# What is the sum of interior angles in a hyperbolic n-gon?

• GeometryIsHARD
In summary, in hyperbolic geometry, the sum of interior angle measures of a polygon is always less than 180 degrees. This can be proven by dividing the n-gon into n-2 triangles, similar to the proof in Euclidean geometry, using the fact that any two distinct points in hyperbolic space can be connected by a line segment. This applies to both triangles and n-gons in hyperbolic geometry.
GeometryIsHARD

## Homework Statement

Given that the sum of interior angle measures of a triangle in hyperbolic geometry must be less than 180 degree's, what can we say about the sum of the interior angle measures of a hyperbolic n-gon?

## The Attempt at a Solution

So in normal geometry an n-gon has to have interior angles of at least (n-2)*180 because an n-gon can be filled in with n-2 triangles that each have interior angles of at least 180's ... is this something like that? Or maybe all n-gon's must have interior angles less than 180 because hyperbolic geometry doesn't obey the normal rules it seems. I'm quite lost. Can somebody here help me understand what's going on?

Hint: Does the proof that the sum of interior angles in Euclidean geometry depend on the parallel postulate or any of its corollaries (besides the corollary that the sum of interior angles of a triangle equals 180)? If not, does the proof still work in hyperbolic geometry if you replace "equals 180" by "less than 180"?

GeometryIsHARD
Hmm, I think I see what you are saying. But how can we relate the Euclid triangle to a hyperbolic n-gon? seems like a pretty far stretch

GeometryIsHARD said:
Hmm, I think I see what you are saying. But how can we relate the Euclid triangle to a hyperbolic n-gon? seems like a pretty far stretch

We don't. We relate the hyperbolic triangle with the hyperbolic n-gon.

Ahh i see. But can we relate a hyperbolic triangle to a hyperbolic n-gon the same way we do normal ones? Can a hyperbolic n-gon be divided into n-2 triangles such that the interior angles of the triangle coincide with the interior angles of the n-gon?

Recall that the Euclidean proof starts by drawing line segments from a given vertex of the Euclidean n-gon to every other vertex, unless it's already connected to the given vertex by an edge. This only assumes that every pair of distinct points determines a line segment. There's nothing about parallelism in that, so the same rule applies just as well in hyperbolic geometry. Any two distinct points in hyperbolic space can be connected by a hyperbolic line segment. So you can follow the same procedure.

GeometryIsHARD
So when you say follow the same procedure, you are saying that indeed a hyperbolic n-gon can be divided into n-2 triangles?

Can you draw lines between every vertex in the n-gon? Then you can subdivide it. If you don't believe me, try an example for a pentagon on the Poincarre half plane.

GeometryIsHARD

## 1. What is N-gon hyperbolic geometry?

N-gon hyperbolic geometry is a type of non-Euclidean geometry that studies the properties of polygons in a hyperbolic space. In contrast to Euclidean geometry, where the sum of the interior angles of a polygon is always 180 degrees, the sum of the interior angles of an N-gon in hyperbolic geometry is less than 180 degrees.

## 2. What is a hyperbolic space?

A hyperbolic space is a non-Euclidean space in which the parallel postulate of Euclidean geometry does not hold. This means that there can be multiple lines passing through a point parallel to a given line. Hyperbolic spaces are characterized by their negative curvature, which leads to unique geometric properties.

## 3. How is N-gon hyperbolic geometry used in science?

N-gon hyperbolic geometry has applications in various fields of science, including physics, computer science, and biology. It is used to study the behavior of particles in curved space-time, to design efficient algorithms for routing networks, and to model the growth of biological organisms in non-Euclidean environments.

## 4. What are some properties of N-gon hyperbolic geometry?

Some key properties of N-gon hyperbolic geometry include the fact that the angles of a polygon add up to less than 180 degrees, the existence of multiple parallel lines through a point, and the existence of infinitely many lines perpendicular to a given line passing through a point. It also follows the axioms of hyperbolic geometry, such as the existence of at least two distinct parallel lines to a given line through a point not on the line.

## 5. How is N-gon hyperbolic geometry related to other types of geometry?

N-gon hyperbolic geometry is a type of non-Euclidean geometry, along with elliptic geometry. It is also closely related to spherical geometry, which studies the properties of shapes on a curved surface. While Euclidean geometry is based on the parallel postulate, non-Euclidean geometries do not follow this postulate and have distinct geometric properties.

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