Conformal maps by Robert Bunney

A close up of 4 of the hundredth roots of unity. By Ian Sammis.

100th post!

A Wolfram Demonstration titled Combinations of Sines in the Complex Plane by Stephen Wolfram. He writes "Combinations of two sine functions must always have their zeros on the real line. Combinations of three need not. The height here is the absolute value of the sum of sine functions; the hue is the phase."

These images are from a Wolfram Demonstration by Michael Schreiber titled Complex Exponential Resonance. He writes "Complex exponentials of complex exponentials feature resonance patterns. There is only one polygon; black regions are under an odd number of self-overlays."

An image from an awesome demo by Ed Pegg Jr. that's part of the Wolfram Demonstrations Project. This image corresponds to Gaussian integers raised to the 11th power. The demo allows one to view Gaussian integers raised to any fractional power. The user can choose a numerator and denominator less than 20.

Gaussian integers raised to the 17/6 th power

Klein's j-invariant in the complex plane

The Real part of z^z^z

A plot of the complex valued function (z-2)²(z+1-2i)(z+2+2i)/z³ by Hans Lundmark

Real part of the j-invariant as a function of the nome q on the unit disk

This image by Stephen Schiller has been one of my favorites for years. It shows the distribution of fractions of Gaussian integers with a restriction on the denominator. Explicitly, this is the set of complex numbers (a + bi)/(c + di) where a,b,c, and d are integers, and √(c² + d²) < 25. Source: The Pattern Book, by Clifford Pickover.

Irreducible fractions come up all the time in math problems, but who knew they looked so beautiful all together?
This picture, from Wolfram, is of irreducible fractions in the Complex plane.

"Map the infinite checkerboard to the plane so that black squares map to squares and white squares map to quadrilaterals. Such a map is a disguised version of a discrete analytic function. This is a discrete approximation to the exponential map f(z)=e^z."
source: http://www.math.brown.edu/~rkenyon/gallery/gallery.html

"Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate the leading integer coefficient of the polynomial the number is a root of (red = 1 i.e. the algebraic integers, green = 2, blue = 3, yellow = 4...). Points becomes smaller as the other coefficients and number of terms in the polynomial become larger. View shows integers 0,1 and 2 at bottom right, +i near top."
(http://en.wikipedia.org/wiki/Algebraic_number)