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In recreational mathematics, a polyform is a plane figure constructed by joining together identical basic polygons. The basic polygon is often (but not necessarily) a convex plane-filling polygon, such as a square or a triangle. More specific names have been given to polyforms resulting from specific basic polygons, as detailed in the table below. For example, a square basic polygon results in the well-known polyominoes. Recreational mathematics includes many mathematical games, and can be extended to cover such areas as logic and other puzzles of deductive reasoning. ...
In mathematics, a plane is the fundamental two-dimensional object. ...
Look up Polygon on Wiktionary, the free dictionary For other use please see Polygon (disambiguation) A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ...
In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...
A square as a geometric shape is described and illustrated at square (geometry). ...
A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. ...
Jump to: navigation, search A polyomino is a polyform with the square as its base form. ...
Construction rules The rules for joining the polygons together may vary, and must therefore be stated for each distinct type of polyform. Generally, however, the following rules apply: - Two basic polygons may be joined only along a common edge.
- No two basic polygons may overlap.
- A polyform must be connected (that is, all one piece; see connected graph, connected space). Configurations of disconnected basic polygons do not qualify as polyforms.
- The mirror image of an asymmetric polyform is not considered a distinct polyform (polyforms are "double sided").
In mathematics and computer science, graph theory studies the properties of graphs. ...
In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
Generalizations Polyforms can also be considered in higher dimensions. In 3-dimensional space, basic polyhedra can be joined along congruent faces. Joining cubes in this way leads to the polycubes. A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ...
Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ...
In recreational mathematics, a polycube is a polyform with a cube as the base form. ...
One can allow more than one basic polygon. The possibilities are so numerous that the exercise seems pointless, unless extra requirements are brought in. For example, the Penrose tiles define extra rules for joining edges, resulting in interesting polyforms with a kind of pentagonal symmetry. A Penrose tiling of rhombi A Penrose tiling is pattern of tiles, discovered by Roger Penrose and Robert Ammann, which could completely cover an infinite plane, but only in a pattern which is non-repeating (aperiodic). ...
Types and applications Polyforms are a rich source of problems, puzzles and games. The basic combinatorial problem is counting the number of different polyforms, given the basic polygon and the construction rules, as a function of n, the number of basic polygons in the polyform. Well-known puzzles include the pentomino puzzle and the Soma cube. Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria. ...
A pentomino is a polyomino composed of five (Greek Ï€Îντε / pente) congruent squares, connected orthogonally. ...
The pieces of a Soma cube (with extra coloring) The same puzzle, assembled into a cube The Soma cube is a solid dissection puzzle created by Piet Hein during a lecture on quantum mechanics by Werner Heisenberg. ...
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