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What are the three basic types of geometry?

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There are three basic types of geometry: Euclidean, hyperbolic and elliptical. Although there are additional varieties of geometry, they are all based on combinations of these three basic types. Euclidean geometry is the original form, dating back to 300 BC, and it is the result of the work of the Greek Alexandrian mathematician Euclid, who developed the five postulates, or axioms, upon which his geometric theorems are built.

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Euclid's fifth postulate, known as the parallel postulate, went without an accompanying proof for thousands of years. The parallel postulate assumes that a straight line crossing through two other straight lines and forming two same-side interior angles of less than 90 degrees determines that those two lines, if extended far enough, will eventually meet on the side of the interior angles. This assumption, however, did not take into account the idea of curved space, which was first conceptualized by Albert Einstein in his 1915 General Theory of Relativity.

The idea of space existing with either a positive or a negative curvature introduced the idea of non-Euclidean geometry, in which the parallel postulate would not always hold true. In curved space, it cannot be assured that the two lines in question will ever meet, regardless of how far they might be extended. The geometry based on space with a negative curvature became known as hyperbolic geometry. Elliptical geometry refers to the type of geometry based on space with a positive curvature.

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  • Q:

    Why is geometry so important?

    A:

    Geometry is defined as the area of mathematics dealing with points, lines, shapes and space. Geometry is important because the world is made up of different shapes and spaces. It is broken into plane geometry, flat shapes like lines, circles and triangles, and solid geometry, solid shapes like spheres and cubes.

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  • Q:

    What is a counterexample in geometry?

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    A counterexample, in geometry as in other areas of mathematics and logic, is an example that one uses to prove that a particular statement is false. A simple example from primary mathematics uses the statement "the inverse of a number is never an integer," and its counterexample would be 1/4. The inverse of 1/4 is 4, which is an integer. For geometry, finding counterexamples involves a few more calculations.

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  • Q:

    What is a converse in geometry?

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    The converse in geometry applies to a conditional statement. In a conditional statement, the words "if" and "then" are used to show assumptions and conclusions that are to be arrived at using logical reasoning. This is often used in theorems and problems involving proofs in geometry.

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  • Q:

    Who invented geometry?

    A:

    The inventor of geometry was Euclid, and his nickname was The Father of Geometry. Euclid obtained his education at Plato's Academy in Athens, Greece and then moved to Alexandria.

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