OT hypercube (was: Con-other)

--- Eugene Oh <un.doing@...> wrote:

> I'm very curious, though, about the animation near
> the top of the article
> you provided: It looks as though the structure is a
> cube inside another à la
> Russian Dolls, both with vertices of variable
> length, and folding in on
> itself. When you said that the walls of a tesseract
> are cubes, I sort of got
> the impression that it was supposed to be like a
> stubby cross. The walls
> don't look at all like cubes to me!
> Eugene
>
That's because you're looking at a *picture* of a
tessaract, not an actual tesseract.

Think of a cube.  It has 6 faces, they are all square,
all the same size, and all the edges meet at 90 deg.

Now imagine/draw a picture of a cube. The faces are
not all square, not all the same size, and not all the
angles are at 90 deg, but you can understand what you
are looking at is a 2D representation of a cube.

If you 'unfold' a cube (3D) to make it flat (2D), the
most typical arangement is a cross shape (although you
could unfold it differently). If you think about this
shape you can see that the sides of some of the
squares in the cross don't touch when the cube is
unfolded, but if you fold the cube back together you
realise what looked like two different sides in
different places on the flat cross, are in fact the
same edge in the same place on the cube.

If you unfold a tesseract (4D) to make it 'solid'
(3D), you could end up with the same cross shape as an
unfolded cube, but instead of flat squares you have
solid cubes - kind of like you had arranged some kid's
building blocks on the table.  In the same way that
different sides of different squares in the unfolded
cube are actually the same edge when the shape is
folded up, different cube faces of different cubes in
the unfolded tesseract are in fact the *same* face of
the tesseract when you put it back together.

Once you get the idea, you can go on and try to
imagine 5D 'cube', where all the 'faces' are
tesseracts....

P

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