The discussion of dancing pendulums in javascript at SSSF led onto Rotating Hypercubes, which resulted in this javascript animation by forum regular mollwollfumble:
4D Rotations (click to view in a new window, right click to download javascript file)
See the SSSF thread for discussion on how to do these things in javascript.
Also see:
Background on the geometry of hypercubes (contains animated applets, but they don’t seem to work in IE9)
John Baez on the geometer H.S.M. Coxeter
First 31 pages of Coxeter’s Regular Polytopes
And a nice animated gif from http://www.123opticalillusions.com/:
Update 27-Jun-11:
As noted in the comments, we had covered rotating 4D hypercubes here before (in a spreadsheet provided by Lori Miller), and I’d forgotten all about it! The video link below shows what it looks like, but for a good look at the workings, click on the link: Hypercube.xlsx, then download the file, and have a look at the range names.
Newton Excel Bach, not (just) an Excel Blog 
Nice demo, the lighting effects add to the realism.
In the recent Excel hypercube download I had trouble recalculating in Excel 2007, maybe this is due to being saved on the skydrive?
To make animation automatic, just enter in the immediate window “do:[c6]=[c6]:doevents:loop”. Changing 4D settings creates some mind-bending rotations in which the inner and outer cubes change position with each of the other six “3-Cells” (analogous to configuration changes of 2D faces in 3D rotations).
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Lori – I downloaded the file, and didn’t have any trouble running it, but then I started a video screen capture program (CamStudio), and the F9 key stopped working. I got it to work eventually with a short VBA routine:
Sub Animate()
Dim i As Long
For i = 1 To 1000
Application.Calculate
DoEvents
Next i
End Sub
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Hi Lori
I’m sorry to say I’d quite forgotten about your hypercube when I wrote that post. When I get some time I’ll have a look so we can add an in-Excel animated hypercube to the list.
Also the gif animation at the bottom is taken from another web-site, so I’ll edit the post to show the link.
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Wanted to let you know, I used more than one of the tesseracts at Blajeny.com.
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