells numbered in the order in which they were latched in their construction. Not only does this method provide a predefined order for all the chiral polynominoes, but is also can be used to define a "normal" orientation (that is we can define the orientation in which a polynomino appears to be it's unright position).
One interesting thing about this sequence is that it is set up so that the "towers" will always get to be No.1 in their order. So The Nod is No.1, the Doon is No.1, and so is the Tropel and Tetril. L and J type pieces will always get to follow the towers immediately (For example Treeb follows Tropel, and Led and Jed follow after Tetril). After this the sequence seems to become somewhat random. Mirrors sometimes appear consecutively, but other times can be greatly seperated. The boxish polynominoes seem to appear in different locations at different orders. The very last polynomino will always be a "zigzagger" (such as the Treeb, and Saw).
I'm not sure if their is a canical order in which polynominoes appear. An order is certainly not neccessary, but it helps in cataloging them. It also gives each one an identification number, which comes in handy. For example, the Zaw can be IDed as 4-5. Meaning it's the 5th order 4 polynomino. These ID numbers can serve as temporary names when a catchy nonsense word isn't forthcoming. This is the main reason I decided to give them an order. I could have coarse choosen an arbitrary order, but having an algorithm that puts them in sequence "automatically" seems much more appealing.
Conclusion
I think that the appeal of polynominoes is their simplicity and variety. The simplicity of the shapes is visually appealing, and the possibilities of linking them together can be addictive. The appeal of the variety is that there are many possibilities, but these possibilities are strictly limited and finite. In otherwords, they are ennumerable.
Perhaps the strangest thing about the polynominoes is that mathematicians have not yet devised an efficient algorithm to calculate the function Poly(n) other than mere exhaustion of the possibilities (which invariably has to be done by a computer using lots of processing power and memory!). There are so many higher order polynominoes that we would never be able to know them all. Because of the difficulties involved in calculations we may never know exact values of Poly(n) above a certain threshold. Likewise their are polynominoes so complex and of such a high order that we could never conceive of them ! In the end we are confronted with the inexhaustability of abstract ideas, and simultaneously the exhaustability and limitations of our reality. Although we can never know all the polynominoes, we can still increase our knowledge of them through study. Just like everything, human beings have a tendency to push forward regardless of the difficulty. One thing we can say about humans is, if it can be done, it will be eventually.
With that I conclude for now. I will continue to expand this site and add more information over time, but I also have other interests, which means that I will be limiting the amount of time spent working on this. None the less, I at least want to classify all of the polynominoes up to order 6. I also would like to add more information on how to play the various order games, and also provide other information. Visit again sometime to see what progress I have made.