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.)
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 120cell {5,3,3}:
The 600cell {3,3,5}:
Finally, a mere 24cell {3,4,3}:
But wait; is all that too boringly static?
Then watch these convex regular 4polytopes rotate for you:
(Forget “Dancing with the Stars”.
Try “Twirling with the Topes”, or “Pirouetting with the Polys”!)
4Simplex or 5Cell {3,3,3}  4Cube or 8Cell {4,3,3}  4Cocube or 16Cell {3,3,4} 
24Cell {3,4,3}  120Cell {5,3,3}  600Cell {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 3Simplex {3,3}  3Cube {4,3}  Octahedron or 3Cocube {3,4}  


How’s that for a nice break from work :)
Sorry I haven’t found rotating views of the two biggies —
the 120cell {5,3,3} and 600cell {3,3,5}.
You might note that, for the 4polytopes,
the two polytopes on the left are each selfdual,
while in each row the two righthand polytopes are dual one to the other.
For the 3polytopes, the {3,3} is selfdual,
while the other polytopes in each row are dual one to the other.
You can check out all six convex regular 4polytopes,
or take a look at all sixteen (notnecessarilyconvex) 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.
Thanks, Wikipedia!
These are hardly the only ways of
trying to depict these fourdimensional 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, prizewinning article is
“The Story of the 120Cell” 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}.
Labels: geometry, mathematics
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