Our website opens with an image of two spherical tetrahedra that are duals, (objects that result when a dot is put in the center of its faces and then the dots are connected). In the past, the tetrahedron was considered a self-dual. However, the image shows two distinct forms, the one on the left is left-handed and the one on the right is right-handed. These two forms are not superimposable. They are mirror images that do not align with one another, even if you try to rotate or flip them. These concepts open a portal to a new form of spherical mathematics based on the Platonic solids that are unique 3D shapes where all faces, edges, and angles are the same. We employ group theory (a branch of mathematics that studies symmetry and how objects can be transformed), and matrices (that are grids of numbers used in mathematics to represent transformations or movements) in constructing this mathematics. We demonstrate that matrices also have a dual form that are mirror images of one another.

The first matrix illustrates the main diagonal and is defined as the identity, which means it does not change anything, when multiplied with another matrix. The second matrix is the mirror image of the first. We give a geometric meaning to the secondary diagonal, linking it to imaginary numbers, (numbers that are multiples of the square root of -1, used in advanced mathematics). Because there are two different forms of the tetrahedron, we suggest that there are actually six Platonic solids, grouped into three pairs of duals. Using group theory, we incorporate an observer into the mathematics with the identity element and the tetrahedron. In a Euclidian model, the tetrahedron is a 3D shape with four triangular faces. The vertical line that faces you divides the surface into a left and a right triangle corresponding to your left and right side. When you rotate the tetrahedron 180°, it is now dividing the surface into an upper and lower triangle corresponding to the back of your head and feet. Our goal is for the observer to share the same orientation as the viewer of the images on the screen.

In Chapter Two, we tell the story of how I finally was published and includes the paper, Optimal Spherical Packing of Circles and Hilbert’s 14th Problem. Where the image of the dual tetrahedra symbolizes an observer facing us. This is one of the reasons why the image and matrices are incorrect. We correct the matrices and images in later papers. My original paper was changed from the Internet to make it more readable, otherwise, it remains completely original. For those not versed in mathematics, focus on the history and concepts rather than the detailed mathematics. A forthcoming video will further illustrate the duality of the tetrahedron.