For the Museum of Mathematics

By Glen Whitney

Although there have been two different Math Mondays touching just tangentially on the wonderful world of string art, it has never covered the basics of the area, and it’s high time. (Why is it high time? Hint: in preparation for non-basics to come…) The fundamental idea is that you create one or more series of connection points, regular in some way, and then stretch taut strings between the connections (again, typically in some orderly fashion), producing (hopefully!) an interesting, beautiful, and/or mathematically significant overall pattern.

The usual starting point is 2*n* nails driven into a board, at the points (1, 0), (2, 0), (3,0), … , (*n*-2, 0), (*n*-1, 0), (*n*,0) and (0,1), (0,2), … , (0, *n*-1), (0,*n*) in some convenient standard coordinate system, like so: (ignore the embroidery floss for the moment)

Then you run a string from (1,0) to (0,*n*), from (2,0) to (0,*n*-1), from (3, 0) to (0, *n*-2), and so on, up through a string from (*n*, 0) to (0, 1). Note that the *x* coordinate of the start of each segment plus the *y* coordinate of the end of each segment always adds up to *n* + 1. Note you don’t have to tie off each segment separately; the most efficient use of string is to tie off at (1,0), run to (0,*n*), pivot around to (0,*n*-1) and run from there to (2,0), over to (3,0), back to (0,*n*-2), down to (0,*n*-3), back to (4,0), over to (5,0), and so on until the pattern is done. Here’s what it comes out looking like: (It’s a great exercise in intuitive mathematics to guess what curve appears as the envelope of all of the line segments you’ve strung. If you don’t know the answer, try to guess before you scroll down below the picture and read on.)

To kill some space before the reveal of what curve this is, here are a few variations on this basic stringing technique. Of course, any technique for creating many straight lines will do, such as pulling embroidery floss through holes in cardstock, or drawing many lines with a straightedge, or projecting many laser beams, or whatever else you can come up with…

OK, here’s the answer to the math question: the envelope of these lines is a parabola (in fact, that’s true of all of the curves in all of the variations above). It’s also a great exercise in formal mathematics to determine the equation for the parabola that all of the lines in the first example shown are tangent to (hint: it’s not of the form *y = *some second-degree polynomial in *x,** *because that form only gives you parabolas whose axes are vertical lines…), and to prove that they really are tangents.

Of course, it’s possible to get other envelopes of lines if you start with other arrangements of connection points (or for that matter, if you connected the points in a different pattern). For example, with a circle of nails, always stringing a constant number of positions around the circle, you get a star of lines whose envelope is a smaller circle:

Alternatively, if you move one end of the segment a single position around the circle each time but move the other end of the segment *two* positions, you get a cardiod: (Reader exercise: what is a cardiod? What’s its equation? Why does it show up from this process?)

So those are the basics of string art. If you’ve got an example of string art of particular mathematical significance, let mondays@momath.org know!

## 7 thoughts on “Beginning of a String”

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John Honniballsays:This type of string art, or curve-stitching as it’s also known, is a useful source of ideas for pen plotter drawings:

http://www.flickr.com/photos/anachrocomputer/9262709103/

While looking for plotter inspiration, I found a web-site with some detailed information on curve-stitching:

http://www.deimel.org/rec_math/curve_stitch.htm

Alex Cochranesays:Blimey – the maths was way beyond me but love the piece and pictures (even if I didn’t understand it)

MattWsays:The hard and fast definitions are beyond my knowledge, but my understanding is that parabolas do not have asymptotes, while curves like hyperbolas do. That would mean that the curves are not parabolas. I can believe my understanding is incomplete or naive- please elaborate.

Glen Whitneysays:The resolution to your conundrum is simple — if you extend the pattern (i.e., continue to holes of the form (-1,0), (-2,0), … and (0,-1), (0,-2), …, you will see that the axes are not asymptotes for the curve which is the envelope of these lines; in fact, the curve crosses both axes. You are right: a parabola has no asymptotes. These envelope curves formed by the basic string art between two lines are parabolas, and they (perforce) have no asymptotes. The fact that when you only do the envelope line segments with both endpoints having positive coordinates the axes look like asymptotes of the envelope, is just a trick of the eye.

Stepaniesays:It makes an adjustment when you come around such a nice post

really meant for visitors instead of search engines, nice read!

Jeff Williamssays:I’ve just recently gotten into string art and discovered cardioids along the way. I’m a little confused by your description of how to form them though since I’m trying to learn. In my wanderings, I found this website (http://xahlee.info/SpecialPlaneCurves_dir/Cardioid_dir/cardioid.html), which describes the formula a little differently. If the points on the circle, n, are numbered from zero to (max_n-1), then it describes that each point on the circle is joined to the second point defined by modulo(n*2, max_n). Although the second point iterates by 2, your description doesn’t actually follow this rule.

In the first, regular circular string art, the relation between the first and second points is defined by a constant, x, and the artist can choose x to be any value of their choosing as long as x < max_n/2 (not that all of them would look very good). In the second one, there is no choice to be made. For every circle, the above formula describes the relation between the points. So I was just a little confused by your description, and I thought other readers might be too. Hope this helps.

Sosumisays:Yes, I’m four years behind the time but I would like to know where I can obtain the pattern for this. Needless to say, I’m math ignorant but I can make string art if I have the instructions.