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2.2 Geometry

2.2.1 Euclidean Geometry

For more than 2000 years, Geometry meant Euclidean Geometry, and the idea that space itself may be curved was only discovered in 1828 by Carl Frederich Gauss. His work was continued by his colleague, Riemann and others, who pioneered the subject of differential geometry. This branch of geometry also extended the concept of two and three dimensional space to any number of dimensions.

In Euclidean geometry, space is empty, three dimensional and 'flat'. Traditionally, this was assumed to be the natural structure of the space in which we exist. Straight lines in Euclidean space are infinitely long, and Euclidean space would, therefore, be infinite in extent. Over the past 100 years, it has become clear that the space of the universe is not Euclidean (flat), but curved in the shape of a hyper-sphere. This is the central concept of this work. However, the universe is so large that its structural curvature, even of intergalactic distances, is almost impossible to measure.

As well as Euclidean space, theoretical geometry now allows two other kinds of space, Elliptical space and Hyperbolic space, which are offered in Cosmology as alternatives to Euclidean space. However, Euclidean space has one property which makes it unique. Lines, areas and volumes, acquire minimal values in Euclidean space. Hence, a straight line in Euclidean space is the shortest distance between two points. Likewise, in Euclidean space, an area inside a closed line is minimum, and a volume bounded by a closed area is minimum. As a consequence of this, the theorem of Pythagoras is only valid in flat (Euclidean) space.

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