Hyperbolic Geometry

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Hyperbolic Geometry
This is an animation of a square rotating in hyperbolic geometry as represented by the Poincaré Disk Model.

Contents

Basic Description

The geometry with which most people first learned to visual basic shapes such as lines, triangles, and squares is the traditional geometry that most of us are used to, formally called Euclidean geometry. In two-dimensions, Euclidean geometry is viewed in a flat, infinite plane. However, there also exists non-Euclidean geometry, examples of which include the elliptic and hyperbolic geometries.


Triangles in Different Geometries
Triangles in Different Geometries


One way to describe the difference between these three geometries is by comparing each geometry's definition of a triangle. In Euclidean geometry, triangles must have three angles that total to 180 degrees. However, in elliptic geometry, the angles in a triangle must sum to greater than 180 degrees, and in hyperbolic geometry, the angles must sum to less than 180 degrees.


Click here for a summary of the Euclidean, Hyperbolic, and Elliptic Geometries.

A More Mathematical Explanation

Euclidean geometry is governed by a list of axioms know as the Five Postulates of Euclid, who was a G [...]

Euclidean geometry is governed by a list of axioms know as the Five Postulates of Euclid, who was a Greek mathematician who lived around 300 BC. These five rules are enough to describe an entire geometry, and if we take away only one of these postulates, we will emerge with completely new geometries. This is the case with hyperbolic (and elliptic) geometry. Hyperbolic (as well as elliptic) geometry follows all of the postulates of Euclid except the final one, known as the parallel postulate, which defines parallel lines and will be addressed below.

Poincaré Disk Model

Octagon as seen in the Poincaré Disk Model
Octagon as seen in the Poincaré Disk Model

Since hyperbolic geometry is non-Euclidean, this geometry cannot be viewed in a traditional flat plane. There are various models that can be used to represent the hyperbolic plane, but the Poincaré Disk Model will be used in this discussion. The image featured at the top of this page shows a triangle rotating in the Poincaré Disk Model and the animation to the side is an octagon in the model.

To learn more about the other models commonly used when dealing with hyperbolic geometry, please click here.

Parallel Lines

First, we must give the definition of a line in hyperbolic geometry. A line is depicted in the Poincaré disk model as an arc (segment of a circle) that intersects the boundary disk at 90 degree angles (see first image above).

In Poincaré disk model, two lines that are parallel to each other in hyperbolic geometry are simply two lines that never intersect each other. However, unlike Euclidean geometry, there are an infinite number of lines parallel to a single line l. Also, parallel lines in Euclidean geometry remain the same distance from each other, while parallel lines in hyperbolic geometry actually curve away from each other.

Let's say we have a line l again and a point P that is not one the line l, and we want to find a line that goes through point P and is parallel to line l (see middle image above). In Euclidean geometry, there would only be one line that would fit this requirement; however, in hyperbolic geometry, there are any infinite number of lines that would pass through point P and be parallel to line l.

Infinite Boundary

Circles in the Poincaré disk model
Circles in the Poincaré disk model

You might wonder how we are able to place the entire infinite Euclidean plane in what seems like a finite circular area. However, the Poincaré disk model is not simply a circle, and, in fact, the distances within the Poincaré disk model are not consistent within the entire disk boundary. As we get closer and closer to the outer boundary, the distance increases so that the boundary edge is infinite.

In the image, all of the shapes depicted are circles. The circles seem to become distorted as they near the edge of the disk because the space is compressed. This allows us to create beautiful hyperbolic tilings using the Poincaré disk model.

Triangles

A triangle in the Poincaré disk model
A triangle in the Poincaré disk model


Triangles in hyperbolic geometry do not have angles that add up to 180 degrees. In fact, their three angles always add up to a number less than 180 degrees and that summation is not a set number for every triangle.






Elliptic Geometry

The Earth as an Elliptic Model with triangles drawn
The Earth as an Elliptic Model with triangles drawn

Elliptic geometry is another common non-Euclidean geometry. However, there are no parallel lines in elliptic geometry because lines tend to curve towards each other and will always intersect. Also, triangles in elliptic geometry will have angles with a sum above 180 degrees. A good model of elliptic geometry is a sphere, or for a real-life example, the globe. As we can observe from the image, a triangle has angles that deviate from the traditional 180 degree definition when the sphere is observed from a distance (angles 90-50-90). However, this changes when we place a much smaller triangle on a close-up position on the elliptic model. The elliptic plane becomes fairly flat and the triangle contains angles that nearly sum to 180 degrees (90-40-50).




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Related Links

Additional Resources

Jos Leys' article about creating a futuristic hyperbolic chamber, http://www.josleys.com/article_show.php?id=83
More basic information about hyperbolic geometry, http://euler.slu.edu/escher/index.php/Hyperbolic_Geometry
To see animations of shapes and lines represented in other models, http://www.mi.sanu.ac.yu/vismath/sazdanovic/space/main.htm
To download Cinderella (a free software) to draw geometry figures in Euclidean, Hyperbolic, and Elliptic geometries,






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