Dissection (geometry)
Encyclopedia
In geometry, a dissection problem is the problem of partitioning a geometric figure
(such as a polytope
or ball
) 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.
The Bolyai-Gerwien theorem states that any polygon
may be dissected into any other polygon of the same area. It is not true, however, that any polyhedron
has a dissection into any other polyhedron of the same volume. This process is possible, however, for any two honecombs
(such as cube
) in three dimension and any two zonohedra of equal volume (in any dimension).
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
(such as a polytope
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
or ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
) 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.
The Bolyai-Gerwien theorem states that any polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
may be dissected into any other polygon of the same area. It is not true, however, that any polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
has a dissection into any other polyhedron of the same volume. This process is possible, however, for any two honecombs
Honeycomb (geometry)
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
(such as cube
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. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
) in three dimension and any two zonohedra of equal volume (in any dimension).
External links
- David EppsteinDavid EppsteinDavid Arthur Eppstein is an American computer scientist and mathematician. He is professor of computer science at University of California, Irvine. He is known for his work in computational geometry, graph algorithms, and recreational mathematics.-Biography:Born in England of New Zealander...
, Dissection Tiling.