Teacher resources and professional development across the curriculum

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Unit 8

Geometries Beyond Euclid

8.1 Introduction

"It should be known that geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence."

-Ibn Khaldun (1332 - 1406)

Math encompasses far more than the study of numbers. At its heart, it is the application of logic in the search for order in the world around us. A fundamental question in this search is how to divide up, or describe, space. The study of this problem, geometry, has been of importance for thousands of years. The geometers of Ancient Egypt established geometric concepts and rules that form the basis of a discussion that has continued into the modern age.

The Egyptians were concerned with a variety of everyday geometric challenges, from how to divide up lands that had been flooded, to the construction of the pyramids. In fact, the need to measure and divide up land helped bring the word “geometry” into existence—“geo” meaning “earth,” and “meter” meaning “to measure.” The meanings of both of these roots have been expanded throughout the centuries so that now, the “earth” aspect can be thought of as encompassing all of space in general, and the “measure” element can be thought of as “divide into regular sections.” Thus, a more useful, modern-day definition of geometry is “the study of how to break space up into regular sections.”

As with other mathematical ideas, the geometric concepts of the Egyptians did not stay confined to North Africa, but rather spread across the Mediterranean. Points, lines, circles, and planes formed the vocabulary of a new kind of thinking, one that was tied to empirical observations, and yet could exist without them. The Greeks latched onto this notion of conceptual mathematics, and soon complicated geometric ideas were being constructed with only the most basic of theoretical tools. Much of this knowledge, accumulated over centuries, was collected and expanded upon by the great mathematician Euclid of Alexandria around 300 BC. His comprehensive collection of geometric knowledge, entitled The Elements, went on to become the authoritative math book throughout the world, with over a thousand editions since its initial printing in 1482.

Of central importance to Euclid were his postulates. These were statements that could not be proven and had to be agreed upon as a starting point. His five postulates described a world of straight lines and flat planes. The shapes he focused on were idealized versions of shapes found in nature. The geometric world Euclid described was, and still is, a wondrous achievement of logical construction. It is a world that behaves self-consistently, lending credence to the idea that it is a model of the “real” world. This idea, that statements about the real world can be made on the basis of reason alone, has guided much of western thought for centuries.

Euclid saw only part of the picture, however. Still, his geometry (which, throughout the remainder of this discussion, will be referred to as “Euclidean geometry”) withstood centuries of scrutiny by the best minds of the day. It was not until the 1800s that Euclid’s view of the world was shown to be inadequate as a model of the real world. The insights that have come to form the basis of the modern study of geometry do not conform to Euclid’s postulates—they do, however, lead to logical ways to describe the world as we know it, and space in general. We are no longer challenged with questions of how to divide plots of land; instead, our new tools enable us to ask, and answer, bigger questions. In fact, we can use the techniques of modern, non-Euclidean, geometry to understand the very fabric of reality.

In this unit we will see how Euclid elegantly combined the mathematical knowledge of his day into a logically self-consistent system. We will then examine how the close scrutiny of one of his fundamental assumptions led to an entirely new kind of geometric thinking. From there we will explore this modern view of geometry to see how one can replace Euclid’s straight lines with curves and what that means for our understanding of the universe.

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Next: 8.2 Euclidean Geometry


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