Is Projective Geometry the Ultimate Form of Kleinian Geometry?

[Symbreak]

Does anyone know of further geometries beyond projective geometry? Kleinian geometry seems to imply projective geometry is the most general, as objects are more congruent and there are more groups of transformations to underly the congruence. But it seems ad hoc and arbitary to have a limit on congruence - could we not say the most general geometry is that limited to 1-dimensional objects, where all things are points and in one state of congruence?

 

[robphy]

Riemannian Geometry, Finslerian Geometry, Symplectic Geometry, Algebraic Geometry, ...
... http://mathworld.wolfram.com/Geometry.html

 

[tacman]

The Kleinian view of geometry is a fascinating topic that has been studied and debated by mathematicians for centuries. It is true that Kleinian geometry does imply that projective geometry is the most general, as it allows for a wider range of transformations and congruence than other geometries. However, this does not mean that projective geometry is the only possible geometry beyond Euclidean geometry.

In fact, there are many other geometries that have been developed and studied, such as hyperbolic geometry, elliptic geometry, and non-Euclidean geometry. These geometries have their own unique properties and rules, and cannot be reduced to a one-dimensional system as you suggest.

Additionally, the idea that projective geometry is limited to 1-dimensional objects is not entirely accurate. While it is true that projective geometry can be used to study points, lines, and planes, it can also be extended to higher dimensions and more complex objects. This flexibility and versatility is one of the reasons why projective geometry is often considered the most general geometry.

In conclusion, while Kleinian geometry does suggest that projective geometry is the most general, it is not the only geometry beyond Euclidean geometry. There are many other geometries that have been developed and studied, each with its own unique properties and applications. It is important to continue exploring and studying these different geometries in order to gain a deeper understanding of the world around us.