Lots of tiny bubbles.

Contents

Domain

The basic domain is the quadruple unit square,

$$\max{(|x|,|y|)} \le 1, \ z = x+iy $$

Colors

I could use the HSV colormap.

hsv_bubbles(@(z) z)

But I prefer "periodic parula", theparulacolormap concatenated with its reverse,[parula; flipud(parula)].

bubbles(@(z) z)

Powers

z^2

bubbles(@(z) z.^2)

z^3

bubbles(@(z) z.^3)

z^9

bubbles(@(z) z.^9)

1/z

bubbles(@(z) 1./z)

sqrt(z)

A complex number has two square roots. One is in the right half plane.

bubbles(@sqrt)

-sqrt(z)

And the other is in the left half plane.

bubbles(@(z) -sqrt(z))

Trig functions

sin(z)

bubbles(@sin)

cos(z)

bubbles(@cos)

tan(z)

bubbles(@tan)

cot(z)

bubbles(@cot)

Exponentials

exp(z)

These polar angles are between -1 and +1 radian.

bubbles(@exp)

exp(pi*z)/exp(pi)

These fill out the entire complex plane.

bubbles(@(z) exp(pi*z)/exp(pi))

log(z)

bubbles(@log)

Polynomials and rationals

z^3 - z

bubbles(@(z) z.^3-z)

.5/(z^5-z/5)

bubbles(@(z) .5./(z.^5-z/5))

Essential singularity.

exp(-1/(8 z^2)

bubbles(@(z) exp(-1./(8*z.^2)))

Quiz

What is happening in the animation at the top of this post? If you think you know, or even if you just know part of the answer, submit a comment. I'll have some sort of prize for the first, or best, solution.

Code

Also available athttps://blogs.mathworks.com/cleve/files/bubbles.m.

typebubbles

function bubbles(F) % bubbles. Color portrait of complex-valued function F(z), % ex. bubbles(@sin) % bubbles(@(z) .5./(z.^5-z/5) ) if nargin < 1 F = @(z)z; end axis(1.5*[-1 1 -1 1]) axis square box on cla m = 256; colormap = [parula(m);flipud(parula(m))]; n = 25; s = -1:2/(n-1):1; [x,y] = meshgrid(s); z = x + y*1i; circle = exp((0:32)/16*pi*1i)/n; w = F(z); r = abs(w); theta = angle(w)+pi; scale = 20; for k = 1:n for j = 1:n t = m*theta(k,j)/pi/scale; idx = ceil(scale*t+realmin); color = colormap(idx,:); p = w(k,j) + r(k,j)*circle; patch(real(p),imag(p),color) end end titleF = char(F); if (titleF(1) == '@') titleF(1:4) = []; end title(titleF) snapnow end

Get the MATLAB code

Published with MATLAB® R2019b