# Point Research Materials

This page contains a list of user images about Point which are relevant to the point and besides images, you can also use the tabs in the bottom to browse Point news, videos, wiki information, tweets, documents and weblinks.

Point Images

Rihanna - Take A Bow
Music video by Rihanna performing Take A Bow. YouTube view counts pre-VEVO: 66288884. (C) 2008 The Island Def Jam Music Group.
Key & Peele: Substitute Teacher
A substitute teacher from the inner city refuses to be messed with while taking attendance.
Mortal Kombat: Legacy - Season 2 Trailer
Watch Season 1 of Mortal Kombat Legacy here: http://www.youtube.com/channel/SWVkIoQKmEa4I The Mortal Kombat Legacy continues in Season 2 as Liu Kang, Kung La...
David Guetta - Just One Last Time ft. Taped Rai
"Just One Last Time" feat. Taped Rai. Available to download on iTunes including remixes of : Tiësto, HARD ROCK SOFA & Deniz Koyu http://smarturl.it/DGJustOne...
Steve Jobs vs Bill Gates. Epic Rap Battles of History Season 2.
Download This Song: http://bit.ly/KzLBGB Click to Tweet this Vid-ee-oh! http://bit.ly/Nt9lg8 Hi. My name is Nice Peter, and this is EpicLLOYD, and this is th...
MACKLEMORE & RYAN LEWIS - CAN'T HOLD US FEAT. RAY DALTON (OFFICIAL MUSIC VIDEO)
Macklemore & Ryan Lewis present the official music video for Can't Hold Us feat. Ray Dalton. Can't Hold Us on iTunes: https://itunes.apple.com/us/album/cant-...
Draw My Life- Jenna Marbles
This video accidentally turned out kind of sad, ME SO SOWWY IT NOT POSED TO BE SAD WHO WANTS HUGS AND COOKIES? Also, FYI for anyone attempting this, it takes...
F*@#ing Ben Affleck
Jimmy reveals that he is f*@#ing Ben Affleck.
Fast Food Lasagna - Epic Meal Time
LIKE/FAV We got 45 burgers, a whole bunch of liquor and bacon.... this is Fast Food Lasagna. Buy TSHIRTS!! Click Here! http://shop.epicmealtime.com/ Like on ...
Draw My Life - Ryan Higa
So i was pretty hesitant to make this video... but after all of your request, here is my Draw My Life video! Check out my 2nd Channel for more vlogs: http://...
Jack Sparrow (feat. Michael Bolton)
Buy at iTunes: http://goo.gl/zv4o9. New album on sale now! http://turtleneckandchain.com.
Giant 6ft Water Balloon - The Slow Mo Guys
Follow on Twitter! - https://twitter.com/#!/GavinFree Watch this one in HD! The slow mo guys are well aware that water balloons are always good in slow motio...
Katy Perry - Wide Awake
Official music video for "Wide Awake," the final chapter from 'Teenage Dream: The Complete Confection' on iTunes: http://smarturl.it/katyperry. Written by Ka...
Rihanna - Where Have You Been
Buy on iTunes: http://www.Smarturl.it/TTT Amazon: http://idj.to/svJVGM Music video by Rihanna performing Where Have You Been. ©: The Island Def Jam Music Group.

In geometry, topology, and related branches of mathematics, a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics dealing with set theory, an element is sometimes referred to as a point.

## Points in Euclidean geometry

A finite set of points (blue) in two dimensional Euclidean space.

Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". In two-dimensional Euclidean space, a point is represented by an ordered pair (x, y) of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is easily generalized to three dimensional Euclidean space, where a point is represented by an ordered triplet (x, y, z) with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms, (a1, a2, … , an) where n is the dimension of the space in which the point is located.

Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points; As an example, a line is an infinite set of points of the form $\scriptstyle {L = \lbrace (a_1,a_2,...a_n)|a_1c_1 + a_2c_2 + ... a_nc_n = d \rbrace}$, where c1 through cn and d are constants and n is the dimension of the space. Similar constructions exist that define the plane, line segment and other related concepts.

In addition to defining points and constructs related to points, Euclid also postulated a key idea about points; he claimed that any two points can be connected by a straight line. This is easily confirmed under modern expansions of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts of the time. However, Euclid's postulation of points was neither complete nor definitive, as he occasionally assumed facts about points that didn't follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these assumptions.

## Alternative approaches

Although the notion of a point is generally considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e.g. noncommutative geometry and pointless topology. A "pointless" or "pointfree" space is defined not as a set, but via some structure (algebraic or logical respectively) which looks like a well-known function space on the set: an algebra of continuous functions or an algebra of sets respectively. More precisely, such structures generalize well-known spaces of functions in a way that the operation "take a value at this point" may not be defined. A further tradition starts from some books of A. N. Whitehead in which the notion of region is assumed as a primitive together with the one of inclusion or connection.

## References

• Clarke, Bowman, 1985, "Individuals and Points," Notre Dame Journal of Formal Logic 26: 61-75.
• De Laguna, T., 1922, "Point, line and surface as sets of solids," The Journal of Philosophy 19: 449-61.
• Gerla, G., 1995, "Pointless Geometries" in Buekenhout, F., Kantor, W. eds., Handbook of incidence geometry: buildings and foundations. North-Holland: 1015-31.
• Whitehead A. N., 1919. An Enquiry Concerning the Principles of Natural Knowledge. Cambridge Univ. Press. 2nd ed., 1925.
• --------, 1920. The Concept of Nature. Cambridge Univ. Press. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at Trinity College.
• --------, 1979 (1929). Process and Reality. Free Press.
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