This says rectangles don't exist in hyperbolic geometry. But according to this you can model the euclidean plane in Hyperbolic geometry. Wondering why you can't model the rectangle, I thought if you can do euclidean geometry in 2d plane you can get rectangle. The question is how to get (or introduce something to get) a rectangle in hyperbolic 2d plane.
Asked
Active
Viewed 1,173 times
4
-
I suppose you could say that the Poincare disc model is a model of the hyperbolic plane in the Euclidean plane. But that doesn't mean there are Euclidean triangles with angle sum $<\pi$. – Angina Seng Mar 01 '19 at 05:07
1 Answers
4
You have to be careful with your definitions. Yes, a Horosphere in Hyperbolic 3 space is an isometric model of the Euclidean plane, but the euclidean lines in this model are not hyperbolic lines. Thus, any model of a euclidean rectangle has four right angles but its sides are not hyperbolic lines, and therefore it is not a hyperbolic rectangle. See my answer to the recent MSE question 3119728 "Description of Model of Euclidean Geometry found in the Hyperbolic Plane".
The same kind of situation arises in using a sphere in Euclidean 3 space as a model of the elliptic plane. The elliptic lines in the model are great circles of the sphere and are not euclidean lines.
Somos
- 37,457
- 3
- 35
- 85