First, I should disclose that I am no mathematician, I am an architect and artist with a crush on geometry :) So please be lenient if my question is non-sense!
I'm interested in exploring the possibility of rewriting trigonometry and other functions in a tetrahedral coordinate system. I've been thinking about how the progression of dimensions from 0D to 3D could lead to the tetrahedron being a natural next step instead the cube inferred by the cartesian coordinate system, and how this could potentially simplify calculations and reveal new insights into mathematical concepts.
Starting from 0D, a point is defined by a single coordinate. Moving to 1D, a line is defined by two coordinates, and in 2D, a plane is defined by three coordinates. When we move to 3D, a cube is defined by eight coordinates, and we use a Cartesian coordinate system to represent these coordinates.
However, a the tetrahedron has four vertices and six planes, and would seem thus more "logical" or rigorous in the progression. The tetrahedron has several interesting properties, such as its symmetry and its ability to divide space into four equal volumes. Because of these properties, could the tetrahedron provide a more symmetrical representation of space than the cube, simplify certain calculations and make certain mathematical concepts easier to understand?
Therefore, I'm wondering if it is possible to rewrite trigonometry and other functions in a tetrahedral coordinate system, and how this could potentially be done. I'm also interested in any existing literature or resources on tetrahedral coordinates and their properties, and any potential challenges or drawbacks to using a tetrahedral coordinate system.
Thank you for your insights and suggestions!