Proving a Special Case of the Coxeter-Hadwiger Conjecture

An orthoscheme or Pythagorean simplex is a solid in n-dimensional Euclidean space whose faces are right triangles. In 1956, Hadwiger asked whether an n-dimensional general (not necessarily Pythagorean) simplex can always be decomposed into a finite number of Pythagorean simplexes. Tschirpke proved in 1994 that this division is always possible in 5D space. Coxeter proved that a 3D Pythagorean simplex can be split into three smaller ones. In a 2024 paper, I generalized Coxeter’s trisection to prove that the dissection of an n-dimensional Pythagorean simplex into n pieces of the same type is possible if each leg of the original solid is equal to the unit distance. In the present paper, I extend this proof to an n-dimensional Pythagorean simplex with legs of arbitrary measure. This means the proof of the Hadwiger conjecture in the special case of a Pythagorean simplex.