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Topic: Polyomino

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	Polyominoes
		The dark square in the third Polyomino is connected to the square above and the one to the left.
		In this section, the connectivity of the Polyominoes is illustrated by by placing equal triangles on each side of a common connecting edge.
		One interesting characteristic of this way of illustrating Polyomino connectivity is that the connectivity pattern is itself a Polyomino (or set of Polyominoes).
	www3.sympatico.ca /diharper/poly.html (884 words)

	polyomino
		The polyomino is a generalization of the domino, which can be thought of as a polyomino with n = 2.
		Related to polyominoes are polyiamonds (formed from equilateral triangles) and polyhexes (formed from regular hexagons).
		The three-dimensional analog of polyominoes uses cubes instead of squares; an example is the Soma Cube.
	www.daviddarling.info /encyclopedia/P/polyomino.html (731 words)

	PlanetMath: polyomino
		A polyomino consists of a number of identical connected squares placed in distinct locations in the plane so that at least one side of each square is adjacent to (i.e.
		Fixed polyominoes (which are also called lattice animals) are considered distinct if they cannot be translated into each other, while free polyominoes must also be distinct under rotation and reflection.
		This is version 7 of polyomino, born on 2005-06-14, modified 2006-09-30.
	planetmath.org /encyclopedia/Domino.html (344 words)

	IFS Attractors: polyominoes
		Polyominoes are a set of mostly irregular polygons which are constructed by joining squares together at a common edge.
		The rectangular polyominos, being rectangles, are rep-tiles with a rep-number of 4.
		Sources: The IFSs for rectifiable polyominoes are independently invented (I assume that these are well-known, as the follow from the widely studied problem of tiling a rectangle with a individual polyomino) except that the tilings of a rectangle from which the IFSs for the y-pentomino and R/G- and y-hexominos were taken from M.
	www.meden.demon.co.uk /Fractals/polyomino.html (880 words)

	Introduction
		A polyomino is a set of simply connected squares, which means that a chess rook on one square could get to any other square in a finite number of moves.
		The numbers of distinct polyominoes with 1 through 10 squares are known to be, respectively, 1, 1, 2, 5, 12, 35, 108, 369, 1285 and 4655.
		To consider a projective dual of a polyomino as a graph, consider the points that are the centers of the squares of the polyomino as the vertices and the lines connecting them as the edges.
	www.saintanns.k12.ny.us /depart/math/Lawrence/intro.html (772 words)

	Jospeh Malkevitch: Student Research Problems in Mathematics (Site not responding. Last check: )
		A polyomino W is called vertically convex if every vertical line cutting the interior of a square intersects W in a connected collection squares.
		A polyomino W is called horizontally convex if every horizontal cutting the interior of a square intersects W in a connected collection of squares.
		A polyomino is called L-convex if for any pair of square in the polyomino there is a path with at most one bend joining these two squares.
	www.york.cuny.edu /~malk/problems.html (629 words)

	Tiling Rectangles With Polyominoes
		A polyomino is a shape that consists of unit squares pasted together.
		Of course the polyomino might already be a rectangle, as illustrated by the domino.
		An L polyomino looks like the capital letter L. In other words, the shape is a rectangle with a rectangular notch removed from one corner.
	www.eklhad.net /polyomino (1470 words)

	Polyominoes
		For polyominoes not having a reflective symmetry, we may differ (or not) their "left" and "right" form.
		From their symbols we could directly make conclusions about the symmetry: every reversible word denotes polyominoes with a sense-reversing symmetry (they don't have "left" and "right" form); irreversible symbols correspond to the polyominos appearing in the "left" and "right" form (e.g.
		Polyominoes (either fl or white) appearing in Lunda designs will be called Lunda polyominoes.
	members.tripod.com /~modularity/pol.htm (587 words)

	The Geometry Junkyard: Polyominoes
		Activities associated with polyominoes, aimed at the level of primary (or elementary) school mathematics.
		Mario Szegedy describes an algorithm for determining whether a (possibly disconnected) polyomino will tile the plane by translation, in the case where the number of squares in the polyomino is a prime or four.
		Tiling rectangles and half strips with congruent polyominoes, and Tiling a square with eight congruent polyominoes, Michael Reid.
	www.ics.uci.edu /~eppstein/junkyard/polyomino.html (1858 words)

	Cambridge College :: Mathematics Institute
		A polyomino is a shape formed from a finite number of square tiles of the same size, by gluing the tiles together, edge to edge (like the tiles in a Scrabble game).
		A tile in a polyomino is called "removable" if removing it would leave a polyomino.
		For example, if the polyomino consists of six tiles in a line, the two end tiles are removable and the four others are not.
	www.cambridgecollege.edu /math/potm.cfm?pid=22 (300 words)

	welltris (Site not responding. Last check: )
		Mixing polyomino classes (so as to permit polyominoes made up by a variable number squares) can also be selected, as well as, the 2 sets of diagonal poly- ominoes that do not have pieces with holes.
		Polyominoes with diagonal or with a higher number of squares earn more points.
		The ability to rotate polyominoes, is available only when the number of squares is >= 4.
	www.tux.org /~bagleyd/welltris.html (1204 words)

	Miscellaneous Polyomino Explorations
		The number of squares within the polyomino, (dark squares,) on each row, column, and main diagonal is 5, and the number within the polyomino in each 3×3 cell corresponds to a number in a 3×3 magic square, where all rows, columns, and diagonals sum to 15:
		Also, since shrinking polyominoes preserves the number of sides, the unshrinkable (n2)-sided polyominoes* could be looked at as canonical members of equivalence classes, where all polyominoes that eventually shrink to the same unshrinkable polyomino are equivalent.
		For polyomino "convexity", we add the restriction that the two points we choose have to be on the same vertical or horizontal line.
	www.xprt.net /~munizao/mathrec/miscpoly.html (1403 words)

	Polyomino, polyhex and polyiamond tiling
		A polyomino (word derived from domino) is a geometric plane figure made of the union of finitely many edge-connected squares from the regular square lattice.
		Sets of the twelve pentominoes in wood or plastic are widely available; traditional problems are to fill a 3 by 20, 4 by 15, 5 by 12 or 6 by 10 rectangle with them; the complete sets of solutions to these problems were determined by computer in the 1960s.
		Tiling the polyominoes of orders 1 through 6 is easy, and John Conway determined the tiling properties of the heptominoes with the aid of Conway’s criterion.
	www.srcf.ucam.org /~jsm28/tiling (1897 words)

	TopCoder (Site not responding. Last check: )
		During this marathon, we were asked to write a program that would select polyominoes of a chosen size and try to place them on a board in way that maximizes the score.
		The polyominoes are generating randomly and, even for small sizes, there were too many possible polyominoes of one size to test whether they all can fit somewhere on the board or not.
		The main idea is to try and check if the polyomino doesn't cover any of the occupied cells on the board after moving its point of origin over each cell on board, and scoring each possible position to get the best one after considering all placements.
	www.topcoder.com /longcontest/?module=Static&d1=match_editorials&d2=intel_mtcs_9 (1213 words)

	Counting horizontally convex polyominoes
		A polyomino of area n is called an n-omino.
		We will consider two polyominoes to be the same if one is a translate of the other.
		A polyomino P is horizontally convex if each horizontal line meets P in a single line segment, or not at all.
	www.cs.uwaterloo.ca /journals/JIS/HICK2/chcp.html (1222 words)

	Recent problems and results on polyomino enumeration. (Site not responding. Last check: )
		A polyomino is the obvious extension of a domino to more an arbitrary, finite number of cells in the square grid forming a simply connected region.
		A polyomino is column-convex (row-convex) if its intersection with any vertical (horizontal) strip in the square grid is either empty or connected; a polyomino is convex if it is both row and column convex.
		The enumeration of convex polyominoes is a case in point as a challenging problem.
	www.ii.uib.no /undervisning/seminar/lista04/abstracts/rogers.shtml (128 words)

	GeneratingPolyominoes - TmrWiki
		The name "polyominoes" was first given to these shapes by the mathematician Solomon W. Golomb, whose book Polyominoes is a fascinating introduction to the subject of polyomino puzzles and techniques for solving them.
		Implementing such an algorithm in Haskell involves generating a list of candidate polyominoes of rank n, based on a list of known polyominoes of rank n-1, and removing from that list all polyominoes that are found to be the same as another included polyomino after they have been translated, rotated and/or reflected.
		In order to compare two candidate polyominoes and determine whether they are the same, we introduce some functions that will convert any candidate polyomino into a normalised, "canonical" form.
	www.haskell.org /tmrwiki/GeneratingPolyominoes (1568 words)

	Cutting Polyominoes
		A polyomino may be viewed as a set of squares connected by their sides.
		We are interested in polyominoes without holes and that have exactly one edge in each grid line that intersects them.
		is a vertex of the inflated polyomino and none of the other vertices of the inflated polyomino belongs to the rectangle (either to its interior or boundary);
	acm.uva.es /p/v9/919.html (371 words)

	[No title]
		Polyomino is a fun logic game in which the objective is to fill puzzles shapes with pentaminoes.
		Polyomino proposes as many as 100 puzzles with different sizes and shapes of grid, all classified by categories and increasing difficulty.
		In this version, the ten first puzzles are offered so that the user can learn the game and evaluate the program.
	membres.lycos.fr /polyomino (283 words)

	AT&T Worldnet Service - Directory
		Joseph Myers and John Berglund found a polyhex that must be placed in two different ways in a tiling of a plane, such that one placement occurs twice as often as the other.
		Joseph Myer's tables of polyominoes and of polyomino tilings, in Postscript format.
		Alon, Bóna, and Spencer show that one can't cover very much of an n by p(n) rectangle with staircase polyominoes (where p(n) is the number of these shapes).
	www.att.net /cgi-bin/webdrill?catkey=gwd/Top/Science/Math/Recreations/Polyominoes (1391 words)

	[No title]
		Polyomino Polyomino Direction type Coord = (Int, Int) instance Show Direction where show GoUp = "Up" show GoRight = "Right" show GoDown = "Down" show GoLeft = "Left" show (BackThen d) = "Back, then " ++ show d -- Lower rank polyominoes.
		forms -- This enables us to assert two useful properties of polyominoes: -- Two polyominoes are equal if their normal forms are equal, -- and not equal if their normal forms are not equal.
		instance Eq Polyomino where (==) p1 p2 = (normalForm p1) == (normalForm p2) (/=) p1 p2 = not (p1==p2) -- Two polyominoes may be compared by comparing their normal forms.
	codepoetics.com /code/polyomino.hs (512 words)

	Polyomino 3.0 Download - Polyomino is a collection of 100 puzzles in which the objective
		Polyomino 3.0 Download - Polyomino is a collection of 100 puzzles in which the objective
		Download (4255 K) Polyomino is a collection of 100 puzzles in which the objective is to fill entirely the grid of the puzzle with the adequate pentaminoes.
		Place Polyomino pieces in picture-like grid shapes.The polyomino pieces are introduced at a rapid pace, so you have to be quick and accurate in placing the pieces.
	www.downloadjunction.com /product/software/44338 (561 words)

	The Mathematics of Polyominoes
		Polyominoes Enumeration - results from D. Redelmeier, W. Lunnon, Kevin Gong, Uwe Schult, Tomas Oliveira e Silva, and Tony Guttmann, Iwan Jensen and Ling Heng Wong (2000).
		Enumeration of 2-d polyominoes (unique with respect to translation) which fit into rectangles of various sizes.
		Enumeration of 2-d polyominoes (unique with respect to 3-d rotation) which fit into rectangles of various sizes.
	kevingong.com /Polyominoes/Math.html (227 words)

	polyom
		Thinking more on Fill-Agree by Kadon Enterprises, I asked Michael Reid about all the ways one could drill a hole through a solid L tromino.
		A polyomino is a figure made of solidly connected squares.
		Gerard's Universal Polyomino Solver has a nice Java applet for solving Pentomino problems.
	www.mathpuzzle.com /polyom.htm (826 words)

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