INTERESTING

Most straight lines in Hyperbolic Geometry appear curved to us, according to our experience built Euclidean Geometry. If you could somehow be transported inside a world of Hyperbolic Geometry, then all of the "straight lines" shown in this simulation would appear perfectly straight to your vision

Moreover, each model has the property of preserving angles from the Lobachevskian plane to the model, but only Poincare models are conformal representations  of angles with respect to their Euclidean measures with the angle between two arcs of circles understood in the usual way.

One of the most interesting things about hyperbolic geometry is how so many properties differ from Euclidean geometry. One of the striking differences is that the sum angles in a triangle measure less than 180°.
The defect of a triangle is defined as:

δ(Δ)=π-α-β-γ

where α,β,γ
are angles of the triangle. Similar proposition applies to all polygons. A way to see the measure of the defect is by movements in Lobachevskian plane along the edges of a given polygon.

 Next animation shows "translations" of a given triangle (red) along the edges of another triangle (blue). The angle between the starting and the last position of the triangle we are "translating" represents the defect of the (blue) triangle.

When we construct Hyperbolic triangles, and measure the defect of each, we find that for small triangles the defect is small (the angle sum is almost 180°) and vice versa.
In fact, as the perimeter of triangle approaches zero,the angle sum approaches 180°. This is consistent with the idea that a relatively small piece of Hyperbolic Space looks, and behaves very much like Euclidean Space.
Our choice was to present this feature on regular octagon:

 Poincare Disk Poincare Halfplane

and to You we leave the question:

IS THE WORLD THAT WE ARE LIVING IN JUST A SMALL PIECE OF HYPERBOLIC SPACE?