Place a cup lip-down on a piece of cardboard. Trace around the lip of the cup with a pencil onto the cardboard. Repeat to make two traced disks.
Cut the disks out with a razor.
Hold both pieces of cardboard together. Make a pencil mark on each piece, near the outer rim of the cardboard disk. Make two more marks directly across from the first ones, so that if a line were drawn between the marks it would evenly cut the disk in half. Using a ruler and pencil, draw that line.
Make two pencil marks near the rim on both disks that are halfway between the first two marks. There should be four pencil marks near the rim on each piece, arranged in such a way that if straight lines were drawn connecting them, they would form an equal armed cross cutting the circular disks into quarters. Using the ruler and pencil, make these lines.
Make pencil marks near the rim, halfway between the existing lines, on each disk. Using the ruler and the pencil, draw lines connecting the opposite pairs of these marks so that eight equal-sided lines now converge in the center of the cardboard disks.
Wrap copper wire around the pencil, so that it forms a spiral approximately 4 inches long. It should be loose enough that it can be pulled off of the pencil once wrapped. Pull the wire back off the pencil; you should have a spring-shaped piece of wire.
Hold the pencil tip to the central point at which the eight lines intersect in the middle of one of the pieces of cardboard. Push the pencil through the cardboard until the cardboard rests, impaled, near the eraser.
Wrap several layers of cellophane tape around the pencil, abutting against the cardboard disk, on both sides of the disk. This is to keep the disk from moving.
Push the coiled copper wire back onto the pencil until the end hits up against the cardboard.
Push the pencil through the center of the other cardboard disk. Push the cardboard down the pencil until it pushes up against the coiled copper spring.
Wrap several layers of cellophane tape around the side of the second cardboard disk facing the pencil tip.
Thread string through the eye of a needle.
Push the needle through any of the holes on either disk. Pull the string through with the needle. Pull the needle and string all the way to the pencil mark in the same position on the other piece of cardboard. The string should make a straight, taut line between the two cardboard disks.
Pull the string out of the eye of the needle. Tie the string to itself, where it enters the cardboard, so that it can be pulled tight from the other end.
Pull no less than 4 feet of string off of the spool. Cut the string with the scissors. Thread the end of the cut string through the eye of the needle.
Push the needle and string through the pencil mark next to the one that the string was first pushed through. Pull all the string through.
Push the needle into the next disk, through the mark directly across from where the string is coming from. Pull the string tight so that there are now two perfectly parallel string lines connecting the cardboard disks.
Repeat Steps 16 and 17 all around the disks, until there are eight parallel lines of string connecting the two disks.
Tie the end of the string to the string adjacent to it, so the string will stay tight and not come undone. Cut the excess off of the end.
Grasp the two disks and hold the pencil horizontal. The parallel strings should resemble a bird cage. Slowly twist the disk closest to the pencil tip. As the strings tighten, the disk will naturally be pushed against the copper spring; let this happen so that the disks get closer as they are twisted. Observe the curved space created by the strings as the disks are twisted. It should appear as if the strings are getting cinched by an invisible belt about their center. Those curved spaces on either side are the hyperbola. Twist the disks back and forth to create hyperbolas with different curves.