three cubes

Three cubes

Imagine two cubes C₁ and C₂ exactly coinciding in space.

Now shrink C₂ by the golden ratio—ie, multiply its edge lengths by the factor 1/φ = (Sqrt[5]-1)/2 = .6180339...
We'll call this new, smaller cube C₃. We'll suppose also that it is positioned in space with the same center as C₁, and with sides parallel to C₁.

Now choose one of the four body diagonals D of this configuration (ie, one of the "internal" diagonals that passes simultaneously through the common center of the cubes C₁ and C₃ and also through two pairs of their opposite vertices).

Imagine rotating C₃ about the axis D until the six of its total eight vertices not located on the body diagonal simultaneously first meet the six sides of the outer cube C₁.

It's not hard to build this configuration with the Zometool modelling kit. The smaller cube sits inside the larger one in an interestingly jaunty way:

Two cubes, with rotational diagonal D indicated by two yellow struts,
and red struts lying in the same plane as the faces of C₁

Alan Schoen saw this model, calculated the rotation angle
cos^-1((3 φ - 2)/4) = 44.4775... degrees

and asked what happens when this construction is iterated to third, fourth, fifth etc cubes. In a 11 May 2003 email he wrote (in part)

Hi Thane:

I did a little more work on your cubes, and I've attached a drawing of six cubes in the sequence. I've also drawn a nine-cube sequence, but the innermost one is pretty small! The eighth cube is rotated with respect to the zeroth one by the angle 355.82... degrees, so it almost looks as if they're parallel...

The illustration below is a live Java applet that shows the three cube configuration. (If the image looks garbled, try clicking and dragging on the picture to rotate it with your mouse).

Figure A: Three cube configuration

Note on LiveGraphics3D images

The image above is a Java applet that uses the LiveGraphics3D package by Martin Kraus. By clicking and dragging your mouse over an image, you can rotate the object. If you release a click while dragging, you can start the object spinning about a fixed axis. And by holding down the shift key and dragging up or down, you can move the object closer or farther away.

When the three cube configuration is viewed down the correct body diagonal, it should look like this:

A cube can also be inscribed on eight appropriate vertices of the regular pentagonal-faced dodecahedron. If in the original two cube construction, the rotated inner cube C₃ is not shrunk by the golden ratio, then both it and the original cube can be simultaneously inscribed in a single dodecahedron (using 16 of its vertices).

Two inscribed cubes (black and grey) in a dodecahedron
Stereo image by Alan Schoen

The five cube compound (image by Alan Schoen)
(five cubes inscribed in the vertices of the regular pentagonal dodecahedron)

It seems to be impossible to build the three cube compound (Figure A) using Zometool (email to Alan Schoen, 12 May 2003):

Hi Alan,

The odd thing is that the Zometool models don't allow the "twice rotated in the same direction" three cube configuration to be built. The node-spheroids have some kind of handedness issue with them (perhaps), ie all I can do is build the three-cube system that has the rotation reversed on the third cube, so that its sides end up parallel to the first cube. I don't really understand what is going on with that (nor do I really understand how the node design is set up in Zometool, exactly, anyway). That's actually something I would like to understand about this configuration, why the Zometool models can't build it, even though its points are clearly in the space lattice spanned by the strut-basis, in some sense...

For example, suppose cube1 is fixed, cube2 is shrunk by the golden ratio, and rotated by your angle around a fixed body diagonal D (just as before). Choosing a different body diagonal D' of cube2 I can build a three cube configuration with Zometool in which cube3 faces aren't parallel to cube1, but it is impossible if I choose the same diagonal D.

Thane

Would anyone like to share insights on these mysteries? Send email to thane@best.com

Note added 15 May 2003: The mystery solved?