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Geometry: Points, Lines, Planes, Solids and Dimensions



 



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The dimensions can be used to describe the basic objects in geometry such as the point, line and the plane. A dimension can be thought of as an extension in a certain direction. In multiple dimensions, each direction is independent and does not overlap with any other.

Zero Dimension

Point

A point is an object that has zero or no dimension. It does have a specific location in one or more dimensions, however. A point is usually represented by a dot in diagrams like the period at the end of this sentence. In geometric notation, a point is usually labeled with a capital letter such as A, B, or C, for example.

A point is a location or place having zero dimension.

One Dimension

Line

A line is an object in geometry that has one dimension. It extends in two opposite directions without limit and therefore has infinite length. A line also contains an infinite number of points. It has the quality of being straight which means that it never changes direction along the points. A point that is part of a line is said to be collinear with respect to that line.

A line has one dimension and all of its points are collinear.

Ray

A ray is also one-dimensional but can only extend to infinity in one direction. While one end continues without limit, the other end is terminated with an endpoint.

A ray has one dimension, one endpoint and extends in one direction.

Line Segment

A line segment exists in only one dimension but does not extend to infinity in either direction. It is part of a line that is terminated by two endpoints. It also has a finite distance which can be measured.

A line segment has one dimension, two endpoints and does not extend to infinity.

Lines, rays and line segments are usually labeled two ways, either with a single lower case letter such as j, k, l, m or n for the line itself or by two points on the line such as AB where A and B are the two different points.

A line, ray or line segment can be labeled two ways: a lower case letter for the line itself or by its labeled points.

Distance

In geometry, the distance between two objects is the amount of extension or separation of the objects along one dimension. Distance is normally used to measure or quantify the length between two points whether the points are isolated, on a line or part of some other geometric figure. In the case of a line segment, it is the length between the endpoints. Some of the other terms used to describe distance other than length include width, height and depth. Some of the units used to measure distance include inches and meters.

Distance measures how far apart objects are such as the length of a line segment between two endpoints.

Two Dimensions

Planes

A plane is an object in geometry that has two dimensions and can be though of as extending in four directions similar to the way that a line extends in two. A line has length but no width while the plane has both. In addition, just as a line can contain an infinite number of points, a plane can contain an infinite number of lines.

Planes can be described by three non-collinear points A, B and C which are points that are not contained in a single line. Planes are also described as being flat. Points are defined as being coplanar if all the points exist inside the plane.

A plane is defined by three points, has two dimensions and extends to infinity. All the points in the plane are coplanar.

Surface

A two dimensional subset of a plane is sometimes a region or a surface depending on its context although the terms region and surface usually describes a bounded subset that does not extend to infinity in one or more dimensions. The term region, however, can be used to describe objects existing in other than two dimensions.

Surfaces are two dimensional structures in the plane or space. They can be bounded or unbounded.

Area

The measure of a region or surface is called area. It is usually applied to the interior of two dimensional shapes such as triangles and circles. The units for area is usually the square of those used for length although special names are sometimes used.

The amount of the bounded interior of a two dimensional object in geometry is called the area.

Three Dimensions

Space

Just as a point corresponds to zero dimensions, a line to one dimension and a plane to two dimensions, space occupies three dimensions. Space can be defined as the infinite extension of the three independent directions and their collinear opposites. These directions and their opposites can also be described as perpendicular or orthogonal since a projection from any one of them does not fall on or overlap with any other direction. Space can also be described as the set of all points in three dimensions.

Space can be represented by three perpendicular axes with each axis as a single dimension.

Solids

Geometry also takes into account objects in the three dimensions of space called solids. One of the properties of solids is called volume which is the amount of space taken up by a solid.

Three dimensional solids occupy space.

Geometry Postulates

Geometry also contains many postulates. They are based on the point, line and plane which are considered to be undefined. The postulates themselves are unproven statements that are fundamental to the understanding of geometry. In the cases presented here, they link and bind the fundamental geometric structures of the point, line and plane together.

One of these postulates states the relationship between two points and a line.

Line Postulate
If there are two points, then there is exactly one line that passes through them.

The line postulate says exactly one line passes through two points.

In some cases, to make a distinction between Euclidean and Non-Euclidean geometry the following definition is used:

Line Postulate
If there are two points in a plane then there is only one line that passes through them.

A variation of the line postulate says exactly one line passes through two points in a plane.

Another postulate deals with the extension of a line segment or a ray.

Extension Postulate
A line segment or ray may be extended to infinity as a regular line.

The extension postulate says that a line can be formed by extending a ray or line segment to infinity.

Some postulates deal with planes such as the following one:

Plane Postulate
If three points are not collinear, meaning that they are not on the same line, then there is exactly one plane that contains them.

The plane postulate says that exactly one plane contains three non-collinear points, meaning that the three points are not on the same line.

 




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