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Foundations of Geometry: Thales and Euclid

It was Egyptian architects and Babylonians who built temples, tombs and pyramids clearly geometric, and the first navigators of the Mediterranean used basic geometric techniques to orient themselves. Do my geometry homework found out these civilizations made practical use of numbers without being clear about the concept of numbers or mathematical theories, and used the practical properties of lines, angles, triangles, circles and other figures without using a detailed mathematical study.

Thales of Miletus, in the 6th century BC, was the one who initiated Greek geometry as a mathematical discipline, the first mathematical discipline.

The book “The Elements” by Euclid, from 350 a. C. is the first written treatise on geometry. For Euclid and for many generations of subsequent mathematicians, geometry was the study of the regular forms that could be observed in the world. Currently, do my geometry homework I saw this study is called Euclidean Geometry or Metric Geometry.

Archimedes and Apollonius were also important figures in the geometry of the ancient world. The first analyzed exhaustively the conic sections, apart from his famous calculation of volumes of revolution figures. Apolonio worked in the resolution of tangencies between circles, as well as in conic and other types of curves.

In the Middle Ages, mathematical science has a boom in the Arab and Hindu world, but is more focused on astronomy. It is not until the Renaissance that the new needs of art and technology push humanists to study geometric properties in order to obtain new instruments to represent reality.

These systems of representation (which make up what we now call descriptive geometry and which we will see in other examples of this series while do my geometry homework for me) are no more than formal ways to translate three-dimensional reality into flat documents, and today they are essential methods for transmitting information between the different estates necessary for the execution of any project.

Geometry presents diverse fields, such as analytical geometry, descriptive geometry, and topology, geometry of spaces with four or more dimensions, fractal geometry, and non-Euclidean geometry.

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The geometry

Geometry is the branch of mathematics that deals with the study of the properties of figures in the plane (points, lines, circles, triangles, rectangles, squares, areas, polygons, etc.) or in space (cubes, pyramids , polyhedra in general, three-dimensional spaces etc.)

Plane geometry

Plane geometry is a part of the geometry, that we study while do my geometry homework for me, and it deals with those elements whose points are contained in a plane. Flat geometry is considered part of the Euclidean geometry, because it studies the geometric elements from two dimensions.

Axioms

The traditional presentation of Euclidean geometry is done in an axiomatic format. An axiomatic system is one that, from a certain number of propositions that are presupposed “evident” (known as axioms) and through logical deductions, generates new propositions whose value of truth is also logical.

Postulates

Euclid raised postulates in his system:

  • Given two points you can draw one and only one line that joins them.
  • Any segment can be extended continuously in any direction.
  • You can draw a circle with center at any point and any radius.
  • All right angles are congruent.
  • If a line, when cutting two others, forms internal angles less than two right angles, those two prolonged lines indefinitely intersect the side where the angles are less than two straight (see fifth postulate of Euclid).

This last postulate is known as the postulate of parallels. This postulate seems less obvious than the other four, and many geometers, including Euclid himself, have tried to deduce it from the previous ones. When they tried to reduce it to absurd by denying it, two new geometries emerged: the elliptical, also called Riemannian or Riemannian geometry (given a line and a point outside it, there is no line that passes through the point and is parallel to the given line) and hyperbolic or Lobachevski (there are several parallel lines that pass through a point outside a given) with we still use in line with do my geometry homework help.

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  • Geometric shapes

    • We call geometric bodies to the figures that are to be represented in three-dimensional space. The geometric bodies always occupy a space.
    • The spatial geometry is based on a system formed by three axes (X, Y, Z):
    • Orthogonal (perpendicular 2 to 2)
    • Normalized (the lengths of the basic vectors of each axis are equal).

    Classes of solids

    These bodies can be of two kinds:

    Polyhedrons, solids those are all flat.

    • Platonic solids
    • Prisms
    • Pyramids

    No polyhedrons or round bodies, those solids that have at least one face with a curved surface.

    • Spheres
    • Cylinders
    • Bulls
    • Cones
    • Properties

    Solids have properties, such as

    • Volume
    • Surface area
    • Likewise, bodies that are hollow can contain other bodies in their interior in an amount that is called capacity.

    GEOMETRIC FIGURES

    The advance of the geometry homework depends strongly on the advance in the definitions, the properties of the triangles are possible to enunciate without making reference to these, but it would be a long and tedious and useless process.

    Fundamental figures: Point, Straight and Plane.

    On the line you can see: Segments and vectors

    In the plane, a line determines two half-planes; its intersection determines the convex figures: strip, angle, triangle, quadrangle and polygon.

    Using the concept of distance: define: the circle and the sphere.

    Using the concept of semi-space is defined: the dihedral, the prismatic space, the trihedral, the polyhedron angle, and the polyhedra. Among the latter we find as particular cases: the tetrahedron, the prism, the pyramid and the parallelepiped.

    The concept of a circle in space gives rise to: the cone and the cylinder.

    RELATIONSHIPS AND PROPERTIES

    In line with geometry homework example, between two or more figures there can be different relationships, two lines can be parallel, perpendicular or oblique (they are cut at a point forming non-right angles).

    In space, they can also be warped (or crossed). One of the most important concepts in geometry is that of congruence or equality.

    GEOMETRY CLASSES

    Taking into account geometry homework sample, more axioms other geometries are obtained (in which everything said so far is valid). If we take for granted the axiom of Euclid’s parallelism, we obtain the Euclidean geometry also known as plane geometry.

    Adding to these the axioms relative to space, we obtain spatial geometry (the latter are merely extensions of the axioms relative to the plane). The descriptive geometry is the one in charge of which the problems make possible the resolution of the problems of the geometry of the space by means of operations carried out in a plane.

    If we add other axioms, whether they are different parallelism postulates or the existence of sets of points greater than the plane (and smaller than space), non-Euclidean geometries are obtained.

    Using the knowledge of other areas (and therefore their respective axioms), we obtain: analytical geometry, the methods of algebra and mathematical analysis.

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