The Mathematics of Beauty > Plane

Plane - from Latin: plantum - "flat surface," A flat surface that is infinitely large and with zero thickness.

Clearly, when you read the above definition, such a thing cannot possibly really exist. Imagine a flat sheet of metal. Now make it infinitely large in both directions. This means that no matter how far you go, you never reach its edges. Now imagine that it is so thin that it actually has no thickness at all. In spite of this, it remains completely rigid and flat. This is the 'plane' in geometry.

point	line	Plane	Solid
Zero dimensions	One dimension	Two dimensions	Three dimensions

It fits into a scheme that starts with a point, which has no dimensions and goes up through solids which have three dimensions:

It is difficult to draw planes, since the edges have to be drawn. When you see a picture that represents a plane, always remember that it actually has no edges, and it is infinitely large.

The plane has two dimensions: length and width. But since the plane is infinitely large, the length and width cannot be measured.

Just as a line is defined by two points, a plane is defined by three points. Given three points that are not collinear, there is just one plane that contains all three. source

shape
n.
1.
a. The characteristic surface configuration of a thing; an outline or contour. See Synonyms at form.
b. Something distinguished from its surroundings by its outline.
2. The contour of a person's body; the figure.
3.
a. A definite distinctive form: "The bomb gave the shape of life, outer and inner, an irreversible charge; a sense of fatefulness would now lie on all things" (Alfred Kazin).
b. A desirable form: a fabric that holds its shape.
4. A form or condition in which something may exist or appear; embodiment: a god in the shape of a swan.
5. Assumed or false appearance; guise.
6. A ghostly form; a phantom.
7. Something, such as a mold or pattern, used to give or determine form.
8. The proper condition of something necessary for action, effectiveness, or use: an athlete in excellent shape.
v. shaped, shap·ing, shapes
v.tr.
1. To give a particular form to; create.
2. To cause to conform to a particular form or pattern; adapt to fit.
3.
a. To plan to bring about the realization or accomplishment of; devise.
b. To embody in a definite form: shaped a folk legend into a full-scale opera.
4.
a. To adapt to a particular use or purpose; adjust.
b. To direct the course of: "He shaped history as well as being shaped by it" (Robert J. Samuelson).
v.intr.
1. To come to pass; happen.
2. To take on a definite shape or form. Often used with up or into. source