Evaluated with the Help of Geometry in Three Structures

This term comes from the words “gê,” which means “earth,” and “métron,” which means “a measure.” is, along with arithmetic, one of the older subfields that fall under the umbrella of mathematics. It is concerned with the qualities of space that are related to distance, form, size, and the position of figures in relation to one another. A geometer is a member of the mathematical community who specialises in the topic of geometry.

An example of how the theorem of Desargues, which is a result in both Euclidean and projective geometry, can be shown

Before the 19th century, practically all of the study of geometry was devoted to Euclidean geometry[a], which is characterised by the use of essential concepts such as point, line, plane, distance, angle, surface, and curve. 

The field of geometry had a significant expansion as a result of various discoveries made throughout the 19th century. One of the earliest discoveries of this kind is called Gauss’ Theorema Egregium (which literally translates to “extraordinary theorem”), and it argues, in a nutshell, that the Gaussian curvature of a surface is independent from any particular embedding in a Euclidean space. This finding has led to the development of the theory of manifolds and Riemannian geometry. It suggests that surfaces are capable of being investigated in their own right, that is, as independent spaces.

Since that time, the field of geometry has seen a significant expansion of its scope, which has led to the division of the discipline into a large number of subfields that are based either on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are ignored—projective geometry, which only considers the alignment of points and disregards distance and parallelism

Geometry

which was initially designed to represent the physical world, has applications in practically all of the sciences, as well as in art, architecture, and other activities that are related to graphics. Moreover, geometry was initially developed to model the physical world.

In addition, there are applications of geometry in fields of mathematics that appear to be unrelated to geometry. For instance, the methods of algebraic geometry are crucial in Wiles’s demonstration of Fermat’s Last Theorem. This was a problem that had been expressed in terms of elementary arithmetic, and it remained unanswered for several centuries.

History

In the 15th century, a European and an Arab both engaged in the practice of geometry.

The history of geometry can be traced back to ancient Mesopotamia and Egypt in the second millennium before the common era for recording historical events. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes that were developed to meet some practical need in surveying, construction, astronomy, and various crafts. These early geometric principles were developed to meet some practical need in these areas. Egyptian Rhind Papyrus, which was written between 2000 and 1800 B.C., Moscow Papyrus, which was written around 1890 B.C., and Babylonian clay tablets, such as Plimpton 322, are the first known texts on geometry (1900 BC). For instance, the Moscow Papyrus provides a formula for estimating the volume of a frustum, which can also be referred to as a truncated pyramid. Later clay tablets, which date from 350 to 50 BC, reveal that Babylonian astronomers used trapezoid algorithms to compute Jupiter’s position and speed within time-velocity space. These geometric processes, including the mean speed theorem, were 14 centuries ahead of their time and foresaw use of the Oxford Calculators. Ancient Nubians developed a system of geometry that included early forms of solar clocks. This culture was located to the south of Egypt.

Dimension

The Koch snowflake, whose fractal dimension is log4/log3, and whose topological dimension is 1, has these dimensions:

Mathematicians and physicists have been using higher dimensions for nearly two centuries, but conventional geometry only allowed for dimensions 1 (a line), 2 (a plane), and 3 (our ambient world conceived of as three-dimensional space). The configuration space of a physical system is an example of a mathematical application for higher dimensions. This space has a dimension that is equal to the number of degrees of freedom that the system possesses. As an illustration, the structure of a screw can be characterised by a set of five coordinates.

Conclusion 

In the latter part of the 19th century, it became apparent that non-Euclidean geometries, which do not adhere to the parallel postulate, are capable of being created without the introduction of any contradiction. A well-known example of the application of non-Euclidean geometry is the geometry that forms the basis of general relativity.