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Schönhardt polyhedron

 

 
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Schönhardt polyhedron
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the Schönhardt polyhedron is the smallest non-convex
Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
 polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 which cannot be triangulated
Triangulation (advanced geometry)

In advanced geometry, in the most general meaning, triangulation is a subdivision of a geometric object into simplex. In particular, in the plane it is a subdivision into triangle s, hence the name....
 into tetrahedra
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
 without adding new vertices. It is combinatorially equivalent to the octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
, and has concave dihedral angle
Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
s on three edges forming a perfect matching of the graph of the octahedron; beyond this qualitative description, the precise vertex coordinates are not critical. Among any four vertices of the Schönhardt polyhedron, at least two must be non-adjacent; the line segment connecting these two vertices lies entirely outside the polyhedron.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the Schönhardt polyhedron is the smallest non-convex
Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
 polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 which cannot be triangulated
Triangulation (advanced geometry)

In advanced geometry, in the most general meaning, triangulation is a subdivision of a geometric object into simplex. In particular, in the plane it is a subdivision into triangle s, hence the name....
 into tetrahedra
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
 without adding new vertices. It is combinatorially equivalent to the octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
, and has concave dihedral angle
Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
s on three edges forming a perfect matching of the graph of the octahedron; beyond this qualitative description, the precise vertex coordinates are not critical. Among any four vertices of the Schönhardt polyhedron, at least two must be non-adjacent; the line segment connecting these two vertices lies entirely outside the polyhedron. Therefore, there is no tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
 defined by four Schönhardt polyhedron vertices that is contained within the polyhedron.

It was shown by that this construction can be generalized to other polyhedra, combinatorially equivalent to antiprism
Antiprism

An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
s, that cannot be triangulated. Another example of such polyhedron is Jessen's icosahedron
Jessen's icosahedron

Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron is a convex set polyhedron with the same number of vertices, edges and faces as the regular icosahedron....
, which is notable by itself. In a different direction, constructed a polyhedron with no internal diagonal
Diagonal

A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
s. used Schönhardt's polyhedron as the basis for a proof that it is NP-complete
NP-complete

In computational complexity theory, the complexity class NP-complete is a class of problems having two properties:* Any given solution to the problem can be verified quickly ; the set of problems with this property is called NP ....
 to determine whether a non-convex polyhedron can be triangulated.

See also

  • Jessen's icosahedron
    Jessen's icosahedron

    Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron is a convex set polyhedron with the same number of vertices, edges and faces as the regular icosahedron....


External links

  • , D. Eppstein
    David Eppstein

    David Arthur Eppstein is an English-born American professor of computer science at University of California, Irvine and a mathematician. His is known for his work in computational geometry, Graph theory and recreational mathematics....
    . Includes a rotatable 3d model of the Schönhardt polyhedron.