Mobile-friendly online Mandelbrot formula explorer with WebGPU acceleration, deep zoom, custom fractal formulas, Buddhabrot rendering, and permalinks to bookmark or share.
Orbit Trap Settings
Animation Settings
Empty: 1000. Maximum: target Max iterations. Density: analyzed starting value to target.
A small offering to the infinite ☕✨
Fuel the Lab
Every click opens another tiny universe: spirals, storms, and impossible coastlines waiting at the next zoom. Infinity is generous like that.
The developer, regrettably, is finite. If this explorer has stolen a pleasant minute from time, a coffee helps keep the math glowing and the laboratory machinery humming. 🔭
How to use the iteration function
Enter the next value of z for each iteration. For example, the standard Mandelbrot set is z*z + c.
| Expression | Meaning | Example |
|---|---|---|
z | Current complex value being iterated. | z*z + c |
c | Constant complex value from the pixel position, or the Julia constant in Julia mode. | z*z + 0.5*c |
n | Current iteration index as a real value. | z*z + c + 0.01*sin(n*z) |
zAt(k) | The z value at iteration k. It is 0 until that iteration has been reached. | z*z + c - 0.2*zAt(20) |
zDelay(k) | The z value from k iterations ago. It is 0 while there is insufficient history. | z*z + c + 0.2*zDelay(5) |
WebGPU supports static non-negative integer values for k. Other valid history expressions use CPU rendering. | ||
i | Imaginary unit. | z*z + i*c |
pi, e | Mathematical constants. | rotate(z*z, pi/6) + c/e |
+ - * / ^, () | Arithmetic operators, power, and grouping. | z^3 - 0.2*z + c |
|x| | Complex magnitude returned as a real value. | z*z + c/(1+|z|) |
Re(x) | Real part of a complex value. | Re(z)^2 + i*Im(z) + c |
Im(x) | Imaginary part of a complex value. | Re(z) + i*Im(z)^2 + c |
arg(x) | Angle of a complex value in radians. | z*z + c + 0.1*arg(z+c) |
abs2(x) | Squared magnitude, useful for damping without a square root. | z*z + c/(1+abs2(z)) |
conj(x) | Complex conjugate. | conj(z)^2 + c |
exp(x) | Complex exponential. | exp(0.45*z) + c |
sin(x) | Complex sine. | sin(z) + c |
cos(x) | Complex cosine. | cos(z) + c |
tan(x) | Complex tangent. | tan(0.5*z) + c |
sinh(x) | Complex hyperbolic sine. | sinh(0.4*z) + c |
cosh(x) | Complex hyperbolic cosine. | cosh(0.25*z) + c |
tanh(x) | Complex hyperbolic tangent. | tanh(z*z) + c |
sqrt(x) | Principal complex square root. | sqrt(z*z + c) + c |
atanSqrt(x) | Square root using atan(Im(x)/Re(x)) for the half-angle. | exp((z^2 + z) / atanSqrt(c^3)) |
log(x) | Base-10 complex logarithm. | z*z + 0.2*log(z+c) |
ln(x) | Natural complex logarithm. | z*z + 0.1*ln(z+c) |
zeta(x) | Riemann zeta function, analytically continued except for its pole at x = 1. | zeta(c) + z |
fract(x) | Fractional part of the real and imaginary parts. | fract(z*z) + c |
mod(x,y) | Component-wise modulo. | mod(z*z, 1) + c |
rotate(x,a) | Rotate a complex value by angle a radians. | rotate(z*z, pi/2) + c |
fold(x) | Mirror into the first quadrant by taking absolute real and imaginary parts. | fold(z)^2 + c |
boxFold(x,r) | Fold values outside the box radius r back toward the center. | boxFold(z, 1.2)^2 + c |