# NonEuclid: Interactive Javascript Software for Creating Straightedge and Collapsible Compass Constructions in the Poincaré Disk Model of Hyperbolic Geometry

If our browser supported the HTML5 canvas and WebGL, there would be a nice animation here :(

NonEuclid allows the curious explorer to gain experience in Hyperbolic Geometry and to empirically investigate questions such as: "Does the Euclidean geometry method for constructing an equilateral triangle work in Hyperbolic geometry?" or "In Hyperbolic Geometry, are the base angles of an isosceles triangle congruent?"

Aside from being interesting in itself, a study of hyperbolic geometry can, through its novelty, be helpful to high school geometry students. For example, when asked to prove that the opposite sides of a rectangle have the same length, many beginning geometry students will be confused about what to do. Students sometimes think: "Why I am being told to prove what I learned in kindergarten is just part of the definition of what it means for a figure to be a rectangle." The strangeness of hyperbolic geometry helps such students think about and understand the difference between what is part of an object's definition and what is a theorem about an object.

Hyperbolic Geometry also has practical aspects such as orbit prediction of objects within intense gravitational fields. Hyperbolic Geometry is used in Einstein's General Theory of Relativity and Curved Hyperspace.

## Authors: Joel Castellanos, Dept. of Computer Science, University of New Mexico Joe Dan Austin, Dept. of Education, Rice University Ervan Darnell, Dept. of Computer Science, Rice UniversityFunding for NonEuclid has been provided by: The Center for High Performance Software Research (HiPerSoft), Rice University, and The Institute for Advanced Study / Park City Mathematics Institute

Polar Coordinates:

The above new Javascript version is still under development. The older Java version is: NonEuclid.jar To run this, download, and either double-click or use the command:
java" -jar NonEuclid.jar

### Contents:

 3) Activities: - Exploring properties in Hyperbolic Geometry of Adjacent Angles, General Triangles, Isosceles Triangles, Equilateral Triangle, Right Triangles, Congruent Triangles, Rectangles, Squares, Parallelograms, Rhombuses, Polygons, Circles, and Tessellations of the Plane. 4) The Shape of Space: - Curved Space, Flatland, Ourland, and Mercury's Orbit. 6) Axioms and Theorems: - Euclid's Postulates, Hyperbolic Parallel Postulate, SAS Postulate, Hyperbolic Geometry Proofs. 8) X-Y Coordinate System: - A description of how an x-y coordinate system can be set up in Hyperbolic Geometry. 9) Disk and Upper Half-Plane Models: - An informal development of these two models of Hyperbolic Geometry.

### References, Appendices, and Supporting Information:

 For The Teacher: Why is it Important for Students to Study Hyperbolic Geometry? References & Further Reading. Wikipedia: Euclidian Compass-and-straightedge Construction. Hyperbolic Geometry Art by Clifford Singer Back when NonEuclid and the Internet were young, some of the young Clifford Singer's art was hosted on this website. Here you will find the original scans form the early 1990s as well as links to Clifford's newer works in mathematically inspired art.

For more information, questions, bug reports, or comments send e-mail to Joel Castellanos joel@unm.edu