Fitting designs (continued)

Further to Fitting designs - which looked at the history of the classic puzzle of making an object that will fit three holes of different shapes: square, circular, and triangular - Emily from the largely mathematics/nature blog Ephemeral Curios commented:

These might also be the inspiration for Douglas Hofstadter's "trip-lets"-- the blocks on the cover of Godel, Escher, Bach that cast three different letter shadows.

Thanks! Yes, the GEB "trip-let" (left) is identical in concept to the object with three orthogonal letter shapes objects spelling "PSM" in Popular Science Monthly for July 1927 (see Arthur L Smith's A Square Peg in a Round Hole).

But I'm not clear if (as is quite possible) it was independently conceived; Hofstadter wrote in the intro to GEB:

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.

Does "idea came to me" mean the whole trip-let concept, or just the idea of using it for the cover?

There are plenty of interesting spin-offs. Googling found a nice page by Humberto José Bortolossi of the Mathematics Department, Universidade Federal Fluminense - Triplets - which has rotatable models, along with an intriguing reference to the generalisation of the idea. Objects can exist with an arbitrary number of specified shadows when illuminated from different directions. This is not merely theoretical, but has actually been applied to the creation of the "digital sundial" designed by Scharstein, Scharstein and Krotz-Vogel: see U.S. patent 5,590,093 and Digital Sundials International.

Another good page is The magic of trip-lets at mariano tomatis blog, which shows the design (not trivial to achieve) of an "EMC" triplet for the Eseential Magic Conference. It also raises the possibility of trip-let type objects in the past - that is, objects designed to look different from different directions - via the example of a 17th century carved crucifix by the Franciscan monk Fra Innocenzo da Petralia, deemed miraculous for its ability to show different expressions on the face of Christ (suffering, dying and dead) according to viewing angle. This looks to me an equivalent of the "Noh Mask Effect": the ability of masked actors to convey expression change in Noh drama by tilting the head to give different viewpoints of their rigid masks. See Michael J Lyons, Noh Masks & Facial Expression Perception, which has several research papers. The concept seems to be pretty widespread: see, for example, Perceiving faces of Buddha statues, Same miraculous statue; different expressions, and How Moving Light Changes Expression on Face of Small Statue.

- Ray