# dimensions

Anton Sherwood (dasher@netcom.com)
Thu, 15 Jan 1998 23:56:50 -0800 (PST)

Anton wrote
: The Cartesian product of a line (d=1) and a line (d=1) is a plane (d=2).
: The Cartesian product of a line (d=1) and a point (d=0) is a line (d=1).

danny <calyk@aol> writes
: and the product of a point and a point is a line

Nope, still a point: 0+0=0.

: if a point isnt a dimension and a line is, what is a point?

Neither *is* a dimension. A point is an entity with no dimension
(no extent, as Euclid said); a line is an entity with one dimension.

: i very much think that a point is a dimension. dimensions dont start
: with a line, they start with a point i think, or maybe something else.

What do you think a dimension is? How do you start one?

: if you had three points it would be a plane, and 4 would be 3d

Nope. Three points *determine* a unique plane - that means that (in
Euclidean or similar geometry) there's only one plane which *contains*
those three points. It does not mean that those three points, all by
themselves, *make* a plane.

: ....(if they were randomly placed within 3d space, but what if they were
: randomly placed within a 4d space?

Then they determine a unique two-dimensional subspace (called a plane)
within 4-space, just as two points determine a unique line whether
in 1-, 2- or 3-space.

: What is a 4d space? time? it seems that would also include movement
: and growth)

Spacetime is, indeed, an example of a four-dimensional continuum.
Other examples are more abstract.

: It seems a point would exist within all dimensions.

Yes, an N-dimensional entity can be embedded in a continuum of N-or-more
dimensions; since a point is 0-dimensional, it can be embedded in 0-space
(itself), 1-space, 2-space, 3-space, ...

-==-=-==-==-=-==-=-==-==-=-==-==-=-==-=-==-==-=-==-==-=-==-=-==-==-=-==-=-==-==

Kennita writes:
: the answer to danny's original question is "no", because you need no
: coordinates at all to determine the position of a point in a point, so
: if you're counting coordinates or writing equations, the 0-dimension
: never shows up.
: Sigh, that doesn't sound clear enough either.

Clear enough that I wish I'd said it.

To state the position of a point in normal space requires 3 measurements.
To state the position of a point in a plane requires 2 measurements.
To state the position of a point in a line requires 1 measurement.
To state the position of a point in a point ...

Anton Sherwood *\\* +1 415 267 0685 *\\* DASher@netcom.com