TITLE:
The Coxeter Trisection and the Hadwiger Conjecture in Multidimensional Spaces
AUTHORS:
István Lénárt
KEYWORDS:
Generalized Pythagoras Theorem, Description of a Pythagorean Simplex, Pythagorean Unit Simplex, Coxeter Partition of a Simplex in -Dimensional Space, Relation to the Hadwiger Conjecture
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.12,
December
31,
2024
ABSTRACT: In this article, I consider the right triangle as the simplex in the Euclidean plane, and extend this definition to higher dimensions. The
n
-dimensional simplex has one hypotenuse and
(
n−1
)
legs (catheti). The
(
n−1
)
legs define an orthogonal path of edges in the solid with perpendicular adjacent edges along the path. The length of the hypotenuse and the volume of the solid can be calculated without the Cayley-Menger determinant, by direct extension of the corresponding right triangle formulas. I give a proof of the existence of these shapes, describe the distribution of right angles in them, give an algebraic proof of the Coxeter trisection of a right tetrahedron into three smaller right tetrahedra, and generalize this construction to
n
-dimensional spaces. Finally, I investigate the connection between the Coxeter partition and the Hadwiger conjecture on the partition of the simplex into orthoschemes, which I call Pythagorean simplexes.