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The foundations of Euclidean geometry essentially rest on five geometrical postulates. The first four of these express "obvious" geometric relations among objects, such as "through any two points, there passes one and only one line", and have always been considered more or less "self-evident". The fifth postulate (the "parallelism" postulate) has not: centuries of mathematical effort were devoted to attempts to prove it from the other four. These efforts were ultimately unsuccessful: the fifth postulate was eventually determined to be independent of the others, and perfectly consistent geometries were constructed where it is false.
The fifth postulate is equivalent to the following statement: through any point not on a given line, there passes one and only one line which does not intersect the first. This statement can fail in either of two ways: there is either no such line, or there are more than one. The Poincaré universe is a "toy universe" where the latter situation prevails.
In this section of the course, we will discuss the Poincaré universe as a subset of the complex plane. We'll use what we've learned about Möbius geometry to develop the properties of this universe, and investigate where and how these properties differ from Euclidean ones.
Before starting our tour, here are links to some preliminary information.
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If you haven't encountered a BinderBaby booklet before (or just need a reminder), you can find out how to navigate one. |
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The live sketches and Poincaré tools of the prototype are powered by Geometer's Sketchpad. Here's how you get Sketchpad and set up your browser to use it. |
| Once you've done that, you can get tools from the Poincaré Toolbox by clicking on anywhere you find it, or get a copy of the empty Poincaré universe to play with by clicking on . |
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Further info about this demo.