Geometry is the oldest of the branches of mathematics that can be traced at least during the times of Pythagoras, Euclid and other mathematics philosophers of the ancient Greek. So, what does geometry mean? The word geometry comes from two Greek words, "geo" and "metron," which mean earth and measurement, respectively. Therefore, it's the branch that is characterized by the study of size, shapes, and properties of space and relative position of figures. Geometer is the name of the person who works in the field of geometry.
The following are essential concepts. They include:
These are the fundamental concept in Euclidean geometry. The Euclid defined a point as the object that has no part. However, a point has many other definitions based on the nested or algebra sets. In various areas of geometry such as the differential, topology, and analytical geometry, every object is considered to be build based on a point. However, some studies of geometry have no reference to a point.
This is a two-dimensional surface that is flat in shape and extends far. They are used mainly in every aspect such as the study in topological surface with no reference to angles or distances. Planes can be studied in the form of affine space where ratios and collinearity are studied rather than the distances. Using sophisticated analysis techniques, planes can be studied as complex planes.
In his Elements book, Euclid's approach was abstract. He introduced some axioms that expressed independent primary characteristics of planes, points, and lines. Through mathematics reasoning, Euclid was able to deduce other properties which came to be well known as the synthetic or axiomatic geometry. In the 19th century, Nikolai Ivanovich Lobachevsky discovered the non-Euclidean geometries with led to increased interest in the subject in the 20th century. David Hilbert used this type of axiomatic reasoning in the attempt of the provision of modern geometry.
A curve is a 1-dimension object that is either a line or not. The plane curve is a 2-dimensional space curve while the space curve is a 3-dimensioned. In the study of topology, curves can be defined as a function of an interval of another space and real numbers. The same definition is also used in differential geometry. Algebraic curves are used in algebraic geometry. The algebraic curves are the algebraic varieties of the first dimension.
This is the inclination of two lines that meet each other and not straight to each other. In modern, angles are formed by two rays which share a common endpoint. The endpoint is known as the vertex of the angle.
Angles are used in the study of triangles and polygons. The basis of the study of these angles of the unit circle or triangle is the important studies in trigonometry. In calculus and differential geometry, the space curves or plane curves' angles are determined through the calculation of derivatives.
This is the generalization of surface and curve concepts. In differential, a manifold can be described as space is diffeomorphic to the space of Euclidean while in topology, the neigh hood of the topological space is homeomorphic. These manifolds are majorly used in the study of string theory and general relativity in physics.
A surface such as a paraboloid or a sphere is a two-dimensional object which is assembled by homeomorphisms or diffeomorphisms in topology and differential geometry respectively. Surfaces are also described by polynomial equations in algebraic geometry.
This is a length that lies equally to its points. In modern geometry, a line can be described as the type of geometry. For instance, in analytical geometry, a line is defined as a point set where its coordinates should satisfy a certain given linear equation.
There are specific symmetric shapes such as regular polygons, platonic solids, and circles. These symmetric shapes were studied even before the Euclid time. Symmetry was realized in the second half of the 19th century.
There are many different fields of study in this field. In this article, we will describe some of them in details.
In this field, all that is studied are solutions sets of polynomial equations known as algebraic varieties. Theses algebraic varieties can sometimes be manifolds although the most have a non-smooth singular point. Since they are algebraic, various tools are available to study from the abstract. The study of these varieties in algebra can be studied not limited to complex or real numbers.
This type of studies manifolds that resemble ordinary n-dimensional spaces created from only complex numbers. Due to the rigid nature of its holomorphic analysis, many of the manifolds are less successful in classifications. This type has some relationship with the algebraic one.
The type studies manifold equipped with a metric structure of Riemannian. The Riemannian metric is a simple rule that governs the measuring of angles and the lengths of curves between the vectors of a tangent. This type of manifold is characterized by curvature, which makes the laws of the classical Euclidean geometry. For example, one of the laws of classical Euclidean geometry is that the sum of the interior angles of a triangle is more or less than π for positive or negative curvature respectively.
The manifold of this type is equipped with an additional symplectic form structure. The symplectic form is the opposite of the Riemannian metric, which expresses different characteristics from that of Riemannian manifolds. This can be well described through the Darboux theorem which notes that symplectic manifolds are relatively the same, although to some extent they can be different which is far from the reality of the Riemannian. The symplectic manifolds are mostly applicable in physics such as in the study of classical mechanics. Sympletic geometry is topological.
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