Question

Please keep it simple for Hyperbolic Geometry (the response I was given was great, but it was well beyond what we needed)

Illustrate with a picture showing that the following are simply not true in Hyperbolic Geometry

c). The angle sum of any triangle is 180

d). Rectangles exist

Answer 1

c)

A polygon in hyperbolic geometry is a sequence of points and geodesic segments joining those points. The geodesic segments are called the sides of the polygon.

A triangle in hyperbolic geometry is a polygon with three sides, a quadrilateral is a polygon with four sides, and so on, as in Euclidean geometry. Here are some triangles in hyperbolic space:

From these pictures, you can see that:

The sum of the angles in any hyperbolic triangle is less than 180°.

Ideal Triangle

An ideal triangle consists of three geodesics that touch at the boundary of the Poincaré disk.

The three geodesics are called the sides of the ideal triangle. Since the boundary of the disk isn’t part of hyperbolic space, the sides of an ideal triangle are infintely long and never actually meet. However, the do get closer together as they head towards the edge. The three points on the boundary are called the ideal vertices of the ideal triangle, and play a similar role as the vertices of an ordinary triangle. Since the sides are all perpendicular to the Poincaré disk boundary, they make an angle of 0° with each other.

ANSWER FOR D:

It can be done knowing that a rectangle can be broken into 2 triangles and a triangle is less than 180 so a rectangle will be less than 360 making the rectangle not have 4 right angles