Geometry
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
Contents
Introductory Videos
https://youtu.be/51K3uCzntWs?t=842 \\ https://youtu.be/j3QSD5eDpzU
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, equivalent to the following statement,
- “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.
