Having come to the end (finally!) of the voluminous tome
that is Gödel,
Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter, I decided
to try my hand at reproducing the GEB-EGB trip-let that forms the cover image
(at least on the copy of the book that I was reading). This self-imposed
exercise came as a welcome change of pace from the mathematical-typographical
exercises that Hofstadter presents (punishes?) his readers with. Hofstadter
describes the cover image thus:
Cover: A "GEB" and an
"EGB" trip-let suspended in space, casting their symbolic shadows on
three planes that meet at the corner of a room. ("Trip-let" is the
name which I have given to blocks shaped in such a way that their shadows in
three orthogonal directions are three different letters. The trip-let idea came
to me in a flash one evening as I was trying to think how best to symbolize the
unity of Gödel,
Escher, and Bach by somehow fusing their names in a striking design. The two
trip-lets shown on the cover were designed and made by me, using mainly a band
saw, with an end mill for the holes; they are redwood, and are just under 4
inches on a side.)
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| GEB-EGB trip-let as depicted on the cover of my copy of Gödel, Escher, Bach |
The trip-lets are simple yet interesting, hence the appeal
of a reproduction. My reproduction would not be in wood, however, as I lack the
tools and familiarity of experience that would make such an undertaking
enjoyable, but instead a virtual reproduction. I fired up Autodesk Fusion 360
and set about modelling away. The trip-lets themselves took little time to
model, but establishing the rendering setup and coming to grips with Fusion
360's peculiarities in this regard took me quite a bit of time. I found myself
limited in my ability to accurately reflect the material and I have to say in
the end I much prefer the original to the virtual approximation. However, there
is a pleasingly recursive aesthetic in the image, being that it is an image of
a reproduction of a three-dimensional space that attempts to mirror a
photographic reproduction of a real three-dimensional space. My virtual
trip-lets may be thought of as being isomorphic to Hofstadter's real trip-lets
(in a high-level conceptual sense), and this I feel is very much in keeping
with the theme of the book.
 |
| Final rendering of my GEB-EGB trip-let |
Some notes on the technical details follow. I used Fusion
360's cloud rendering option, however this left a fair amount of noise in the
final image. There is no way to improve the quality via cloud rendering and
rendering on my local machine at high quality was projected to take a number of
days. I applied a median noise filter using Paint.NET to eliminate the noise as
best I could. As well you will notice that the cast shadows have peculiar
notches in them. I attribute this to the relatively low number of ray-tracing
passes performed by the renderer as these notches are not in the geometry and a
simplistic shadow-casting rendering contains no such artifacts. My trip-lets
are 25 millimetres on a side (roughly one inch) and are cubes, rather than
Hofstadter's four-inch trip-lets. Theoretically such differences in scale could
be made imperceptible by careful selection of lens parameters. Arguably though
there would be a closer correspondence in the wood-grain texture had I kept the
original scale, as I could not find a way to scale the texture size in Fusion
360. I used no global scene lights, and illuminated the scene using models
placed in the scene and assigned to them emissive materials. These lights
consist of a central white "LED" surrounded by an annular
"LED" with a yellow filter over it. A virtual shutter covers the
light directly, permitting only a circle (for the "B") and ovals (for
"GE" and "EG") of light to be projected onto the virtual
room surfaces. The lights were placed sufficiently far away to get the desired
shadow sizes on the surfaces. Finally I notice that the letter-forms are not as
faithful to the originals as I would like (the "G" is too boxy, the
proportions of the "E" and "B" are slightly off), so
perhaps a re-do is eventually in order (sigh).
Hofstadter, Douglas R. (September 1980) Gödel, Escher, Bach: An Eternal Golden Braid. Vintage Books Edition. Reprint of the ed. published by Basic Books, New York. April 1979.
References
Hofstadter, Douglas R. (September 1980) Gödel, Escher, Bach: An Eternal Golden Braid. Vintage Books Edition. Reprint of the ed. published by Basic Books, New York. April 1979.
