The rationale behind this question is computer rendering of the Mandelbrot set using binary floating point (a subset of the dyadic rationals): interior and exterior points are relatively easy to classify, the boundary is trickier, and it may be easiest to implement if there is a finite list of cases to test against.

These points are exactly on the boundary of the Mandelbrot set and have dyadic rational coordinates (both real and imaginary parts are exactly representable in binary floating point given sufficient precision):

  • $\frac{1}{4}$, cusp of the period $1$ cardioid;
  • $-\frac{3}{4}$, bond of period $1$ cardioid with period $2$ bulb;
  • $-\frac{5}{4}$, bond of period $2$ bulb with period $4$ bulb;
  • $-\frac{7}{4}$, cusp of period $3$ island;
  • $-2$, Misiurewicz point at tip of antenna;
  • $\pm 1i$, Misiurewicz points at tip of branches near period $3$ bulbs;
  • $\frac{1}{4} \pm \frac{1}{2} i$, bonds of period $1$ cardioid with period $4$ bulbs;
  • $-1 \pm \frac{1}{4}i$, bonds of period $2$ bulb with period $8$ bulbs.

Question: is this an exhaustive list?

  • $\begingroup$ mrob.com/pub/muency/rationalcoordinates.html $\endgroup$
    – Adam
    Commented Jun 14, 2017 at 18:03
  • $\begingroup$ thanks @Adam, there was one pair of points there missing from my list, now added $\endgroup$
    – Claude
    Commented Jun 17, 2017 at 14:25
  • $\begingroup$ I converted comment to answer. You can accept it. $\endgroup$
    – Adam
    Commented Jul 11, 2017 at 7:31

1 Answer 1


Here is a list made by Robert P. Munafo



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