   SEARCH HOME Math Central Quandaries & Queries  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 what is the distance of the center of each cube from the center of the tesseract ? 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     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.