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 (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. Additionally, to avoid set-theoretic issues related to the Banach–Tarski paradox 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 ...

Polygon dissection problem

The Bolyai–Gerwien theorem states that any 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 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 in 1902. It seeks a dissection from equilateral triangle into a

. Dudeney provided a hinged dissection with four pieces. In 2024, Erik Demaine, Tonan Kamata, and Ryuhei Uehara published a preprint claiming to prove that no dissection with fewer pieces exists.

References

External links

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

Polygon dissection problem

Equilateral-triangle squaring problem

See also

References

External links