Stereoscopic Animated Hypercube v2.0

Put on your 3D glasses glasses (red on the left). (Free glasses here.)
Click the Detach button and make the graphics window bigger.
3D rotation is by left-button mouse-dragging.
4D rotation is by right-button mouse-dragging (or hold down the shift key, then drag).
Play with the controls!
Troubleshooting here. If this OpenGL version doesn't work, you might try the non-OpenGL version.
Original 1996 Hypercube here. Disambiguation here.
My Java Notes here.


screen shot

 


More Java applets here.

August 17, 2009:
Better explanations will follow on a later date.
For now, just a few remarks...

The Object choice:

The applet is capable of displaying various 4D polytopes in addition to the hypercube. All the regular convex 4D polytopes are supported, as well as several compounds. I'll probably add some non-convex polytopes when I get around to it.

The A x B choice displays a Rectangular Product (see below).

What you're seeing:

The edges of the 4D polytope are represented as a sort of wireframe in 4D, only instead of thin wires, I have 4D struts. The struts are 4D hypercylinders. When their shadows are cast onto a 3-space (from a 4D point light source), they appear as 3D cylinders.

Actually, if the 4D point light source is not too distant, the 3D cylinders' thickness should vary along their length (they should be truncated cones), since the parts of the hypercylinders that are nearer to the 4D point light source should make larger shadows than those that are farther away from it.

The applet compensates for this 4D perspective effect, displaying constant-thickness cylinders. However you can un-check the Compensate checkbox to see the struts in their natural form with 4D perspective.

There is also a 4d Struts checkbox that utilizes a different kind of 4D struts (not hypercylinders). I'll add an explanation of that later (after I refresh my memory about how it works).

Rectangular Products (A x B):

The Pick AxB button displays the Rectangular Product Picker window.

The A x B polytope (in 4 dimensions) is formed as the rectangular product of two lower-dimensional polytopes (see Coxeter01, p124). There are two families of them, the 2 x 2s and the 3 x 1s. The 2 x 2 polytopes are rectangular products of two polygons (polygons are 2-dimensional polytopes). The 3 x 1 polytopes are rectangular products of a 3-dimensional polytope and an edge (a 1-dimensional polytope).

The Picker window has 3 parts:

The notation used in the formula should become obvious as you play with the Picker. Some polygons: The last one is an example of the {p/q} notation used in Coxeter01 (p93) for star polygons. In the pentagon {5}, consecutive vertices occur every 1/5 turn around a circle. In the pentagram {5/2}, consecutive vertices occur every 2/5 turn around the circle.

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