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Topic: Polyiamond

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	Polyiamond - Biocrawler (Site not responding. Last check: 2007-10-20)
		A polyiamond (also polyamond or simply iamond) is a polyform in which the base form is an equilateral triangle.
		The word polyiamond is a back-formation from diamond, motivated by the fact that this word is often used to describe the shape of a pair of equilateral triangles placed base to base.
		The basic combinatorial question is how many different polyiamonds with a given number of triangles exist.
	www.biocrawler.com /encyclopedia/Polyiamond (152 words)

	Tessellations
		An enantiomorphic polyiamond is one which cannot be superimposed on its reflection, its mirror image.
		I propose the following classification of polyiamond tessellations which is based on the operations performed on the polyiamond being tessellated..
		A single or multiple of a polyiamond may be combined to form a figure which is capable of tessellating the plane using only the translation operation.
	www.mathpuzzle.com /Tessel.htm (790 words)

	Polyomino, polyhex and polyiamond tiling
		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.
		Polyiamonds (word derived from diamond) are shapes analogous to polyominoes, but made up of equilateral triangles taken from the regular tiling by such triangles.
		Both polyiamonds and polyhexes are widely used in recreational mathematics, similarly to polyominoes.
	www.srcf.ucam.org /~jsm28/tiling (1897 words)

	Rectifiable Polyiamonds
		Obviously a polyiamond which can form a triangle can also form a trapezium or hexagon and one which will form a trapezium will also form a parallelogram.
		The triangle, parallelogram, and hexagon formed are the smallest possible with this polyiamond.
		Now if we try to use the pentiamond to fill a triangle we see we shall need to fill a B type space at some point and since the pentiamond has only P and A type edges this is not possible.
	www.recmath.org /PolyPages/PolyPages/Rectifiam.htm (1990 words)

	tertris
		Mixing polyiamond classes (so as to permit polyiamonds made up by a variable number triangles) can also be selected, as well as, the 1 set of diagonal polyiamonds that do not have pieces with holes.
		Polyiamonds with diagonal or with a higher number of triangles earn more points.
		The ability to rotate polyiamonds, is avail- able only when the number of triangles is >= 8.
	gwyn.tux.org /~bagleyd/tertris.html (1193 words)

	Rectifiable Polyiamonds
		The term rectifiable is probably not the best one to use for polyiamonds since they will not, of course, form rectangles.
		Obviously a polyiamond which can form a triangle can also form a trapezium or hexagon and one which will form a trapezium will also form a parallelogram.
		Now if we try to use the pentiamond to fill a triangle we see we shall need to fill a B type space at some point and since the pentiamond has only P and A type edges this is not possible.
	www.recmath.com /PolyPages/PolyPages/Rectifiam.htm (1990 words)

	Polyiamond -- from Wolfram MathWorld
		Discrete Mathematics > Combinatorics > Lattice Paths and Polygons > Polyiamonds
		A generalization of the polyominoes using a collection of equal-sized equilateral triangles (instead of squares) arranged with coincident sides.
		The bottom row of 6-polyiamonds are known as the chevron, signpost,
	mathworld.wolfram.com /Polyiamond.html (252 words)

	IMACS03 conference abstract: Yang
		The min perimeter of a polyiamond with area n is whichever of ceil(sqrt(6n)) + {0, 1} has the same parity as n.
		The max area of a polyiamond with perimeter p is round(p^2/6) - {0, 1}, where we choose 0 iff p = 0 mod 6.
		These results are related as follows: we use max polyiamonds (polyiamonds with max area and fixed perimeter) to get a lower bound on min perimeter, and we use this result to construct min polyiamonds (polyiamonds with fixed area and min perimeter).
	www-math.cudenver.edu /IMACS03/abs/yang.html (294 words)

	Polyiamond at AllExperts
		As with polyominoes, fixed polyiamonds (where different orientations count as distinct) and one-sided polyiamonds (where mirror images count as distinct but rotations count as identical) may also be defined.
		The number of free polyiamonds with holes is given by ; the number of free polyiamonds without holes is given by ; the number of fixed polyiamonds is given by ; the number of one-sided polyiamonds is given by.
		In the case of only mirror symmetry we can distinguish having the symmetry axis aligned with the grid or rotated 30Â° (requires at least 4 and 3 triangles, respectively); ditto for 3-fold rotational symmetry, combined with mirror symmetry (requires at least 18 and 1 triangles, respectively).
	en.allexperts.com /e/p/po/polyiamond.htm (392 words)

	Eric's TriSolve Application (Site not responding. Last check: 2007-10-20)
		A polyiamond is a puzzle piece made of equilateral triangles.
		Sets of polyiamond pieces can be put together to form a multitude of shapes.
		Polyiamond puzzle sets are sold by Kadon Enterprises.
	www.edp.org /TriSolve (197 words)

	Math Forum - Geometry Problem of the Week (Site not responding. Last check: 2007-10-20)
		Polyiamonds are figures made by fitting together two or more equilateral triangles.
		The simplest polyiamond, a 'diamond', consists of two equilateral triangles.
		There are twelve different polyiamonds that contain six equilateral triangles each.
	mathforum.org /pow/solutio89.html (700 words)

	Mirror curves (Site not responding. Last check: 2007-10-20)
		Trying to discover their common mathematical background, they appear two questions: how to construct such a perfect curve - a single line placed uniformly in a regular tiling, this means, how to arrange the set of mirrors generating it, and how to classify the curves obtained.
		In principle, any polyomino (polyiamond or polyhexe) with mirrors on its border, and two-sided mirrors between cells or perpendicular to the internal cell-edges in their midpoints, could be used for the creation of the corresponding curves.
		For their construction in some polyomino (polyiamond or polyhexe), we propose the following method.
	www.mi.sanu.ac.yu /vismath/mir/mir.htm (919 words)

	IFS Attractors: polyiamonds
		A polyiamond is a plane figure composed of a number of equilateral triangles joined by common edges.
		The only polyiamond which I have created such a construction is the triamond.
		In general any bar polyiamond consisting of an even number of equilateral triangles is a rep-tile, with the same construction as is shown for the diamond, bar tetriamond, bar hexiamond (below) and bar octiamond (below).
	www.meden.demon.co.uk /Fractals/polyiamond.html (249 words)

	Other Polyforms
		If we try dividing each triangle of a polyiamond in two then we tend to get some very complex pieces but if we include the restriction that each cut be the same on both sides of a join then we produce pieces such as those below derived from the tetriamonds, pentiamonds and hexiamonds.
		We can define the diameter of a polyiamond as the smallest triangle into which the polyiamond will fit.
		Under this definition there are 17 diameter 3 polyiamonds which have a total area of 104 unit triangles.
	www.geocities.com /alclarke0/PolyPages/PolyX/PolyX.htm (975 words)

	Activity 7
		Polyominoes and polyiamonds are connected arrangements of squares and triangles, respectively.
		They can be used to explore both reflectional and rotational symmetry, they can be arranged to form other shapes, and they can be extended into three-dimensional shapes to form solid objects.
		Two polyiamonds are the same if one can be flipped, turned, or moved so as to coincide with the other.
	homepage.mac.com /efithian/Geometry/Activity-07.html (735 words)

	Other Polyforms
		Because, in order to produce a polyform, it is usually necessary to start with one of the three regular tiling patterns (squares, equilateral triangles or hexagons) most of these are either subsets or derivations of the main polyforms - polyominoes, polyiamonds and polyhexes.
		If we try dividing each triangle of a polyiamond in two then we tend to get some very complex pieces but if we include the restriction that each cut be the same on both sides of a join then we produce pieces such as those below derived from the tetriamonds, pentiamonds and hexiamonds.
		Under this definition there are 17 diameter 3 polyiamonds which have a total area of 104 unit triangles.
	clarkjag.idx.com.au /PolyPages/PolyX/PolyX.htm (1014 words)

	Kadon Enterprises, More About Polyforms
		Solomon Golomb started the trend in 1953 with "polyominoes." Shapes made of equilateral triangles became "polyiamonds," from diamond, made of two triangles.
		Shapes made of hexagons are simply "polyhexes," and shapes made of isosceles right triangles are "polytans," from their resemblance to tangrams.
		Aad van de Wetering from the Netherlands has written a fine and quick program for solving figures with polyominoes, polyiamonds and polyhexes.
	www.gamepuzzles.com /morepoly.htm (413 words)

	Kadon Enterprises, More About Polyforms
		Shapes made of equilateral triangles became polyiamonds, from diamond, made of two triangles.
		Shapes made of hexagons are simply polyhexes, and shapes made of isosceles right triangles are polytans, from their resemblance to tangrams.
		Aad van de Wetering from the Netherlands has written a fine and quick program for solving figures with polyominoes, polyiamonds and polyhexes.
	gamepuzzles.com /morepoly.htm (604 words)

	[No title]
		This program creates data suitable for the {\mc DANCE} routine, given the description of a board to be covered and a set of polyiamond shapes.
		Each piece name consists of at most three characters; the name should also be distinguishable from a board position.
		First comes the name, followed by two integers $s$ and $t$, meaning that the shape should appear in $s$ rotations and $t$ transpositions.
	www-cs-faculty.stanford.edu /~knuth/programs/polyiamonds.w (456 words)

	Wade E. Philpott Fonds - Accession No.: 2813-2858.7
		Radio Shack TRS-80 microcomputer 1978 2824 technical reference handbook Polyomino and polyiamond problems 1976 2825 71 p.
		Typescript and photocopy [mostly undecipherable] Polyiamonds File 6.0 and Heptiamonds 1964-1975 2830.10 File 6.7 89 p.
		(9) Polyomino and polyiamond 2842.1.9 problems, Part I (Journal of recreational mathematics, v.10, no.1, 1977-78; p.2-14, 98-105).
	www.ucalgary.ca /lib-old/SpecColl/phil281.htm (1346 words)

	Science Math Recreations Polyominoes (Site not responding. Last check: 2007-10-20)
		Flexagons - Folded paper polyiamonds which can be unfolded to show hidden faces.
		Polyiamond Exclusion - Colonel Sicherman asks what fraction of the triangles need to be removed from a regular triangular tiling of the plane, in order to make sure that the remaining triangles contain no copy of a given polyiamond.
		This Geometry problem of the week asks whether a six-point star can be dissected to form eight distinct hexiamonds.
	www.pacosdrivers.com /odp/scat.asp?ID=Science/Math/Recreations/Polyominoes (1948 words)

	The Geometry Junkyard: Polyominoes
		A famous open problem asks whether any animal in three dimensions can be transformed into a single cube by adding and removing cubes, at each step remaining an animal (it is known that removal alone does not always work).
		Other related figures include polyiamonds (collections of equilateral triangles), polyabolos (collections of half-squares), and polyhexes (collections of regular hexagons).
		Folded paper polyiamonds which can be "flexed" to show different sets of faces.
	www.ics.uci.edu /~eppstein/junkyard/polyomino.html (1858 words)

	The On-Line Encyclopedia of Integer Sequences (Site not responding. Last check: 2007-10-20)
		Maximal number of triangles in polyiamond with perimeter n.
		W. Yang, R. Meyer, Maximal and minimal polyiamonds, manuscript, 2002.
		a(10) = 16 because the max number of triangles in a polyiamond of perimeter 10 is 16.
	www.research.att.com /~njas/sequences/A069813 (137 words)

	Trexle - Polyominoes
		The Poly Pages - Information on the various polyforms - polyominoes, polyiamonds, polycubes, and polyhexes.
		Some Variations on the Theme of Polyomino - Looks at patterns that illustrate how the squares of polyominos are connected to each other.
		Xominoes - Livio Zucca finds a set of markings for the edges of a square that lead to exactly 100 possible tiles, and asks how to fit them into a 10x10 grid.
	www.trexle.com /Directory/Top/Science/Math/Recreations/Polyominoes (621 words)

	AT&T Worldnet Service - Directory
		Folded paper polyiamonds which can be unfolded to show hidden faces.
		Brown examines the number of polyominoes up to order 12 for various cases involving rotation or reflections.
		Colonel Sicherman asks what fraction of the triangles need to be removed from a regular triangular tiling of the plane, in order to make sure that the remaining triangles contain no copy of a given polyiamond.
	www.att.net /cgi-bin/webdrill?catkey=gwd/Top/Science/Math/Recreations/Polyominoes (1391 words)

	> Science> Math> Recreations> Polyominoes / Career Websites by WritingCareer.com / Careers for Writers / Resources for ... (Site not responding. Last check: 2007-10-20)
		Flexagons - - Folded paper polyiamonds which can be unfolded to show hidden faces.
		Polyiamond Exclusion - - Colonel Sicherman asks what fraction of the triangles need to be removed from a regular triangular tiling of the plane, in order to make sure that the remaining triangles contain no copy of a given polyiamond.
		Tesselating Locking Polyominos - - Bob Newman examines the history of the subject and presents his minimal solutions.
	www.writingcareer.com /careerwebsites/index.php?browse=/Science/Math/Recreations/Polyominoes (2214 words)

	Erich's Undergraduate Research Problems
		A polyiamond is a union of unit area equilateral triangles where triangles meet along two entire sides.
		For example, there are 3 convex polyiamonds with area 8:
		Philosopher's Phutball is a two-player game played on a n x m rectangular playing board.
	www.stetson.edu /~efriedma/research (951 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Attention is focused on a surprising result in constructing convex shapes from tetratans.
		Key words: Polyomino, Polyiamond, Compatibility, Polytans, Tetratans INTRODUCTION The problem of compatibility of polyforms is attractive, however, very difficult in general and it has been solved only in a few cases.
		Analogously when a square (being the generating element for polyominoes) is replaced by the equilateral triangle or regular hexagon we obtain polyiamonds and polyhexes respectively.
	www.icme-organisers.dk /tsg04/finalpapers/5_ACIF_LT.doc (779 words)

	Polyomino - ExampleProblems.com
		Related to polyominoes are polyiamonds (formed from equilateral triangles), polyhexes (formed from regular hexagons), and other polyforms.
		Some variants of the Sudoku puzzle use polyomino shaped regions on the grid.
		Eric W. Weisstein, "Polyiamond", from MathWorld — a Wolfram Web Resource
	www.exampleproblems.com /wiki/index.php/Polyomino (1794 words)

	Polyiamond Exclusion (Site not responding. Last check: 2007-10-20)
		If a polyiamond of order n tiles the plane, you must remove at least 1/n of the cells, one for each tile.
		Any straight polyiamond of even order n can be excluded with 1/n holes.
		Straight polyiamonds of odd order require more than 1/n holes.
	www.monmouth.com /~colonel/xpoly/xpoly.html (396 words)

	Polyominoes Recreations Math Science
		- Information on the various polyforms - polyominoes, polyiamonds, polycubes, and polyhexes.
		- Windows software to solve polyiamond and sliding block puzzles.
		- Polyforms (polyominoes, and polyiamonds) graphics, tables and resources.
	www.iaswww.com /ODP/Science/Math/Recreations/Polyominoes (435 words)

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