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Among dissection puzzles, an example is the Haberdasher's Puzzle, posed by puzzle writer
Dissection Tiling
Discrete geometry Euclidean geometry Geometric dissection Polygons Polyhedra Polytopes Mathematical problems {{geometry-stub
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, 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 patt ...
(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 ...
or ball
A ball is a round object (usually spherical, but sometimes 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 for s ...
) 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 ...
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 ...
and Tarski's circle-squaring problem, 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 or n ...
. For instance, they may be restricted to being the closures of disjoint open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s.
Polygon dissection problem
The Bolyai–Gerwien theorem states that anypolygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
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 (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
has a dissection into any other polyhedron of the same volume using polyhedral pieces (see Dehn invariant). This process ''is'' possible, however, for any two honeycombs (such as cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
) in three dimension and any two zonohedra
In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...
of equal volume (in any dimension).
A partition into triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s of equal area is called an equidissection. Most polygons cannot be equidissected, and those that can often have restrictions on the possible numbers of triangles. For example, Monsky's theorem states that there is no odd equidissection of a square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
.
Equilateral-triangle squaring problem
Among dissection puzzles, an example is the Haberdasher's Puzzle, posed by puzzle writer Henry Dudeney
Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games. He is known as one of the foremost creators of mathematical puzzles.
Early life
Dudene ...
in 1902. It seeks a dissection from equilateral triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
into a square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
. Dudeney provided a hinged dissection with four pieces. In 2024, Erik Demaine
Erik D. Demaine (born February 28, 1981) is a Canadian-American professor of computer science at the Massachusetts Institute of Technology and a former child prodigy.
Early life and education
Demaine was born in Halifax, Nova Scotia, to mathe ...
, Tonan Kamata, and Ryuhei Uehara published a preprint claiming to prove that no dissection with fewer pieces exists.
See also
*Hilbert's third problem
The third of Hilbert's problems, Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedron, polyhedra of equal volume, is it always possible t ...
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 algor ...
Dissection Tiling
Discrete geometry Euclidean geometry Geometric dissection Polygons Polyhedra Polytopes Mathematical problems {{geometry-stub