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 ...

, the Schönhardt polyhedron is the simplest non-convex

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 ...

that cannot be triangulated into

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

without adding new vertices. It is named after German mathematician

Erich Schönhardt Erich Schönhardt (born 25 June 1891 in Stuttgart, Germany, died 29 November 1979 in Stuttgart).Cauchy's rigidity theorem as an example where polyhedra with two different shapes have faces of the same shapes.

Construction

One way of constructing the Schönhardt polyhedron starts with a

triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...

, with two parallel equilateral triangles as its faces. One of the triangles is rotated around the centerline of the prism, breaking the square faces of the prism into pairs of triangles. If each of these pairs is chosen to be non-convex, the Schönhardt polyhedron is the result.

Properties

The Schönhardt polyhedron has six vertices, twelve edges, and eight triangular faces. The six vertices of the Schönhardt polyhedron can be used to form fifteen unordered pairs of vertices. Twelve of these fifteen pairs form edges of the polyhedron: there are six edges in the two equilateral triangle faces, and six edges connecting the two triangles. The remaining three edges form

diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...

s of the polyhedron, but lie entirely outside the polyhedron. The

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

of the Schönhardt polyhedron is another polyhedron with the same six vertices, and a different set of twelve edges and eight triangular faces; like the Schönhardt polyhedron, it is combinatorially equivalent to a

regular octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

. The

symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. Th ...

of the Schönhardt polyhedron consists of three tetrahedra, each lying between one of the concave dihedral edges of the Schönhardt polyhedron and one of the exterior diagonals. Thus, the Schönhardt polyhedron can be formed by removing these three tetrahedra from a convex (but irregular) octahedron.

Impossibility of triangulation

It is impossible to partition the Schönhardt polyhedron into

whose vertices are vertices of the polyhedron. More strongly, there is no tetrahedron that lies entirely inside the Schönhardt polyhedron and has vertices of the polyhedron as its four vertices. This follows from the following two properties of the Schönhardt polyhedron: *Every triangle formed by its edges is one of its faces. Therefore, because it is not a tetrahedron itself, every tetrahedron formed by four of its vertices must have an edge that it does not share with the Schönhardt polyhedron. *Every diagonal that connects two of its vertices but is not an edge of the Schönhardt polyhedron lies outside the polyhedron. Therefore, every tetrahedron that uses such a diagonal as one of its edges must also lie in part outside the Schönhardt polyhedron.

Jumping polyhedron

In connection with the theory of

flexible polyhedra In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such ...

, instances of the Schönhardt polyhedron form a "jumping polyhedron": a polyhedron that has two different rigid states, both having the same face shapes and the same orientation (convex or concave) of each edge. A model whose surface is made of a stiff but somewhat deformable material, such as cardstock, can be made to "jump" between the two shapes. A solid model could not change shape in this way. Neither could a model made of a more rigid material like glass: although it could exist in either of the two shapes, it would be unable to deform sufficiently to move between them. This stands in contrast to Cauchy's rigidity theorem, according to which, for each

convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

, there is no other polyhedron having the same face shapes and edge orientations.

Related constructions

It was shown by that the Schönhardt polyhedron can be generalized to other polyhedra, combinatorially equivalent to

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

s, that cannot be triangulated. These polyhedra are formed by connecting regular ''k''-gons in two parallel planes, twisted with respect to each other, in such a way that ''k'' of the 2''k'' edges that connect the two ''k''-gons have concave dihedrals. For sufficiently small twisting angles, the result has no triangulation. Another polyhedron that cannot be triangulated is

Jessen's icosahedron Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a Convex polyhedron, non-convex polyhedron with the same numbers of vertices, edges, and faces as the regular icosahedron. It is named for Børge Jessen, who studied it i ...

, combinatorially equivalent to a

regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It ...

. In a different direction, constructed a polyhedron that shares with the Schönhardt polyhedron the property that it has no internal

s. The

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

and the

Császár polyhedron In geometry, the Császár polyhedron () is a nonconvex toroidal polyhedron with 14 triangular faces. This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron ...

have no diagonals at all: every pair of vertices in these polyhedra forms an edge. It remains an open question whether there are any other polyhedra (with

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

boundary) without diagonals, although there exist non-manifold surfaces with no diagonals and any number of vertices greater than five.

Applications

used Schönhardt's polyhedron as the basis for a proof that it is

NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by tryi ...

to determine whether a non-convex polyhedron can be triangulated. The proof uses many copies of the Schönhardt polyhedron, with its top face removed, as

gadgets A gadget is a mechanical device or any ingenious article. Gadgets are sometimes referred to as '' gizmos''. History The etymology of the word is disputed. The word first appears as reference to an 18th-century tool in glassmaking that was devel ...

within a larger polyhedron. Any triangulation of the overall polyhedron must include a tetrahedron connecting the bottom face of each gadget to a vertex in the rest of the polyhedron that can see this bottom face. The complex pattern of obstructions between tetrahedra of this type can be used to simulate

Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denote ...

components in a reduction from the

Boolean satisfiability problem In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfie ...

References

External links

Three Untetrahedralizable Objects
D. Eppstein. Includes a rotatable 3d model of the Schönhardt polyhedron. {{DEFAULTSORT:Schonhardt polyhedron Nonconvex polyhedra