Schönhardt polyhedron

In geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices. It is named after German mathematician Erich Schönhardt, who first described it in 1928.

The Schönhardt polyhedron can be formed by two congruent equilateral triangles in two parallel planes, such that the line through the centers of the triangles is perpendicular to the planes. The two triangles should be twisted with respect to each other, so that they are neither translates of each other nor 180-degree reflections of each other.

The convex hull of these two triangles forms a convex polyhedron that is combinatorially equivalent to a regular octahedron; along with the triangle edges, it has six edges connecting the two triangles to each other, with two different lengths, and three interior diagonals. The Schönhardt polyhedron is formed by removing the longer of the three connecting edges, and replacing them by the three diagonals of the convex hull.

Alternatively, the Schönhardt polyhedron can be formed by removing three disjoint tetrahedra from this convex hull: each of the removed tetrahedra is the convex hull of four vertices from the two triangles, two from each triangle. This removal causes the longer of the three connecting edges to be replaced by three new edges with concave dihedral angles, forming a nonconvex polyhedron.

The Schönhardt polyhedron is combinatorially equivalent to the regular octahedron: its vertices, edges, and faces can be placed in one-to-one correspondence with the features of a regular octahedron. However, unlike the regular octahedron, three of its edges have concave dihedral angles, and these three edges form a perfect matching of the graph of the octahedron; this fact is sufficient to show that it cannot be triangulated.

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