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Real-valued Wallpaper Functions

Whenever we draw an image of a wallpaper function, we take each pixel on the screen and set its color value to reflect the value of the function, as discussed below. Without further comment, we assume the reader realizes that this involves assigning Cartesian coordinates to the screen in some way.

The first wallpaper function we ever saw was drawn using Microsoft Excel. We created a table of values of the function and invoked Excel's Chart Wizard to make a rough coloring of a piece of the domain. We didn't have to worry about color values ourselves because Excel did that for us.
Image of a wallpaper function titled "Bats".

For technical reasons, we recomputed this ourselves: Excel inexorably imposed a rectangular grid over the image.

A very attractive way to display wallpaper functions is also very simple to program, based on addressing the screen via RGB color values. Suppose f(x) is scaled to produce a value called Fx that ranges from 0 to 255. It's then very simple to create a grey-scale image by setting the red, green and blue components of the color each equal to Fx.

We get a more interesting variety of colors if we stir things up a bit. Writing (R,G,B) to indicate the triplet of color values, we could use:

(Fx, 196, 255-Fx) [Fx, 196, 255-Fx]
(255-Fx, Fx, Fx) [255-Fx, Fx, Fx]
(Fx, Fx mod 128, 255-Fx) [Fx, Fx mod 128, 255-Fx]
(Fx, 255-Fx, Fx mod 128) [Fx, 255-Fx, Fx mod 128]
(Fx, Fx mod 128, 196) [Fx, Fx mod 128, 196]

Each of these gives a pretty coloring of the range of the function. It is also possible to cycle through this range twice or more over the range of the function, as in the pgg example [Image].

Note that each of these colorings fits nicely with the concept of negation. The first two turn out to be a bit safer from round-off error, as the colors around the value 0 (at the center of the color strip) will not change drastically. To address this potential problem, in patterns with negating symmetries, we have sometimes colored the area where the value of the wallpaper function is nearly zero with a particular color, often black [Image] or white [Image].

A third possibility is to construct a different wallpaper function for each of R, G, and B. This was done for the p6 [Image], p3m1 [Image], and p31m [Image] examples.

The p3m1 example above and a few others (cmm' [Image] and pmg [Image]) where modified using "Hot Wax" function of Paint Shop Pro. We don't know exactly what is happening to the image, but it looks nice.

A time-honored way to picture a real-valued function in the plane is to view its graph. Here we have a 3D view [Image] of the graph of a wallpaper with negating symmetries.


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Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 08 Jul 1998 --- Last modified: Sep 30, 2003 9:27:41 AM
Comments to: CVM@maa.org