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 ...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.展开更多
文摘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.
