TITLE:
The Right Triangle as the Simplex in 2D Euclidean Space, Generalized to n Dimensions
AUTHORS:
István Lénárt
KEYWORDS:
Cycles of Incidence, Quadrirectangular Tetrahedron, Rectangular Pentachoron, Generalization of Pythagoras Theorem, Volume of a Rectangular Simplex, Cayley-Menger Determinant
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.10 No.9,
September
30,
2022
ABSTRACT: The purpose of the research is to show that the general triangle can be replaced by the right-angled triangle as the 2D simplex, and this concept can be generalized to any higher dimensions. The main results are that such forms do exist in any dimensions; meet the requirements usually placed on an n-dimensional simplex; a hypotenuse and legs can be defined in these shapes; and a formula can be given to calculate the volume of the shape solely from the legs by a direct generalization of the Pythagorean Theorem, without computing the Cayley-Menger determinant.