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When you leave math professors to talk about random things...
quadrifolio
He says:

In Dimension 2, a unit square's length to the furthest vertex from the origin is sqrt(1^2 + 1^2) = sqrt(2).
In Dimension 3, a unit cube's length to the furthest vertex from the origin is sqrt(1^2 + 1^2 + 1^2) = sqrt(3).

So as the dimension increases, the unit square's length increases.

Dimension 2:
Suppose you have a square of side 0.5. Its area is 0.25.
Suppose you have a cube of side 0.5. Its volume is less than the area of the square: it's 0.125 [decreases?!]

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