Multibrot
w = z^n + c
Open in the app The dials and keys named below are the app's.
What it draws
w = z^n + c is applied over and over with c held at the point being drawn, and the points whose orbits stay bounded make the set. That is the Mandelbrot construction with the square replaced by whatever power the n slider carries, and the family of sets it sweeps out is called the multibrot sets. The slider opens at 3, one step up from the famous one.
The symmetry count
At a whole n the picture has n − 1 fold rotational symmetry: turn it by 360° / (n − 1) about the origin and it lands on itself. The reason is short. Multiply c by a number u with u^(n − 1) = 1, and every point of the orbit is multiplied by u as well, so it stays bounded or escapes exactly as it did. At the opening value that gives two lobes, 5 gives four, and 10 gives nine bays round the rim. The Mandelbrot set is the member with one, which is no rotational symmetry at all.
Between the whole numbers
A fractional power is read on the principal branch, the one that sends a positive real number to its ordinary positive power. Conjugating a number conjugates that power, so the picture stays mirror symmetric about the horizontal axis at every value of the slider, whole or not. What it loses in between is the rotation: at 3.5 the set is caught between the two-fold form and the three-fold one and has neither.
The two dull values sit low on the slider. At 0 the map ignores z altogether, so no orbit can escape and the whole square belongs to the set. At 1 it adds c forever, and only a speck around the origin survives.
Try
- Drag n to 2 for the Mandelbrot set itself, the one member of the family with no rotational symmetry.
- Drag n to 10: nine identical bays, and more of the square stays bounded than at any whole value from 2 up.
- Drag n to −3: the family turns inside out. Most of the square is now bounded, with a lacy hole where the escaping points used to be.
- Type z^n + 0.3: the constant stops following the point, so the same map draws a filled Julia set instead of the parameter picture, and the slider now sets that set's symmetry.
- Top for the flat picture, straight down on the plane.