# Hyperbolic Geometry: Definition & Postulates

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• 0:00 The History of Geometry
• 0:58 What Is Hyperbolic Geometry?
• 1:42 Hyperbolic Geometry Postulates
• 2:38 Hyperbolic Geometry…
• 3:06 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Hyperbolic geometry studies the geometry of hyperbolic, or saddle-shaped surfaces. In this lesson, learn about the history, postulates, and applications of hyperbolic geometry.

## The History of Geometry

A little over 2,000 years ago, a Greek mathematician named Euclid first wrote down the set of definitions and axioms that we now know as geometry. In Euclid's geometry, he assumed that all surfaces were flat, and he worked out relationships between lines and angles on flat surfaces. Even today, Euclidean geometry is used to understand the geometry of shapes on two-dimensional flat surfaces.

Of course, we know that not all surfaces are flat. What happens to Euclidean geometry when you try to apply it to a curved surface? Some things may still be the same, but some postulates of Euclidean geometry will not be true anymore. For example, consider the parallel postulate of Euclidean geometry. The parallel postulate says that if you have a line and a point outside the line, it's possible to draw only one line that will go through the point and also be parallel to the first line.

## What Is Hyperbolic Geometry?

While the parallel postulate is certainly true on a flat surface like a piece of paper, think about what would happen if you tried to apply the parallel postulate to a surface such as this:

This shape is technically known as a hyperbolic paraboloid, but it's commonly known as saddle shaped. On a saddle-shaped surface like this, lines will never be straight because the surface is curved. Therefore, Euclid's parallel postulate is not true on this surface because you can draw multiple lines through a single point, and they will still never cross the original line.

Any geometrical system in which the parallel postulate is violated is called a non-Euclidean geometry. The geometry of saddle-shaped surfaces like this is one type of non-Euclidean geometry known as hyperbolic geometry.

## Hyperbolic Geometry Postulates

In many ways, hyperbolic geometry is very similar to standard Euclidean geometry. However, there are a few key postulates that differentiate it. We have already seen that the parallel postulate is different. In hyperbolic geometry, it's possible to have two or more lines drawn through a single point that are all parallel to some other line.

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