Mathematics is the language by which science expresses itself. But mathematics has originated from concepts in the human mind that again has its existential condition formed by the human body. The living matter, being the source of all this science, now becomes object of the mathematical modelling of science. This can be considered as a kind of feedback: science has an increasing influence on the living matter that developed it.

My aim is to use mainly geometry in physics and use several appealing geometric models to create several possibilities of the physical world. If experimental verification of our theories in fundamental physics become increasingly difficult or even impossible by technical instruments, we might be forced to choose a model of the world we live in by using arguments that go beyond the classical view of an independent objective universe. If such a universe in itself exists, it is most likely unknowable for us and unreachable for our theories. Our knowledge and theories are necessarily limited by the sensorial capabilities of our body and the brain that is dependent on them.

Euclidean geometry, spherical geometry, elliptic geometry, hyperbolic geometry and projective geometry can serve as mathematical tools to express properties of the real world. There is no true geometry in my opinion. A geometry is a way to say something about the real world and it is very well possible that we need all of them to express the rich properties of the real world. Hyperbolic geometry, for instance, is used in the theory of relativity because it can express the existence of a barrier: the velocity of light limits our experiencing the world. The light cone in space-time is unreachable and impenetrable for material particles. Elliptic geometry has no such barrier. Like in the case of projective geometry there is a horizon that seems to disappear if one tries to get there. It is possible to travel through projective space without ever reaching a border.

The first illustration below shows a torus surface on which 4 equidistant circles lie, each having 4 equidistant points that are 4 of the 16 vertices of a hypercube in stereographic projection. On the vertical line and the circle lie 4 equidistant points each, completing the 16 vertices of the hypercube to the 24 vertices of a 24-cell.

The next illustration below shows 2 torus surfaces on which 5 equidistant circles lie, each having 10 equidistant points that are 100 of the 120 vertices of a 600-cell in stereographic projection. On the vertical line and the circle lie 10 equidistant points each, completing the 120 vertices of the 600-cell.

To see the vertices of the Witting polytope encoded with the icosian representation:

Click Here for the interactive picture of the vertices of the Witting polytope