Difference between revisions of "Geometry"

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'''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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'''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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==Introductory Videos==
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https://youtu.be/51K3uCzntWs?t=842 \
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https://youtu.be/j3QSD5eDpzU
  
 
== Euclidean Geometry ==
 
== Euclidean Geometry ==
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===Parallel Postulate===
 
===Parallel Postulate===
 
{{main|Parallel Postulate}}
 
{{main|Parallel Postulate}}
The fifth [[postulate]] stated in the book,  
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The fifth [[postulate]] stated in the book, is the following statement:
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:''If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.''
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Though some resources state:
  
 
:''“Through any line and a point not on the line, there is exactly one line passing through that point parallel to the line”''
 
:''“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.
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This postulate 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.
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There are also other types of geometry that don’t follow this postulate, like hyperbolic geometry and spherical geometry, which say that there are more than 2 parallel lines, or there are no parallel lines, respectively.
  
 
== Non-Euclidean Geometry ==
 
== Non-Euclidean Geometry ==
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*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 [[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 [[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.
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*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 [[ray]]s at their endpoints. It is measure in either [[degree]]s or [[radian]]s, though the less-common [[metric system|metric]] unit [[gradian]] is also used.
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*An [[angle]] is a structure formed by the intersection two [[ray]]s at their endpoints. It is measure in either [[degree]]s or [[radian]]s, though the less-common [[Système international|metric]] unit [[gradian]] is also used.
  
 
== See Also ==
 
== See Also ==
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* [[Topology]]
 
* [[Topology]]
  
[[Category:Geometry]]
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[[Category:Geometry]] [[Category:Mathematics]] [[Category:Topology]]

Latest revision as of 00:52, 25 August 2024

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

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, is the following statement:

If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Though some resources state:

“Through any line and a point not on the line, there is exactly one line passing through that point parallel to the line”

This postulate 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.

There are also other types of geometry that don’t follow this postulate, like hyperbolic geometry and spherical geometry, which say that there are more than 2 parallel lines, or there are no parallel lines, respectively.

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.

See Also