1: An Invitation to Geometry
How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?
-- Albert Einstein
Out of nothing I have created a strange new universe.
-- János Bolyai
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- 1.1: Introduction
- The infinite plane model of the two-dimensional universe works well enough for most purposes, but cosmologists and mathematicians, who notice that everything within the universe is finite, consider the possibility that the universe itself is finite. Would a finite universe have a boundary? Can it have an edge, a point beyond which one cannot travel? A two-dimensional mathematician suggests that the universe looks like a rectangular region with opposite edges identified.
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- 1.2: A Brief History of Geometry
- Geometry is one of the oldest branches of mathematics, and most important among texts is Euclid's Elements. His text begins with 23 definitions, 5 postulates, and 5 common notions. From there Euclid starts proving results about geometry using a rigorous logical method, and many of us have been asked to do the same in high school.
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- 1.3: Geometry on Surfaces- A First Look
- Think for a minute about the space we live in. Think about objects that live in our space. Do the features of objects change when they move around in our space? If I pick up this paper and move it across the room, will it shrink? Will it become a broom? If you draw a triangle on this page, the angles of the triangle will add to 180°. In fact, any triangle drawn anywhere on the page has this property. Thus, Euclidean geometry on this flat page is considered to be homogenous.