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Name:  Patricia
Status: educator
Age: 50s
Location: N/A
Country: N/A
Date: 2000-2001


Question:
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.


Replies:
Pat,

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.

Regards,
ProfHoff


Patricia,

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 difficulties.

Vince Calder



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