String Art Calculus

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String Art

String art is a graphic art form with its roots in Mary Everest Boole's "curve-stitching." It became popular as a mode of visual expression in the 1970's, when artists began to use it to create increasingly complex figures. The basis of all string art, though, is one of the main ideas in calculus: the use of straight lines to represent curves.


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Basic Description

I was sitting in my seventh-grade English class, learning to spell "beneficent," when my friend leaned over to me and whispered, "Hey, check this out. I can make a circle out of straight lines."


I looked over at her desk and watched her draw something very much like the animation you see on the left. It was delightful – something new to doodle in class. I spent much of the rest of the year idly making circles out of straight lines, finding out how many lines it took for the circle to appear, how to space them and what angles to use. I still occasionally find myself making these line-circles in the margins of my notes.


It took several years and some math classes for me to realize that, back in seventh-grade English, my friend and I had been doing calculus. Mathematicians tend to approach the idea of the derivative by starting with a known curve and looking for its tangent lines. This is an important facet of calculus, but it often makes us forget that we can approach derivatives in another way; we can start with a set of tangent lines, and look for the curve they create.


Consider the animation on the right. Given the derivative of a curve, in this case a parabola, and a few points to set its position, we can find the original curve by simply drawing lines – no integration or complicated mathematics needed.


We chose a point near the middle of the range we wish to examine, and draw the tangent line for that point. Then we choose two more points, on on either side of the first, and draw the tangent lines for the new points. We pick a pair of surrounding points again for each of the new points, an repeat the process. As we continue to add lines, the curve to which they are tangent begins to appear. On the right, we see that the more lines we have, the more clearly we see the curve. We have used calculus to make a curve out of straight lines.


This is the process used to create string art. Straight segments of string are stretched at many angles across a flat surface to create curves. Any curve in a piece of string art has tangent lines equal to the lines created by the string. With patience, string art can be used to create almost any two-dimensional shape, including all differentiable functions.


Consider the sine wave that appears in the string art to the left. Even though strings that are tangent to sine at one point often cross sine at another point, the curve still comes out clearly in the intersections of the strings.


You can explore this more by experimenting with some string of your own. Using pins to fasten the ends of the strings to a board, see how different angles and locations of strings create different curves. How many strings are needed for a curve to appear? How many strings can cross the curve before it ceases to look like a curve? If you're still not convinced that playing with string is a way of studying math, you can add axes to your board, identify a handful of points on the curve, and use them to find the equation of the curve you have created.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Calculus

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