# Wallpaper Patterns Generated by Complex Mapping of a Picture

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

One way of visualizing a complex-valued function in the plane is to assign a unique color to each point of a certain region of the complex plane—for example, the color value on a picture of the point with coordinates and . In this way, wallpaper patterns can be produced.

[more]
Contributed by: Izidor Hafner (February 2016)

(Idea by Frank A. Farris)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Let be a finite group of rotational transformations in the complex plane. is a function defined on the complex plane. An average of over can be defined by

.

For example, in the case we can choose the group .

This Demonstration considers also the cases for rotations of order . No periodic wallpaper pattern can be produced for .

Reference

[1] F. A. Farris, *Creating Symmetry: The Artful Mathematics of Wallpaper Patterns*, Princeton: Princeton University Press, 2015 pp. 66–67.

## Permanent Citation