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| Math Central | Quandaries & Queries |
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Question from Dan: I am not a mathematician. This seems to me an intuitively simple enough problem that I very much need an answer to from someone who's mathematics are better than mine. Please help. The question is: for a tesseract of side length = 1 I think I have calculated the distance of each vertex from the center, and of the center of each edge from the center, but the question above baffles me. (anyone not having a clue what I am talking about can brush up here http://en.wikipedia.org/wiki/Tesseract ) Thanks in advance |
We'll use coordinates with the center at (0,0,0,0) and the vertices at
(±1/2,±1/2,±1/2,±1/2)
and use Pythagoras' theorem in its 4-dimensional version
\[d^2 = w^2 + x^2 + y^2 + z^2\]
to get distances.
For a vertex of the tesseract, all four coordinates differ from the center's by 1/2:
\[d = \sqrt{\left[(1/2)^2 + (1/2)^2 + (1/2)^2 + (1/2)^2\right]} = 1\]
(did you get that?)
For edge centers, three coordinates differ:
\[d = \sqrt{\left[(1/2)^2 + (1/2)^2 + (1/2)^2 \right]} = \frac{\sqrt 3}{2}\]
I think you can solve now for square-face centers and cube-face centers!
Good Hunting!
RD
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