# Equations of a space

joe shmo
Posers and Puzzles 06 Jan '12 03:03
1. joe shmo
Strange Egg
06 Jan '12 03:031 edit
Why does a cartesian equation for a space not exist, or does it, and I'm just not aware of it? I suppose I'm overlooking some detail, but what? Is 4-dimensional coordinate system needed to define a 3- dimensional space in a general equation?
2. talzamir
Art, not a Toil
06 Jan '12 20:211 edit
An equation by nature sets two things as being equal. That lowers the degrees of freedom in the variable set by one. For example, if you have a single dimension where x is any real value, you have one degree of freedom (x can be anything). When you set an equation in it, say 2x + 3 = 5, that reduces the degree of freedom by one, to zero; that is, x = 1. In a plane you have two degrees of freedom, and can reduce that by one by setting an equation there; e.g. y = x^2; that is true for x equals any real value, say a, as long as y equals a^2. In space you have three degrees of freedom but nearly any equation reduces that freedom from three to two, giving you a plane.

To get a solution with three degrees of freedom, you could start with four and limit that by one to three.. say, (x,y,z,t) in a 4D space and use the equation t = 0, which is true for any real values of x, y, and z, as long as t is zero.

alternatively.. you could use a non-conditional equation that does not reduce the degrees of freedom. Say, the values x,y,z for which

x + y + z = x + y + z.
3. joe shmo
Strange Egg
09 Jan '12 04:451 edit
Originally posted by talzamir
An equation by nature sets two things as being equal. That lowers the degrees of freedom in the variable set by one. For example, if you have a single dimension where x is any real value, you have one degree of freedom (x can be anything). When you set an equation in it, say 2x + 3 = 5, that reduces the degree of freedom by one, to zero; that is, x = 1. In ...[text shortened]... oes not reduce the degrees of freedom. Say, the values x,y,z for which

x + y + z = x + y + z.