Geometry
From AoPSWiki
Geometry is the field of mathematics dealing with figures in a given space. It is one of the two oldest branches of mathematics, along with arithmetic (which eventually branched into number theory and algebra). The geometry usually studied is
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Euclidean Geometry
- Main article: Euclidean geometry
The most common type of geometry used in pre-collegiate mathematics competitions is Euclidean geometry. This type of geometry was first formally outlined by the Greek mathematician Euclid in his book The Elements.
Parallel Postulate
- Main article: Parallel Postulate
The fifth postulate stated in the book,
- “Through any line and a point not on the line, there is exactly one line passing through that point parallel to the line”
was the subject of a controversy for many centuries, with many attempted proofs. It is much less simple than the other postulates, and more wordy. This postulate is the basis of Euclidean geometry.
Non-Euclidean Geometry
Non-Euclidean geometry are geometries in which the fifth postulate is altered. Types of non-Euclidean geometry include:
Student Guides to Geometry
Main Concepts
- The notion of dimensions is fundamental to geometry. N-space is a term related to this concept.
- A point is a geometric structure with no area, length, width, or dimension. Its only property is space. It is said to be zero-dimensional.
- A line is generally taken to mean a straight line, which is the locus of points on the Cartesian plane satisfying a linear function. It has length and position, but no other properties. It is one-dimensional. A line segment means a finite segment of a line, while a ray is a line infinitely extending in only one direction.
- A plane is a line, but in a Cartesian space. It as length, width, and position. It is two-dimensional. The point/line/plane sequence can be extended to spaces and higher dimensions.
- An angle is a structure formed by the intersection two rays at their endpoints. It is measure in either degrees or radians, though the less-common metric unit gradian is also used.
