Perhaps the digital sundial of Falconer is what you need: http://www.researchgate.net/publication/225574085_Digital_sundials_paradoxical_sets_and_vitushkins_conjecture
Here is a photograph of a working model:
http://apod.nasa.gov/apod/ap120626.html
SolsticeDial http://cs.smith.edu/%7Eorourke/MathOverflow/solsticedial_mari_500.jpg
(Image added by O'Rourke)
