Thread by @ewanbirney

In my voyages in maths with my daughter series - we've been discussing geometry (seed question - can you prove the (n-2)*180 for the interior angles of an n-sided polygon) and this threw up some interesting things.

First off, a discussion of 180 leading to radians + Pi. Thought experiment - if there was a planet of intelligent otters and they used a different number/angle system, they would no doubt not divide the circle into 360, but the concept of a circle, half circle, would be the same

(you might guess my daughter is a fan of otters. We decided they would likely work in base 7 I think due to their tails being the extra "unsymmetric" digit. Don't ask for more details).

We then got into a heated debate of the best proof for (n-2)*180, which I think is best done as n*(half circles) due to each extension having interior + exterior is one line - (1 full circle) for turning around when you go around the polygon.

My daughter prefered - choose a point, sub-divide polygon into n-2 triangles (can't do the last two because they are lines on top of each other to the two neighbourghing points), each triangle interior angles are 180.

My argument - for her proof you need to prove interior angles of a triangle is 180; she was convinced it was the "same thing" as my line is 180... this got us to Euclid's axioms.

We had fun with triangles on a sphere (yes, definitely, one can have a triangle with 3 right angles on a sphere etc); I held off from going to space-time being hyperbolic as I am not sure I could go there.

...but... this lead me to think whether there are other "intuitive" non-eucledian geometries that relax / change the other axioms (beyond the parallel lines one) - is it meaningful to do this - and is there any of these higher geometries useful.

I also get the spherical geometry is (duh!) on a sphere; I think a torus geometry is like a plane where left meets right and up meets down (right?) - is this ... euclidan? And if this is the case, what is the geometry of a two-holed torus?

Thoughts, muses, or other things really appreciated. I no doubt will have to weave in the intelligent otters (living a torus or something like that!) when discussing with my daughter.