Dissection into orthoschemes explained
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. If true, then more generally every convex polytope could be dissected into orthoschemes.
Definitions and statement
In this context, a simplex in
-dimensional
Euclidean space is the
convex hull of
points that do not all lie in a common
hyperplane. 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). The points that form the simplex in this way are called its
vertices.
An orthoscheme, also called a path simplex, is a special kind of simplex. In it, the vertices can be connected by a path, such that every two edges in the path are at right angles to each other. A two-dimensional orthoscheme is a right triangle. 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.
A dissection of a shape
(which may be any
closed set in Euclidean space) is a representation of
as a union of other shapes whose
interiors are
disjoint from each other. 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
dimensions can be dissected into
orthoschemes.
Hadwiger's conjecture is that there is a function
such that every
-dimensional simplex can be dissected into at most
orthoschemes. Hadwiger posed this problem in 1956; it remains unsolved in general, although special cases for small values of
are known.
In small dimensions
In two dimensions, every triangle can be dissected into at most two right triangles, by dropping an altitude from its widest angle onto its longest edge.
In three dimensions, some tetrahedra can be dissected in a similar way, by dropping an altitude perpendicularly from a vertex
to a point
in an opposite face, connecting
perpendicularly to the sides of the face, and using the three-edge perpendicular paths through
and
to a side and then to a vertex of the face. However, this does not always work. 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, proved that every tetrahedron can be dissected into at most 12 orthoschemes. proved that this is optimal: there exist tetrahedra that cannot be dissected into fewer than 12 orthoschemes. In the same paper, Böhm also generalized Lenhard's result to three-dimensional
spherical geometry and three-dimensional
hyperbolic geometry.
In four dimensions, at most 500 orthoschemes are needed. In five dimensions, a finite number of orthoschemes is again needed, roughly bounded as at most 12.5 million. Again, this applies to spherical geometry and hyperbolic geometry as well as to Euclidean geometry.
Hadwiger's conjecture remains unproven for all dimensions greater than five.
Consequences
Every convex polytope may be dissected into simplexes. Therefore, if Hadwiger's conjecture is true, every convex polytope would also have a dissection into orthoschemes.
A related result is that every orthoscheme can itself be dissected into
or
smaller orthoschemes. Therefore, for simplexes that can be partitioned into orthoschemes, their dissections can have arbitrarily large numbers of orthoschemes
