Question:
In the response to the original article, Sam Bowen stated that it
is not difficult to imagine 4-dimensional objects but it is difficult to think
about what they look like. My question is what is the difference between
imagining them and thinking about what they look like? Sure its easy to write
equations for 4- dimensional objects that are representable by functions but
take for example a hyper-cube. How would one go about finding what the
projections of this would be onto a 2-dimensional surface? I have seen the
projection of a hyper-cube onto 2-dimensions but cant imagine what the actual
hyper-cube would 'look like.' On the note of 4 dimensional space-time, I do
realize that we live in a 4 dimensional universe, but I am talking about
objects that are in 'mathematical' four space. It seems that since we
experience time instantaneously and that it seems to be uni-directional that
it is different from the other 3 dimensions. I still think that it is
impossible to 'see' anything that exists in four 'physical' dimensions since
we exist in only 3 'physical' dimensions, the temporal dimension would not
seem to help in realizing what these four dimensional objects really are.

Replies:
I guess we have a semantic problem. I was considering problems
in spaces larger than 3 where the dimensions might be different products or
activities and we might be trying to minimize or maximize some function over
these spaces. Developing some imagination about what the volume in that space
can aid in the seeking of a solution. If you are insisting on four physical
dimensions I would have to agree with your original statement that these
cannot be seen. In relativity there are often needs to calculate integrals
over time and space and that leads to four dimensional integrals and their
evaluation. While this is only mathematics it is very "real".

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