If you click on the title you'll go to the Math Forum, where I'm explaining to a pre-college audience about their non-Euclidean geometry options.
Sure, you're welcome to fiddle with the 5th postulate to get other self-consistent vistas, likewise suitable for exploration and discovery.
Those bothered by "dimensionless points" and/or perhaps more drawn to the "atoms" of Democritus, will be happy to learn of our "geometry of lumps" under development since at least the 1940s (the histories will vary on this point).
Karl Menger's proposal segues nicely with later explorations, and our practice of calling all shapes "4D" in res cogitans, 4D++ in res extensa (which latter we've associated with "energy" ever since Einstein (energy has shape)).
The key paper was in an anthology on Einstein's relativity: 'Modern Geometry and the Theory of Relativity', in Albert Einstein: Philosopher-Scientist , The Library of Living Philosophers VII, edited by P. A. Schilpp, Evanston, Illinois, pp. 459-474.
For further reading:
About Karl Menger (Illinois Institute of Technology)
Small World (Grain of Sand, Aug 11, 2007)
Claymation Station (Grain of Sand, Oct 3, 2008)
Re: 4D vs. 4D vs. 4D (Coffee Shops Network, Feb 7, 2009)
Proposal Re: Dimension (Grain of Sand, Jan 12, 2009)
Egoistic Writing on 4D (Synergeo, Nov 27, 2010)