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
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;![SolsticeDial][1]<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<sub>(Image added by O'Rourke)</sub>

 


  [1]: http://cs.smith.edu/~orourke/MathOverflow/solsticedial_mari_500.jpg