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)leg...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.展开更多
文摘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.
