Eric Weeks - personal pages - misc

Spiral Triangles

A simple way to make interesting spirals. See the bottom of the page for links to other simple algorithms like this one.

weeks@physics.emory.edu

NOTE: I also wrote a page about mathematical spirals (Archimedes' Spiral, etc).

I learned this idea from George Francis when I took his Math 198 class (which, from his web page, it looks like he is still teaching; it's a good class, I recommend it).

The basic idea is to start with a triangle, interpolate to find a point a fraction of the way down a side, draw a line to this point, and use that point as the new corner for the triangle. This will make a picture that looks like this:

Here's the simple awk script that made that picture:

BEGIN { ee = 0.08; x1=0; y1=0; x2=10; y2=0; x3=5; y3=5*sqrt(3); print x1,y1,x2,y2,x3,y3,x1,y1; for (t=0;t<100;t++) { x4 = ee*x2 + (1-ee)*x3; y4 = ee*y2 + (1-ee)*y3; print x4,y4; x3=x2; y3=y2; x2=x1; y2=y1; x1=x4; y1=y4; } } ' | psdraw -l 0.1 -X - - - 10 > spi0.ps

The variable "ee" is the scale factor used to determine how tight the spiral is.

The program psdraw is a utility I wrote to make simple postscript plots. Click here for more information on psdraw.

Since psdraw can handle colors, I could make the following picture:

That was made with this script (using "gawk" to handle trig functions):

gawk ' BEGIN { ee = 0.02; x1=0; y1=0; x2=10; y2=0; x3=5; y3=5*sqrt(3); print x1,y1,0,0,0; # black outline print x2,y2,0,0,0; print x3,y3,0,0,0; print x1,y1,0,0,0; for (t=0;t<100;t++) { x4 = ee*x2 + (1-ee)*x3; y4 = ee*y2 + (1-ee)*y3; print x4,y4,sin(t/4),0,cos(t/4); x3=x2; y3=y2; x2=x1; y2=y1; x1=x4; y1=y4 } } ' | psdraw -l 0.1 -X - - - 10 -C > spi3.ps

Of course, if you write a computer program you can do some more powerful things (such as the picture at the top of the page). I just put in the awk scripts to emphasize how simple this algorithm is. I wrote a C program that fills the triangles, and makes six in a pattern. This program made the picture at the top, and made the following pictures:

Click here to get a copy of this C program. To compile, use "cc -o spiral spiral.c -lm".

NOTE: I just added a page about mathematical spirals (Archimedes' Spiral, etc).

Hey, someone linked to this web page! There is a French web page with lots of math resources that put in a link to this page.

Click here to learn how to convert PostScript pictures to GIF.

Links...

Odds and Ends: These are various tiny ideas that are simple to do which make interesting pictures, or are otherwise interesting (to me). Click one of the pictures below to see others.

Eric's Home Page
Other software
Pictures I've made -- fractals, quasi crystals, cellular automata
My research -- what I get paid for
Graphics Techniques
Questions? Comments? Send me email; email address below.

This page created August 27, 1996.

Current address:
Eric R. Weeks
weeks@physics.emory.edu
Department of Physics
Emory University
Atlanta, GA 30322-2430