Step 1: Materials and Methods
Bike chain - should be reasonably straight, but pretty much any crappy, rusty, bent chain will work. You can combine various chains to make one of any size desired. I suggest cleaning off the grease first.
Chainring - any bike gear will work, but chainrings are bigger and have convenient bolt holes to put your pen in. If you use a cog from the rear gears of a bike, you'll need to affix some cardboard or something to make pen holes.
Pegboard - 1/4 inch pegboard with 1inch hole spacing seems to work perfectly. A bike chain is about 1/4inch thick, so this size works well. You can also use plywood, plexi, or probably even cardboard. Those things don't have conveniently pre-drilled holes, however.
Wire - thin, flexible steel wire works well. Nothing too thick and avoid insulation. You'll probably need at least 5 feet. You could use some sort of glue to put the chain on as an alternative. Hot glue would probably work real nice.
Jigsaw and drill to start the cut.
Step 2: Calculations!
The graphs you can generate will depend on the number of teeth in your chainring and the number of links in your chain. If you think of the graph as a star, the number of points in the star will be the lowest common multiple of the tooth number and link number divided by the tooth number. The number of times you have to trace around is the lowest common multiple divided by the link number.
Examples? of course
The pictures in this instructable are of two chains: 130 links and 84 links. I also have two chainrings: 52 teeth and 48 teeth. That gives the following
130:52 -> 5 points (5x52 = 260 = 130x2)
130:48 -> 65 points (65x48 = 3120 = 130x24)
84:52 -> 21 points (21x52 = 1092 = 84x12)
84:48 -> 7 points (7x48 = 336 = 84x4)
you get the idea
So once you've decided on a chain length, you'll need to cut a circle out of your plywood or pegboard. Your chain will lay along the inner rim of the circle, and so it needs to be precisely cut. The circle size is calculated as follows:
Radius = Chain length / 2 / pi + chain width / 2
Chain length should be measured from pin center to pin center to be accurate.
Chain width is measured at the widest point, with the chain pins perpendicular to your measurement.
You should be able to just calculate the length based on link number, as bike chains are 1/2 inch per link. (Note: you'll need an even number of links, unless you get a special link to connect an odd number of links). Also, the standard width is 5/16 inch. This gives you:
Circle radius = (Number of links) * (1/2 inch) / 2 / pi + (5/32 inch)
Step 3: Cutting
Now you can cut out the center using a jigsaw.
Remember to use old drawers as support when sawing.
Step 4: Affixing the Chain
1. Insert wire through hole.
2. Bring around to the chain.
3. Insert through gap in chain where chainring teeth insert. (do NOT loop around whole chain). Chain links alternate between wide and narrow (inner and outer). If you use wire, only attach to the wide ones. The wire in the narrow ones will catch your chainring teeth as you try to draw the spirograph.
4. Twist wire to secure.
If you secure all of your wire to one side of the chain, it will tend to put a twist it, so I alternately secured to the top or bottom of the chain.
You can probably use glue instead of wire. but that is messier.
Step 5: Making Sweet Art
The procedure is the same as using a normal spirograph. Just put your pen in one of the bolt holes, press out towards the chain, and smoothly trace around until you return to the start.
Combined with a new Sharpie, this should also work to make wall-art, draw on cars, etc.