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From Classical Geometries to Elementary Differential Geometry

Braden Ward

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Published in Curved Spaces: From Classical Geometries To Elementary Differential Geometry

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Geometry, the study of shape and space, has a rich and fascinating history that spans millennia. From the early days of Euclidean geometry to the more recent advent of differential geometry, this field has played a pivotal role in our understanding of the world around us. In this article, we will explore the evolution of geometry, from its classical roots to its modern manifestations in differential geometry.

Classical Geometries

The foundations of geometry were laid by the ancient Greeks. In particular, Euclid's Elements, written in the 3rd century BCE, laid out the principles of Euclidean geometry, which would remain unchallenged for over 2000 years. Euclidean geometry is based on the idea of a flat, two-dimensional space, and its axioms (such as the parallel postulate) describe the relationships between points, lines, and angles in this space.

Curved Spaces: From Classical Geometries to Elementary Differential Geometry

by P. M. H. Wilson

4.8 out of 5

Language	:	English
File size	:	2986 KB
Screen Reader	:	Supported
Print length	:	198 pages
X-Ray for textbooks	:	Enabled

Around the same time that Euclid was developing his geometry in Greece, other civilizations were also exploring geometric ideas. In India, for example, the mathematician Baudhayana developed a set of rules for constructing altars, which included a geometric construction for finding the square root of 2. In China, the Mohists developed a form of geometry that included ideas about infinity and the concept of a "curved line".

Non-Euclidean Geometries

The development of non-Euclidean geometries in the 19th century challenged the long-held belief that Euclidean geometry was the only possible geometry. Non-Euclidean geometries are based on the idea that space can be curved, rather than flat. The first non-Euclidean geometry was developed by the Russian mathematician Nikolai Lobachevsky in 1829, and shortly thereafter, the Hungarian mathematician János Bolyai published similar results.

Non-Euclidean geometries have a number of interesting properties that distinguish them from Euclidean geometry. For example, in non-Euclidean geometries, the parallel postulate does not hold, and the sum of the angles in a triangle can be greater or less than 180 degrees. These properties have led to a wide range of applications in fields such as physics and cosmology.

Differential Geometry

Differential geometry is a more modern branch of geometry that emerged in the 19th century. Differential geometry is concerned with the study of smooth, differentiable surfaces and spaces. One of the key concepts in differential geometry is the differential calculus, which provides a way to calculate the derivatives of functions defined on smooth surfaces. The differential calculus can be used to study a wide range of geometric properties, such as curvature, geodesics, and topology.

Differential geometry has a number of applications in other fields, such as physics, engineering, and computer graphics. For example, differential geometry is used in physics to describe the behavior of fluids and deformable bodies, and in computer graphics to create realistic 3D models.

The evolution of geometry from its classical roots to its modern manifestations in differential geometry is a fascinating tale of human ingenuity and creativity. Geometry has played a vital role in our understanding of the world around us, and it continues to be a source of new insights and discoveries.

Curved Spaces: From Classical Geometries to Elementary Differential Geometry

by P. M. H. Wilson

4.8 out of 5