Today’s fractals are manipulated with reflection or other symmetric effects.
The first two use CoreImage filters to reflect parts of the image. The third and fourth images use a color cycle with transparencies in every other color band, with the layer duplicated and rotated. The third image uses a single duplicate rotated 180 degrees, the fourth uses two duplicates rotates 120 and 240 degrees.
All of these were made with MathPaint, which is now available on the Mac App Store. You can request a 10-day demo to try out MathPaint on your Mac.