Examples

• z * z + c
• z * z + c
• sin(z) * z + c
• z * z/2 + c
• z*z*z+c*c
• z * z + sin(c)

Documentation

These visualizations are generated by iterating a function of complex numbers.

This function is iterated at every point in the complex plane from -2-2i to 2+2i

Imagine the function zn+1 = zn2 + c where Z0 = 1

At point 0+0i,

• z1 = (1)2 + (0 + 0i) = 1
• z2 = (1)2 + (0 + 0i) = 1
• z3 = (1)2 + (0 + 0i) = 1

Clearly, this function will never diverge, so we color (0+0i) black

At point 0+i

• z1 = (1)2 + (0 + i) = 1 + i
• z2 = (1+i)2 + (0 + i) = 2i

The magnitude of z₂ is 2, so we know this iteration will continue to diverge. We color it a different color since it took 2 iterations to diverge.

You can define simple functions of complex numbers using a simple stack machine notation

• z pushes zn to the stack
• c pushes the current coordinate to the stack
• + adds the top two elements of the stack and pushes the result to the stack
• - subtracts the top two elements of the stack and pushes the result to the stack
• * multiplies the top two elements of the stack and pushes the result to the stack
• s take the sine of the top element on the stack
• number pushes the number onto the stack
• v take the top two numbers from the stack and combine them into a complex number

Expressions are compiled to GLSL and run inside a fragment shader on the GPU