z, i, pi, numbers · + − * / ^n (n a whole number) · sin cos exp sinh cosh, cas (= cos + sin), tan cot tanh · diff(p): the derivative in z, newton(p) = z − p/diff(p) · 2z means 2*z · with c: the parameter plane (the pixel is c) · Enter draws, Esc restores · F opens this panel
What you see. The Julia set of the formula in the address, #f=… (cas z = cos z + sin z by
default, change it top left): every point of the plane is iterated, z, f(z), f(f(z)), … Light points settle in
an attracting cycle, the colour says how many steps it takes; dark points escape. A formula with c, such as
z^2 + c, shows its parameter plane instead: the pixel is c, the orbit starts at a critical point, black is what
does not escape (the Mandelbrot set for z^2 + c).
Deep zoom. The centre is computed in arbitrary precision, every pixel on the GPU only as a small difference from it (perturbation), checked against full-precision arb results down to 10120.
Controls. Wheel: zoom at the cursor. Drag: pan. Click: centre the point. S + mouse movement: zoom at the point where S was pressed; D + mouse movement, + and −: zoom at the centre. P: centre the nearest repeating (periodic) point and hold it while zooming at the centre, where the picture repeats forever; P again zooms in on its own until any other input. C: initial view. H: technical readout. F or top left: the formula. K or bottom right: colours by the number of steps or by the attracting cycle reached (the root, for Newton's method). Touch: drag to pan, pinch to zoom, double tap to centre, long press for P. Bottom left: coordinates, a link to this view, back to the initial view, the periodic point.
References. C. Heiland-Allen, Perturbation algebra (2018); C. T. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. AMS 300 (1987).
converging points coloured