# Comments on *Beyond 3d*

## Kenny Liu

The discussion on the development of non-Euclidean geometry reminded me of a book called The Structure of Scientific Revolutions by Thomas Kuhn. In the book he states that scientists work within a given paradigm, or set of rules, and that when puzzles arrive that directly challenge those rules, within a short space of time, a new paradigm is established with new rules that incorporate ways to solve the puzzles while still maintaining consistency with the old data. It seems that this was the case with Gauss, et al. Unsatisfied with the fifth postulate of Euclid, they posited geometries that showed that it was not necessary. By a "simple" realization that two dimensional surfaces did not have to lie flat in space a revolution in mathematics was initiated, all it took was a reevaluation of the underlying framework (assumptions) of geometry at the time.

I already thought that it was established that our three-dimensional space doesn't satisfy Euclidean geometry by the mere fact that light rays curve as they pass massive objects. Wouldn't a triangle in a black hole have angles that sum to something other than 180 degrees, thus proving that space is curved? Obviously this hasn't been done, but it sounds right.

This is kind of off the wall but I was wondering if there were any documented histories of anyone who suddenly woke up and thought that he was backwards, in a mental institution or otherwise. Little kids sometimes insist that their right hands are actually their left hands, maybe before a certain age kids have the ability to journey to the fourth dimension. Just kidding, I don't really believe that. Also, I read in the acknowledgements that you had a theory relating the fourth dimension and the Trinity. I've had my theories, what are yours?

Prof. Banchoff's response