The Right Triangle as the Simplex in 2D Euclidean Space, Generalized to n Dimensions

The purpose of the research is to show that the general triangle can be re-placed 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.


Introduction
Various generalizations of the Pythagoras Theorem have been known for centuries. Among these, the construction most relevant to this article investigates a trirectangular tetrahedron with three faces of right triangles. The three right angles of the triangles meet at one vertex of the tetrahedron (De Gua [1]). This extension of the Pythagoras Theorem can also be implemented in higher dimensions, but with results completely different from those described in this article.
Until recently, the subject of n-dimensional geometries was very alien for me. This conception can be extended to hyperbolic and Euclidean geometry for segments of perpendicular straight lines. Figure 1 shows a spherical Napier pentagram, Figure 2, a hyperbolic rectangular Napier pentagon, both 5-cycles. (All pentagons on Figure 2 are regular hyperbolic Napier pentagons.) The adjacent perpendiculars represent incident elements, while their point of intersection is omitted from the cycle.
It follows that spherical or hyperbolic geometry allows 5-cycles to be simplices instead of 6-cycles of general triangles with alternating vertices and sides.
The next step is to apply the same thought to Euclidean plane geometry. The Euclidean 5-cycle is a right triangle with two vertices and three sides, including the two perpendiculars. But can we proceed to higher dimensions to find shapes   This finding inspired me to explore the shape in 3D Euclidean space and move to higher dimensions.

Simplices in n-Dimensional Euclidean Spaces
The initial idea is based on an axiom of spherical geometry: If two equators are perpendicular, the pole point of one is on the equator of the other, and vice versa. It follows that the sentences "Two great circles are perpendicular" and "A point and a great circle coincide" are equivalent, interchangeable statements about points and great circles in the same construction. Two perpendicular straight lines represent a special case of the incidence relation.
Given a system of geometry in which we define the incidence of a point and a straight line, and the perpendicular property between two straight lines. Consider both cases as special cases of incidence. A cycle of incidence is an ordered series of elements in which any two adjacent elements are incident, including the last and the first.
Any polygon represents a cycle of incidence. A triangle is a 6-cycle, a quadrilateral is an 8-cycle, and so on. However, perpendicular straight lines as incident elements allow for cycles with an odd number of elements. For example, a triangle with one right angle is a 5-cycle in Euclidean, spherical or hyperbolic geometry. The vertex at the intersection of the two legs is not counted, and the cycle consists of five elements, namely, two vertices and three sides, including the two perpendiculars.
In this sense, the right triangle is not a special case of the general triangle. On the contrary, the right triangle, the 5-cycle is the simplex, and the general triangle, the 6-cycle is a composite shape derived from the right triangle.
The main subject of this article is the Euclidean case of the 5-cycle as the simplex, which seems to be the least suitable for the purpose. Regular 5-cycles are excluded here, since regular right triangles do not exist in Euclidean geometry, in contrast with the regular spherical Napier pentagram or the regular hyperbolic Napier pentagon.
The task is as follows: We are looking for a new type of simplex in two-, three-, ... n-dimensional Euclidean geometry. Each face is a right-angled triangle, and the n-dimensional shape consists of (n + 1) number of (n − 1)-dimensional shapes.

The Simplex in Two Dimensions, n = 2
The measure of the angles and sides are all correct on the picture ( Figure 3).
Remark. Right angles in the present paper are indicated as arcs between two perpendicular sides, regardless of their apparent length. (The usual notations of a right angle proved confusing for the drawings.)

The Simplex in Three Dimensions, n = 3
While the 2D configuration is easy to construct, the 3D shape is by no means trivial. We are looking for a tetrahedron of which all four faces are right triangles.
This shape does exist and can be constructed in the Euclidean 3D space by cutting a rectangular cuboid of dimensions a, b, c along the plane of a space diagonal and a face diagonal ( Figure 4 and Figure 5).  Given three non-coplanar and non-concurrent sides, a perpendicular to b, b to c, c to 2D subspace ab.

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In the Euclidean 3D space, the shape is a tetrahedron with six sides, (a), (b), , and four faces of right triangles, b c + , a, and hypotenuse Both ways yield the same result This 3D shape cannot be realized on a 2D flat surface.

Displaying Right-Angled n-Dimensional Shapes on Regular Planar Polygons
In order to proceed to higher dimensions, it is of advantage to turn to regular Petrie polygons (cf. Coxeter [4]). The idea is to display right-angled shapes in distorted form. The vertices, faces and edges of the shape are represented by the sides, diagonals and angles of regular polygons on flat surface. Visualization is more difficult in 2D and 3D cases, but generalization is easier in higher dimensions.
The right-angled 2D triangle is displayed on the sides and angles of a regular 2D triangle with sides and angles distorted (Figure 7, cf., Figure 3). This shape has 3 vertices, 1 face of a right-angled Euclidean triangle, and 3 sides. It can be realized in 2D Euclidean plane.
The quadrirectangular tetrahedron is displayed on a regular 2D square with sides, diagonals and angles distorted (Figure 8, cf., Figure 6). This shape has 4 vertices, 4 faces of right-angled Euclidean triangles, and 6 edges. It can be   realized in 3D Euclidean space, but not on 2D Euclidean plane.

The 4D Right-Angled Pentachoron
This shape has 5 vertices, 10 faces of Euclidean triangles and 10 edges, just as with the general pentachoron. Each face is a right triangle, so the entire shape has a total of 10 right angles. It requires four dimensions to construct, and cannot be realized in 3D Euclidean space.
Enter four independent data a, b, c, d for which the segment of length a is perpendicular to b, b to c, c to d. Any four vertices determine a quadrirectangular tetrahedron. Figure 9 shows the initial position, while Figure 10 shows the completed construction.
Any four vertices determine a right-angled tetrahedron. Table 1 shows the defining equations of the sides of the four right triangles for each tetrahedron, as illustrated on the completed pentachoron on Figure 10     dimensional simplices.
The proof is based on induction: Suppose that the statement is valid for all (n − 1) dimensional right-angled simplices. The detailed proof for arbitrary n is very clumsy to describe, so I illustrate the pattern for the n = 6 case ( Figure 11). =  the legs of the n-dimensional right-angled simplex. By the same logic, the right-angled simplices can be called Pythagorean shapes in n-dimensional space (Pythagorean triangle, Pythagorean tetrahedron, Pythagorean pentachoron, etc.).

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On Figure 12, side  Table   3.
The same pattern can be applied to arbitrary dimension, with

Volume of a Right-Angled Simplex in 3D Euclidean Space
Apply the Cayley-Menger determinant to determine the volume by the length of sides for the 3D tetrahedron:

The Generalized Pythagoras Theorem in n-Dimensional Euclidean Spaces
Theorem 2: Given a right-angled simplex with 1 n + vertices of the defining Petrie polygon in an n-dimensional Euclidean space. Denote

Decomposing a General Tetrahedron into Pythagorean Tetrahedrons
In 2D Euclidean space, on a flat surface a triangle can be decomposed into two Pythagorean triangles by an altitude of the triangle (Figure 13).
In the 3D Euclidean space, a similar method gives six right-angled Pythagorean tetrahedra (Figure 14). Figure 13. Decomposing a 2D triangle into two right triangles. Figure 14. Decomposing a tetrahedron into six Pythagorean tetrahedra.

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On Figure 14, tetrahedron ABCD is divided into six Pythagorean (quadrirectangular) tetrahedra.
The volume of the regular tetrahedron ABCD is equal to six times the volume P V of the Pythagorean tetrahedron ABMO: ( ) The decomposition can be generalized in like manner to higher dimensions.

Conclusions
The article offers a new kind of simplex in n-dimensional spaces and a generali- This option leads to an alternative construction of multidimensional geometries using rectangular simplexes as building blocks, instead of the traditional general triangles. It can simplify theorems and techniques in other areas of multidimensional geometries and their applications, in determinant theory or ndimensional calculus. Moreover, it can be connected with recent tendencies of using Euclidean geometry, "euclidicity" in the four-dimensional space of the Theorem of Relativity (Machotka [5]). Another challenge is whether Pythagorean shapes can be applied in the theory of the Cayley-Menger determinant in spherical and hyperbolic spaces (cf. Tao [6] or Audet [7]).

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.