Euclid Geometry

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Euclid's Elements of Geometry

Based on various theorems and axioms, the study of geometrical figures and shapes is called Euclidean geometry. Generally, this type of geometry is inaugurated for flat surfaces. This geometry helps in explaining the shapes of geometrical figures better, especially the plane figures. Greek mathematician, popularly known as Euclid, employed Euclidean geometry and even described it in his own written book named 'Elements'. Also, he named this geometry on his name 'Euclid Geometry'. He introduced fundamentals of geometry in his book Elements which include geometric figures and shapes as well as he stated major five postulates and axioms which are universal truths. However, these axioms and postulates are not yet proved.


Euclid's Elements

In Alexandria, Ptolemaic Egypt, an ancient Greek mathematician named Euclid wrote his mathematical and geometrical work which are present in a total of 13 books. It is known as Euclid's Elements. These thirteen books in which Elements was divided were responsible for popularized geometry all over the globe. Euclid Elements as a whole is a compilation of postulates, axioms, definitions, theorems, propositions and constructions as well as the mathematical proofs of the propositions. 

 

Plane Geometry is mainly discussed in book 1 to 4th and also 6th. Five postulates were given by Euclid which he referred to as Euclid's Postulates. This geometry is popularly considered as Euclidean geometry. Due to the work of Euclid, today we have a vast and collective source and reference for learning and practising geometry as it laid the foundation for geometry.


Euclid's Window

Euclid's Window (The story of geometry from parallel lines to hyperspace) is a book written by the author popularly known as Leonard Mlodinow. Leonard Mlodinow was an American theoretical physicist who, with the help of Euclid's Window, had delightfully and brilliantly led us on a journey with the help of 5 revolutions in geometry. As in this revolution of geometry, he explained geometry with the Greek concept of parallel lines to the latest notions of hyperspace. 


Euclid's Axioms

Euclid has given seven axioms for geometry which are considered as Euclid axioms. The axioms are listed below:

  • Things that are equal to identical things are also equal to each other.

  • In any case, equals of something are added to equals, then as a whole, they are considered as equals.

  • In any case, equals of something when subtracted from equals, then their remainders come are equal.

  • Things that are coinciding with each other are equal.

  • The whole of something is greater as compared to the part.

  • Things that are double of identical things are equal to each other.

  • Things that are halves of identical things are equal to each other.

Euclid's Postulates

Euclid has given five postulates for geometry which are considered as Euclid Postulates. These postulates include the following:

  1. From any one point to any other point, a straight line may be drawn.

  2. A terminated line can be produced indefinitely.

  3. By taking any center and also any radius, a circle can be drawn.

  4. All right angles are equal.

  5. When on two straight lines, if a straight line falls, then the interior angles made are on the same side of the line which if taken together is less as compared to the two right angles. Also, then the two straight lines, if produced indefinitely, meet together on the same side on which the sum of angles is less as compared to the right angles.

FAQ (Frequently Asked Questions)

Q1. What is the Difference Between Euclidean and Non-Euclidean Geometry?

Euclidean Geometry is popularly known as an axiomatic system. It is considered so because all the theorems in geometry, are originated from a small number of simple axioms. "Geometry" is the term that deals with aspects such as line, angles, points, triangles, shapes and squares. Similarly, Euclidean Geometry deals with geometrical shapes and figures and especially, the plane figures due to which it is also considered as the “plane geometry”. This geometry deals with the relationship between all the things and also their properties.


Non-Euclidean is different from all the figures and surfaces which are not flat falls under the category of Non-Euclidean geometry. Depending upon the nature of the parallel lines, both the geometry are different. 

Q2. What are the Three Types of Geometry?

Mainly, there are three different kinds of geometries in the two-dimensional plane which includes:

  • Euclidean (for flat surfaces)

  • Spherical (for curved surfaces)

  • Hyperbolic

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Euclidean Geometry: The study of the flat surfaces as well as the flat figures or shapes and also straight lines in two dimensions.


Spherical Geometry: It is another type of geometry that deals with curved surfaces. If L is the given line and P which denotes a point but this point  'P' does not lie on the line L. No lines are passing through point P but these lines are parallel to line L.


Hyperbolic Geometry: It is also one of the types of geometry. For example, If L is the given line and P denote a point but this point 'P' does not lie on the line L, then there must be at least two lines that are passing through the point P and these are parallel to the line L.