geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a dissection problem is the problem of partitioning a geometric

figure Figure may refer to: General *A shape, drawing, depiction, or geometric configuration *Figure (wood), wood appearance *Figure (music), distinguished from musical motif *Noise figure, in telecommunication *Dance figure, an elementary dance pattern ...

(such as a

polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...

ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...

) into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a dissection (of one polytope into another). It is usually required that the dissection use only a finite number of pieces. Additionally, to avoid

set-theoretic Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

issues related to the

Banach–Tarski paradox The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...

and

Tarski's circle-squaring problem Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Mikl ...

, the pieces are typically required to be

well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. Th ...

. For instance, they may be restricted to being the closures of disjoint

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...

s. The Bolyai–Gerwien theorem states that any

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

may be dissected into any other polygon of the same area, using interior-disjoint polygonal pieces. It is not true, however, that any

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

has a dissection into any other polyhedron of the same volume using polyhedral pieces (see

Dehn invariant In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissection problem, dissected") into another, and whether a polyhedron or its dissections can Honeycomb (geometry), tile s ...

). This process ''is'' possible, however, for any two honeycombs (such as

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

) in three dimension and any two

zonohedra In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...

of equal volume (in any dimension). A dissection into

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

s of equal area is called an

equidissection In geometry, an equidissection is a partition of a polygon into triangles of equal area. The study of equidissections began in the late 1960s with Monsky's theorem, which states that a square cannot be equidissected into an odd number of triang ...

. Most polygons cannot be equidissected, and those that can often have restrictions on the possible numbers of triangles. For example,

Monsky's theorem In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square does not have an odd equidissection. The problem was posed by Fred Richman in the '' American ...

states that there is no odd equidissection of a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

References

External links

David Eppstein David Arthur Eppstein (born 1963) is an American computer scientist and mathematician. He is a Distinguished Professor of computer science at the University of California, Irvine. He is known for his work in computational geometry, graph algorit ...

Dissection Tiling
Discrete geometry Euclidean geometry Geometric dissection Polygons Polyhedra Polytopes Mathematical problems {{geometry-stub

See also

References

External links