Leírás
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 by the direct extension of the corresponding formulas to a right triangle, without using the Cayley–Menger determinant. 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 quadrirectangular tetrahedron into three smaller tetrahedra of the same type, and generalize this construction to n-dimensional spaces. Finally, I analyze the connection between the Coxeter partition and the Hadwiger conjecture regarding the partition of the simplex into a finite number of orthoschemes, which I call Pythagorean simplices.