Peano(*t*) = *x*+*iy* continuously maps the unit interval
[0,1] onto the unit square [0,*i*+1]. If one (respectively both) of
*x* and *y* are dyadic rationals, there are two (respectively
three) inverse images *t*. Peano can be implemented by a finite
state machine which eats two bits of *t* and emits one bit each of *
x* and *y*. Thus if *t* is rational, then so are *x*
and *y*. I believe I once showed that the Thue constant *T* = .
*0110100110010110*... and 1-*T* are the only solutions to Peano(
*t*)=*t*+*i*.

The following are filled, connect-the-dots polygons generated by
successive values of Peano given *t* in various arithmetic
progressions.

The previous was actually an erroneous rendition of

Peano((14n+9)/256). Thus the white triangle is not equilateral. (Why not?)