Geometry

Geometry is a study of graphics in a given number of dimensions and spaces of a given type. The most common geometry types are spherical geometry such as planar geometry (processing objects such as points, lines, circles, triangles, polygons), solid geometry (objects such as lines, spheres, polyhedra) and spherical triangles and spherical polygons Processing object). Geometry is a part of a quarter of the professors of medieval universities

The point of mathematics points out that life is meaningless without geometry. An older child's joke answers "What will the acorn say when growing up" and "geometry" ("Oh, I am a tree")

Historically, geometric studies came from several recognized facts (axioms or assumptions) and used systems and strict stepwise proofs to establish true statements. However, geometry far exceeds the way of this relatively dry textbook. This is because beautiful and unexpected results of projected geometry have been proved (Schubert's powerful but problematic enumeration geometry goes without saying).

The late mathematician ET Bell states geometry as follows (Coxeter and Greitzer, 1967, p. 1). "Literature is much more integrated than algebra and arithmetic, at least as broad as analysis, geometry is a richer treasure, literature on algebra, arithmetic, analysis has grown widely since the Bell era But the rest of his comments are still there today.

Formally, geometry is defined as a completely local homogeneous Riemannian manifold. Possible geometric shapes are the Euclidean plane, the hyperbolic plane and the elliptical plane. Possible geometric shapes include Euclidean, hyperbolic and elliptical shapes, but also include five other types.

In this article, I will learn about two kinds of non-Euclidean geometry. The first type, called hyperbolic geometry, is the geometry discovered by Bolyai and Lobachevsky. (The great Karl Friedrich Gauss (1777-1855) also discovered this geometry; however, he did not publish his work in fear of it being controversial for establishment. Euclid's fifth hypothesis was replaced by this. The famous Pythagorean theorem depends on parallel hypotheses, so it is Euclidean geometry's theorem. However, in chapter 5 and chapter 6 we meet non - Euclidean variants of this theorem and propose the Pythagorean theorem unified in Chapter 7. This is the closest. Appear in

One of the first challenges of non-Euclidean geometry is to determine its logical consistency. Have you made a system that leads the contradiction theorem by changing Euclid's parallel hypothesis? In 1868, Italian mathematician Enrico Beltrami (1835-1900) showed that we can construct a new non-Euclidean geometry on the Euclidean plane as long as it is Euclidean geometry. It is consistent and non-Euclidean geometry is also consistent. Therefore, non-Euclidean geometry is placed on firm ground.

When I was in Göttingen in 1871, Klein made a big discovery in geometry. He published two papers on so-called non-Euclidean geometry. There, we showed that Euclidean geometry and non - Euclidean geometry can be thought of as being adjacent to a particular conical section in the special case. When the projection plane Euclidean geometry is consistent there is a surprising inference that non - Euclidean geometry is consistent. At the time, the fact that non-Euclidean geometry was still a controversial topic is now disappearing. Its state is the same as Euclidean geometry. Cary has never accepted Klein's view and believes his view is circular.

In 1871, in Göttingen, Klein made a big discovery in geometry. He published two papers on so-called non-Euclidean geometry. We showed that Euclidean geometry and non-Euclidean geometry can be thought of as a metrological space determined by Cayley-Klein metrics. In this view, we conclude that non-Euclidean geometry is consistent only if we terminate with Euclidean geometry and non-Euclidean geometry on the same basis. All controversy over non-Euclidean geometry Cary has never accepted Klein's claim that it is cyclical