# Printing Pictures on the Surface of Polyhedra

## Building a dodecahedron with the cube edges marked

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When
Gareth
asked if I could make a dodecahedron with the five inscribed cubes
marked on its surface, I wasn't immediately sure how best to do it.
All my other polyhedral pictures are basically *spherical*
images, without any specific reference to the polyhedron they'll end
up drawn on; you can re-project them on to a different solid
relatively trivially. But to construct the net of a dodecahedron
with an extra set of lines drawn between its vertices, this seemed
like overkill: what I'd be drawing would be fundamentally
*based* on the vertices I already knew about, and there would
be no need to project it on to a different solid at all.

Nonetheless, it turned out, using the existing spherical-image
mechanism was still the easiest way to go about it, not least
because in order to do it any other way I'd have had to modify my
net-drawing utility again, and it's complicated enough that that
would have outweighed any saving due to conceptual parsimony.

And, in fact, it wasn't too hard to do it directly using the
spherical image code. First I found the polyhedron description file
I was going to use for my dodecahedron; that has the 3-D coordinates
of the 20 vertices already specified. So it was just a question of
identifying the eight subsets of those vertices which formed cubes,
and directly writing a PostScript spherical-image description
consisting of appropriately coloured straight lines between those
points in space. My projection code doesn't change the orientation
of the image relative to the polyhedron, so this naturally produced
the right result when projected on to the same dodecahedron
description file I'd started with.

Of course, geodesics in spherical image space (great circle arcs)
project into straight lines on the resulting solid, so there wasn't
even any difficulty in making the lines straight.

Identifying the actual cubes was a moderately frustrating ten-minute
task with a pencil and paper, in which I figured out the quick and
easy way to do it just as I was finishing the last cube. But it only
had to be done once, and then it was just a matter of tweaking the
colours and line thicknesses.

## Downloads

Here's the PDF net of the cube-marked
dodecahedron.

[go back to the main polyhedral pictures page]

(comments to anakin@pobox.com)

(thanks to
chiark
for hosting this page)

(last modified on Tue Oct 30 00:40:42 2007)