The inwardly mirrored tetrahedron is a model for the social level of emergence which is connected by hyper complex algebras to the level of the octonions. The quaternions are are a model of autopoiesis of the existential living organism and the complexnions are a model of the dissipative ordering of consciousness.
Would like to draw your attention to Special Systems Theory. See
Ben Goertzel also did some papers on Ons at
We had a study group called the Octonion Appreciation Group in the 90s where we collaborated on studying special systems theory which is based on Hyper Complex Algebra. This group included Onar Aam, Ben Goertzel, Tony Smith and Kent Palmer. Onar Aam realized that hyper complex algebras can be modeled as facing mirrors. He created an image of what an inwardly facing Tetrahedron would look like inside via ray tracing. And we attempted to understand the dynamics inside the inwardly mirrored tetrahedron and its vertex figure which has twelve lines intersecting at each point forming a regular polygon which can be seen in these images by Ryan Budney which is a much better rendition than that which we were using back then. I am so happy to have found these images that are key to understanding the theory visually. Onar also produced Mandelbrot type images of the quaternions and octonions at that time.
The basic idea is that the Hyper Complex Algebras are captured in the analogy of facing mirrors so the Reals are a single mirror, the Complex Numbers are two facing mirrors, the Quaternions are three facing mirrors, and the Octonions are four facing mirrors. At the sedenion level which is after the Octonions there can be no regular mirroring configuration and so Onar called this the Funhouse because the mirrors have to be either spaced apart of warped. In the theory I related the real numbers to systems, the complex numbers to Prigogine Dissipative Structures, the quaternions to autopoietic systems of Maturana and Varella, the octonions to reflexive social systems related to reflexive sociology of B. Sandywell and J. O’Malley, A. Blum and others. I see the reflexive tetrahedron as a model of the social. Beyond that the Sedenion is a model of the meta-system.
To answer the question as to what you see: This is a model of interpenetration and intrainclusion as we get in Fa Tsang’s Hua Yen Buddhism. In other words you see a model of the interpenetration of all things. This is happening dynamically in the reflections in the mirrors.
But more important than what you see is the fact that are the Octonion level the associative property as well as the commutative properties are lost in algebra and that means that who sets next to who at the dinner table matters, and also actions cannot be easily reversed. However the division property is still in tact and it will not be lost until we go to the sedenion level which is the next unfolding of the algebras. So more than what you see it is the possible dynamics that is different at these various algebraic levels, and when you lose the associative level then social relations matter, so this is a model of the emergence of the social. What ever you put into this inwardly mirrored tetrahedron is reflected on all sides. So there is closure of appearances which is still regular around each object on all sides. So this is a model of the relation between reality and appearance which is controlled and which has a reference grid which in the reflections is some kind of polytope that has an incidence of twelve edges at each vertex. With the reference grid it is possible to map back and forth between the appearances in the reflections and the actual three dimensional space of the inwardly mirrored tetrahedron. When the mirrors are spaced or warped this becomes much more difficult to transform between the apparent images in the reflections and the actual objects being mirrored within the space of the inwardly mirrored tetrahedron. Each node where 6 lines intersect is a distorted image of what B. Fuller called a vector equilibrium. There is a space-filling lattice of octrahedra and vector equlibria see