The other evening I was telling one of my neices
about hypercubes. Start with a D=1 object, namely
a line:
_______
Then duplicate it, andconnect corresponding ends.
The result is a D=2 object, namely a square:
_______
| |
| |
| |
|_______|
Then duplicate that, and connect corresponding corners.
The result is a D=3 object, namely a cube:
_______
/| /|
_______ / |
| | | |
| |____|__|
| / | /
|_______|/
Now keep going. Duplicate that, and connect corresponding
corners. I'm not going to draw it using ASCII art, but
it's not very hard to draw it using pencil and paper.
If you can project a D=3 cube onto D=2 paper, you can
perfectly well do the same for a D=4 hypercube.
Meanwhile I noticed that my other neice had gone down
to the basement and returned with a hot-glue gun and
a box of toothpicks, and was busy gluing them together.
She made two interlocking cubes, and connected
corresponding corners with toothpicks. The result was
a very nice D=4 hypercube, or rather the projection of
a D=4 hypercube onto D=3 space.
I wanted to play with the hypercube, but she had other
ideas. She wanted to go for D=5. I tried to talk her
out of it, but to no avail.
The result is quite something. It has
-- 32 corners
-- 80 sticks: five sticks meet at each corner
-- 80 faces: they are squares in D=5, but many of them
get squashed into rhombuses by the projection down to D=3
-- 40 cubes: that is, they are cubes in D=5, but many get
squashed into rhombohedrons by the projection
At first glance, it looks like just a jumble of sticks.
But on second glance, you start seeing patterns, then
more patterns, then ..... In that way it symbolizes
science itself: at first science looks like a jumble
of facts, but then you start seeing patterns, then more
patterns, then .....
You can project the hypercube onto D=2 by looking at
its shadow under a far-away light. That brings out
additional regularities that aren't super-easy to see
just by eyeballing it at arm's length.
=====
Glued joints are rigid. We conjecture that it would
be amusing to make all or most of the joints flexible,
perhaps by attaching short threads to the end of the
sticks and joining the threads.
============
Remark: The process of increasing the dimensionality
by duplicating the object and connecting corresponding
corners is a pretty good way to visualize what a wedge
product does in Clifford Algebra, especially if you
add arrowheads to the toothpicks, to show the direction
of circulation.
============
Homework: As a function of D (the number of dimensions
in the hypercube), from D=1 to D=6, calculate the
number of d=0 corners, the number of d=1 edges,
the number of d=2 faces, the number of d=3 hyperfaces,
et cetera. Hint: there's a relationship between rows,
somewhat reminiscent of Pascal's triangle. Duplicate
and connect corresponding parts.....