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Building Blocks of Geometry

Dimension

Problems

Problems

Dimension is a characteristic of all geometric regions, objects, and spaces. The previous sections have probably already made you aware of the concept of dimension. It is roughly the number of directions in which a region or object can be measured. More formally, it is the number of lines required to span a region in space. Examples make dimension much easier to understand.

A point is zero-dimensional. It has no length, width, thickness, or any other physical means of measurement. It only exists as a symbol to identify a single location in space.

A line is one-dimensional. It has the dimension of length. To put it another way, there is only one way that you can move along a line: lengthwise. In a similar vein, there is no way to move within a point. A point is a single location in itself, whereas a line is a collection of points, or locations.

A plane is two-dimensional. It has length and width. (Technically speaking, the property of width is really only length in a different direction). You can move along a plane in two directions, lengthwise and widthwise. You might think that you can actually move along a plane in an infinite number of directions, but actually every direction in which you move can be broken down into a component of length and a component of width.

Figure %: Any ray in a plane can be divided into a component of length and width.

It should now be easier to understand the more formal definition of dimension: the number of lines required to span a region in space:

  • A point is not a region in space, it is only a specific location. Therefore it takes zero lines to span it, and it is zero-dimensional.
  • One line is required to span a line (itself). Therefore a line is one- dimensional.
  • It requires two lines to span a plane, so therefore a plane is two- dimensional. These two lines represent length and width. Any point in the plane can be expressed as a combination of a certain length and a certain width, depending on the location of the point. The span of a line (or many lines) is the region that contains all the points that can be expressed as combinations of that line (or lines). A geometric space can also be spanned by points or planes.

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