Imagine two cubes C1 and C2 exactly coinciding in space.
Now shrink C2 by the golden ratioie, multiply its edge lengths by the factor
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: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,Would anyone like to share insights on these mysteries? Send email to firstname.lastname@example.org
Note added 15 May 2003: The mystery solved?