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Making a colour cube is kind of an obvious thing to do if you're looking around for interesting or significant things to draw on the surface of a polyhedron.
For anyone who doesn't already understand the concept: a colour on a computer is generally represented using three numbers giving the intensities of red, green and blue light that make up the colour. An obvious way to plot these in a graphical representation is to assign one to each of the three coordinates in 3-D space, which gives you a notional cube in which every point represents a different colour. For more information (and links to far more detail than you could reasonably want), see Wikipedia.
Of course, most of the available colours are in the interior of the colour cube, but a model of this type can only show the surface. But that's OK; all the most vivid colours are on the surface, so that's the part that looks the prettiest!
Much like the construction of the cube-marked dodecahedron, this project was most simply done by hand-writing a picture description file which defined a load of small filled rectangles in 3-D space, and made sure the locations of those rectangles matched up to the cube.
When I came to print it, it turned out that it's quite a demanding
test of a colour printer, and shows up interesting features of its
colour response which you might not have expected. On my printer,
for example, there's a clearly visible part of the surface of the
cube which looks like a particularly pure cyan – but it's not
at what ought to be the cyan corner of the cube. Instead, it's
part-way along one edge from that corner. Presumably the printer is
doing some sort of non-trivial internal processing when translating
PostScript setrgbcolor commands into CMYK.
Here's the PDF net of the colour cube.
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chiark
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(last modified on Sun May 7 14:33:22 2017)