Dissection into orthoschemes

In geometry, it is an unsolved conjecture of Hugo Hadwiger that every simplex can be dissected into orthoschemes, using a number of orthoschemes bounded by a function of the dimension of the simplex.

[1] If true, then more generally every convex polytope could be dissected into orthoschemes.

For example, a 2-dimensional simplex is just a triangle (the convex hull of three points in the plane) and a 3-dimensional simplex is a tetrahedron (the convex of four points in three-dimensional space).

A three-dimensional orthoscheme can be constructed from a cube by finding a path of three edges of the cube that do not all lie on the same square face, and forming the convex hull of the four vertices on this path.

That is, intuitively, the shapes in the union do not overlap, although they may share points on their boundaries.

For instance, a cube can be dissected into six three-dimensional orthoschemes.

A similar result applies more generally: every hypercube or hyperrectangle in

Hadwiger posed this problem in 1956;[2] it remains unsolved in general, although special cases for small values of

[2] In three dimensions, some tetrahedra can be dissected in a similar way, by dropping an altitude perpendicularly from a vertex

In particular, there exist tetrahedra for which none of the vertices have altitudes with a foot inside the opposite face.

Using a more complicated construction, Lenhard (1960) proved that every tetrahedron can be dissected into at most 12 orthoschemes.

[3] Böhm (1980) proved that this is optimal: there exist tetrahedra that cannot be dissected into fewer than 12 orthoschemes.

[5] In five dimensions, a finite number of orthoschemes is again needed, roughly bounded as at most 12.5 million.

[6] Hadwiger's conjecture remains unproven for all dimensions greater than five.

Therefore, if Hadwiger's conjecture is true, every convex polytope would also have a dissection into orthoschemes.