Tie dye patterns notes:

A tie-dye pattern is an example of a mathematical function that maps points on one 2-D shape to points on another.

Twisting

If you make a spiral, you take a disk and you twist it up before you dye it, like this:

And then you un-twist it after it's dyed, the colors show you where the points on the big circle twisted to. Here's a picture of un-twisting the spiral and keeping track of some particular points:

notice that the + is always in the red, between the black line and the outside circle,
the * is always in a red section that has the black line on both ends of it
the & is always in a green section that is close to the center
and the @ is always in the blue section where it crosses the black line

When we make a spiral, we don't actually fold the disk over on itself, and so we can imagine that each of the points on the little twisted circle matches up nicely with exactly one point on the big disk, and vice versa.

Folding

Here's an example of a way to fold a disk and then dye it:

If you unfold this one, keeping track of a point, you find that it corresponds to many points in the un-folded disk:

The act of folding means that one point in the folded version corresponds to many in the unfolded version.

Where's the math?

The act of folding or twisting is an example of a function (mapping is another word for function). Mathematically, we say that we have a function (or map) from the big disk to the smaller twisted disk, or from the disk the the small folded shape.

The function from the big disk to the twisted disk can be made (if we have special rubber paper, or if the twists are gentle enough) so that each point on the big disk goes ends up at one point on the little disk and each point of the little twisted disk corresponds to only one point of the big disk. This is a one-to-one function if points match up in pairs.

In the function from the big disk to the small folded shape, each point on the big disk ends up at one point of the little shape, but each point of the little shape corresponds to several (16, in the exampleof the * point) points of the big disk. This function is not one-to-one.

In both cases, the colors tell us something interesting about the function. The red wedge in each of the small shapes corresponds to a larger red shape in the disk. The larger red shape is the pre-image of the red wedge. The colored pictures show graphically where points on the disk go when you do the twisting or the folding function.