Geometric Fourth Edition
In your answer to the question about the uses of the 4th
dimension, you only refered to time as the 4th dimension. Why not
consider the 4th dim. to be a geometric dimension as in a hypercube? If
we consider a 2D world (as in Flatland by Abbott), time still exists but
not as the 3rd dimension. If you do consider the 4th dim. to be
geometric, would there be any uses for it? If so, what would they be.
Other than tradition, there is nothing special about referring to time as the
4th dimension. You may label the time dimension any way you like.
As to a use for a hypercube: Stuck as I am here in 3-D. I cannot think of a
use for one -- other than it's fun to contemplate the model I built and have
hanging in my office.
There are ways to use more than three dimensions geometrically. One is to
create a model of a set of quantities that can be linked quadratically. If
there are four quantities related in such a way that A^2+B^2+C^2+D^2 has to
equal a constant value(k^2), you can use A,B,C,& D as four geometric
dimensions. All possible combinations of the four variables map onto a
four-dimensional sphere of radius k. Any geometric rules that apply to a
4-D sphere also apply to these variables. Geometry is used frequently in
such cases. because the geometric rules have already been worked out.
Dr. Ken Mellendorf
Illinois Central College
Time is used as the "4th dimension" because the theory of relativity says
that physics is described in terms of "space-time" not "space" + "time". The
latter works O.K. for the everyday world, but not in the relativistic world
when things are happening at speeds approaching the speed of light. In such
cases, position, velocity, and time must be considered together. Things like
fields must be independent of the frame of reference. Mathematically
"space-time" does this by a set of equations called Lorenz transformations.
See Feynman's "Lectures on Physics" for the details.
Four or more geometric dimensions do not pose any mathematical problems. All
the geometric operations -- dot product, cross product, gradient, etc. can
be, and have been, generalized to "n" dimensions.
I cannot think of an application, but there are not any mathematical
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Update: June 2012