Mandelbar Set

The Mandelbar set changes one thing in the rule of the Mandelbrot set: before each square, the point is mirrored across the horizontal axis, so (x,y)(x, y) becomes (x,−y)(x, -y).1 A pixel's coordinates give a number to add, called cc, the point starts at zero, and the pixels whose points never run away belong to the set.

That one mirror replaces the Mandelbrot set's heart shape with three matching corners, which is why John Milnor called this set the tricorn.2 In the route below, the dotted line shows the mirror before each jump.

Follow one pixel

Pick a pixel to test it. The point starts at 0. Each jump mirrors it across the horizontal axis and squares it, then adds the pixel's coordinates.

JumpPoint
0(0, 0)

The three corners come from what the mirror and the square do to a turn. Suppose the point and cc are both turned by 120 degrees. Mirroring changes that turn to −120 degrees, and squaring doubles it to −240 degrees, which faces the same way as +120. Adding cc, also turned by 120 degrees, keeps the whole result turned by 120 degrees.

So when you turn a pixel by a third of a circle, its entire route turns with it, jump for jump. If one pixel's route runs away, so does the route of the pixel turned by 120 degrees, and the picture has to look the same after a third of a turn.

Squaring without the mirror turns 120 degrees into 240, so the Mandelbrot set has no such match. A higher power adds a corner: the fourth power with a mirror turns 72 degrees into −288, the same as 72, which gives five corners.3

Rotate the pixel, rotate its route

Drag the point. The other colors start from the same pixel, turned by equal parts of a full circle. Then step through the jumps.

Power 2 · 3 matching routes · Jump 4

The smooth body in the middle is where a route settles on one point, and its edge is where that point sits exactly 1/2 away from zero. To draw that edge, walk the point zz around its circle and compute c=z−zˉ2c = z - \bar{z}^2, where zˉ\bar{z} is the mirrored zz. The curve has three sharp points. Use c=z−z2c = z - z^2 instead, without the mirror, and you draw the Mandelbrot set's heart, with one sharp point.4

Further exploring

Sources and notes

  1. Mandelbar Set.
  2. Remarks on Iterated Cubic Maps, §4.
  3. On Multicorns and Unicorns II: Bifurcations in Spaces of Antiholomorphic Polynomials, §1.
  4. Calculation. A point settles on z=zˉ2+cz = \bar{z}^2 + c when two jumps shrink nearby distances, which needs ∣2z∣<1|2z| < 1. Setting z=12eiθz = \tfrac12 e^{i\theta} gives c=12eiθ−14e−2iθc = \tfrac12 e^{i\theta} - \tfrac14 e^{-2i\theta}.