Geometry is a branch of mathematics dealing with questions of shape, size, the relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.
In two dimensions there are 3 geometries: Euclidean, spherical, and hyperbolic.
What about in three dimensions, which corresponds to space we actually live in? It has been shown that in three dimensions there are eight possible geometries.
There is a 3-dimensional version of Euclidean geometry, a 3-dimensional version of spherical geometry and a 3-dimensional version of Hyperbolic geometry. There is also a geometry which is a combination of spherical and Euclidean, and a geometry which is a combination of hyperbolic and Euclidean.
What is Euclidean Geometry?
The study of plane and solid figures on the basis of axioms and theorems captured by the Greek mathematician Euclid. In its irregular outline, Euclidean geometry is the plane and solid geometry often taught in secondary schools.
In Euclid’s great work, the Elements, the only tools employed for geometrical constructions were the ruler and the compass—a reduction retained in fundamental Euclidean geometry to this day.
What is Non-Euclidean Geometry?
Non-Euclidean geometry is a type of geometry. This geometry only uses some “postulates” (assumptions) that Euclidean geometry is built on. In normal geometry, parallel lines can never meet. In non-Euclidean geometry they can meet, either boundless many times (elliptic geometry), or never (hyperbolic geometry).
The non-Euclidean geometries grew along two different historical threads. The first thread started with the search to understand the movement of stars and planets in the obviously hemispherical sky.
An example of Non-Euclidean geometry can be seen by drawing lines on a round object, straight lines that are parallel at the equator can meet at the poles.
What Is Spherical Geometry?
From early times, people noticed that the shortest distance between two points on Earth were great circle routes. For example, in geography
the Greek astronomer Ptolemy wrote:
“It has been demonstrated by mathematics that the surface of the land and water is in its totality a sphere…and that any plane which passes through the centre makes at its surface, that is, at the surface of the Earth and of the sky, great circles”
Great circles are the “straight lines” of spherical geometry. This is an outcome of the properties of a sphere, in which the shortest distances on the surface are great circle routes.
There are many ways of projecting a portion of a sphere, such as the surface of the Earth, onto a plane. These are known as maps or charts and they must certainly buckled distances and either area or angles.
For example, Euclid wrote about spherical geometry in his astronomical work circumstances. In addition to looking to the heavens, the earliest attempted to know the shape of the Earth and to use this understanding to solve problems in navigation over long distances. These activities are features of spherical geometry.
What Is Hyperbolic Geometry?
Hyperbolic geometry, also called Lobachevskian Geometry, a non-Euclidean geometry that turned down the soundness of Euclid’s fifth, the “parallel,” postulate. Clearly declared, this Euclidean postulate is: through a point not on a given line there is exactly one line parallel to the given line. In this geometry, through a point not on a given line there are at least two lines parallel to the specified line. The doctrine of hyperbolic geometry, however, confess the other four Euclidean postulates.
Although many of the theorems of hyperbolic geometry are uniform to those of Euclidean, others differ. In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant. In hyperbolic geometry, two parallel lines are taken to converge in one direction and split in the other. In Euclidean, the sum of the angles in a triangle is equal to two right angles; in hyperbolic, the sum is less than two right angles. In Euclidean, polygons of different areas can be similar; and in hyperbolic, similar polygons of differing areas do not exist.
Geometry applied to many fields, including art, architecture, as well as to other branches of mathematics.
Mathematics and art are related in a variation. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of panorama geometry.
Mathematics and architecture are associated, since, as with other arts, architects use mathematics for multiple reasons. Apart from the mathematics required geometry when engineers start constructing buildings, architects to create forms considered melodious, and thus to lay out buildings and their surroundings according to mathematical, decorative and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to keep down wind speeds around the bases of tall buildings.