Volumes in Hyperbolic Space

This research started in the summer of 2009 with a fellow student, Michael Fry who had graduated that summer. During those summer months, we spent our time investigating volumes of large Coxeter hyperbolic polyhedron. The main focus was to explore hyperbolic geometry beyond the class room and to see if we could find the smallest possible volume for this class of objects. By the end of the summer, we had found a polyhedron which generated the smallest volume we had seen. The polyhedron we had found was the Lambert cube. Following this research, the fall of 2009 and spring of 2010, were spent on further research by myself. The goal for both semesters was supposed to lead into my being introduced to Hyperbolic Dehn Filling of ideal polyhedra. We did not get to spend as much time on it as needed, and therefore that part of my research will not be included in this research paper.

To have a better and more formal understanding of Hyperbolic Geometry, we were introduced to Curved Spaces, a program which visualizes Hyperbolic Geometry, and Orb, a program which calculates volumes of hyperbolic polyhedra. On a side quest, I tried to combine both concepts of Curved Spaces and Orb. I wanted a program which would allow for the visualization of any kind of hyperbolic polyhedra given the matrices which were generated by Orb. Curved Spaces only had five preprogrammed hyperbolic polyhedra, and to see the polyhedra one is working with makes working with the polyhedra a lot easier. However, I did not have a chance to finish the program, which is why that part is also not included in this research paper. This paper assumes that the reader is familiar and comfortable with Euclidean Geometry. Some concepts therefore will be assumed to be known to the reader and most likely not explicitly stated. Euclidean Geometry, as the name suggests, was named for Euclid (300 B.C.). Euclid established five postulates as the foundation of Euclidean Geometry:

1. Given any two points in space, there exists a unique line segment connecting these two points.

2. Any line segment can be extended indefinitely.

3. Given any point in space, a circle can be drawn with any radius around that point.

4. Any two right angles are congruent.

) Given any point P in space and a line l 1 that does not pass through P , there exists a unique line l 2 through P which does not intersect l 1 . [ab12] Mathematics is meant to describe the world we live in. Euclidean Geometry, or what people most commonly refer to as just Geometry, describes the physical world in fundamental ways. Euclid observed these things and decided that these few postulates are all that is needed in order to successfully describe everything around us. We tend to assume that we live in a Euclidean Geometry. Notions of distance and concepts of triangles are well defined to and by us. We can find the shortest distance between two points by using the Pythagorean Theorem, assuming, of course, that we superimpose a grid on the plane the two points lie in. A triangle is simply defined to be three points in space which are joined by three line segments. We also know that the sum of the interior angles of a triangles sum up to π. There are also fundamental operations called isometries which are the motions of the plane that preserve distances and angles, and whose existence is added to the axioms of Euclidean geometry. There are three isometries: translations, rotations, and reflections.

[ab12] A translation can be thought of as the shifting of the plane by a scalar; translations have no fixed points or is what is called the identity. A rotation fixes one point P in space and rotates space around P by a θ degrees. A reflection fixes a line l in space and reflects every point through l. The only points that are not moved by the reflection are the points that are on l.

When people are first introduced to Non-Euclidean Geometry, it comes as a shock, because people are rarely taught anything outside of Euclidean Geometry. A question to ask yourself is, do we actually live in a world that follows the rules Euclid set down over two thousand years ago, or do we live in another kind of geometry? It is hard to imagine that we live in anything but Euclidean Geometry. At the same time, we are very small and the world is very big. For a very long time, we did not even know that the earth was round! Mathematicians found ways to describe the world we live in, found geometries which followed all of Euclid’s postulates, except the 5th one; and so, Hyperbolic Geometry was born. In Hyperbolic Geometry, all of Euclid’s postulates hold, except the Parallel Postulate. The revised 5th postulate for Hyperbolic Geometry goes as follows: “Given any point P in space and a line l 1 , there are infinitely many lines through P which are parallel to l 1 " [ab12].

Envisioning the hyperbolic plane, H 2 , is for the most part impossible, hence models need to be used in order to work with H 2 or any higher dimensions. In the case

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