Where to go to get lost :-)

Highly recommended!
(Be sure to audition the crystalline soundtrack!)
For its home web page, click here.
(Its Wikipedia page.)

(Note: Once it starts playing,
both the video and audio are considerably sharper
if you click on the lower right “360p”, changing it to “480p”.
(On my Microsoft system, in both Firefox and IE, anyhow.))


















The 120-cell {5,3,3}:

The 600-cell {3,3,5}:

Finally, a mere 24-cell {3,4,3}:

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But wait; is all that too boringly static?
Then watch these convex regular 4-polytopes rotate for you:
(Forget “Dancing with the Stars”.
Try “Twirling with the Topes”, or “Pirouetting with the Polys”!)

4-Simplex or 5-Cell {3,3,3}	4-Cube or 8-Cell {4,3,3}	4-Cocube or 16-Cell {3,3,4}
24-Cell {3,4,3}	120-Cell {5,3,3}	600-Cell {3,3,5}

Stepping down a dimension, here are the five Platonic solids,
i.e., the five convex regular polyhedra in three dimensions.

Tetrahedron or 3-Simplex {3,3}	3-Cube {4,3}	Octahedron or 3-Cocube {3,4}
Dodecahedron {5,3} Icosahedron {3,5}

How’s that for a nice break from work :-)
Sorry I haven’t found rotating views of the two biggies —
the 120-cell {5,3,3} and 600-cell {3,3,5}.
You might note that, for the 4-polytopes,
the two polytopes on the left are each self-dual,
while in each row the two right-hand polytopes are dual one to the other.
For the 3-polytopes, the {3,3} is self-dual,
while the other polytopes in each row are dual one to the other.

You can check out all six convex regular 4-polytopes,
or take a look at all sixteen (not-necessarily-convex) regular polychora —
there is some very interesting Java animation at that site
(in some cases you have to move your mouse onto the figure to see it).
For the Platonic solids, see here.

By the way, if your display doesn’t have enough vertical height
to display all the polytopes without scrolling,
at least on some Windows systems the F11 key will expand the vertical scope.
Interestingly, in my system, the polytopes spin faster
in Firefox than in Internet Explorer.

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Thanks, Wikipedia!

These are hardly the only ways of
trying to depict these four-dimensional figures in two dimensions,
a difficult task at best.
There are many others, each with its own advantages and disadvantages.
The Schlegel wireframes above are dramatic and perhaps aesthetic,
but that isn’t everything.

A fascinating, thorough, prize-winning article is
“The Story of the 120-Cell” by master expositor John Stillwell.
You also might want to check out John Baez at Week 155.

You can also see what Google turns up for the polytopes:
24 {3,4,3}, 120 {5,3,3}, 600 {3,3,5}.