math [1.02-01] the philosophy of lines

A line need not be said to actually exist in space but instead can be thought of as a symbol that exists to refer to the distance between two points in space.

The marker for the line in the examples below is a bright blue line that has actual existence in empirical reality. However, the line itself need not have any substantial dimension and no where in the world of observation can we find lines.

Like a point in space, the line can be imagined to exist between two points. The points and line need no substantial existence to be thought of as real. A line is a mental representation and no lines can be found in empirical reality although we can equate linear things, like wires and rope, to lines.

If the point is all that exists, then there is no dimension to the space we are considering.

If the line is all that exists, then there is one dimension to the space we are considering. If another line is drawn perpendicular to the first line, we can create a two-dimensional plane in the space we are considering.

And, if we draw a line that is perpendicular to the two-dimensional plane, we have created the third dimension in the space we are considering.

dimensional line graphics


Lines in the same plane of two-dimensional space can be said to be (a) parallel, meaning that they will never intersect or (b) perpendicular, meaning that they will intersect. Lines that are neither in the same plane nor intersect are called skew lines.

types of plane lines graphic


A line can be defined as an infinite set of points in a straight path with only a single dimension of length. A line has no height and no width.

A ray is a type of line that begins at a point and projects outward from that point in some direction. If a line continues infinitely in both directions, then a ray continues only in one direction.

A line segment is any portion of a line, denoted by two distinct points on a line.

line segments graphic


In the above examples, line EF is a normal line that extends infinitely in both directions. The notation for this type of line includes the double arrows written above the designations of the two points which the line passes through.

A bidirectional symbol [] is used to denote a normal line, written above two points through which the line travels, as in line EF above.

The line segment is written as a simple straight line above the two end points of the line, as in line segment AB. A single directional line above two points through which it runs is used to denote a ray, as in the ray QR above.

The single directional sign, [] is written above the line segment QR to denote that it continues infinitely in one direction.



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