Parabolas from String Art (Part 6)

Recently, I announced that my paper Parabolic Properties from Pieces of String had been published in the magazine Math Horizons. The article had multiple aims; in chronological order of when I first started thinking about them:

  • Prove that string art from two line segments traces a parabola.
  • Prove that a quadratic polynomial satisfies the focus-directrix property of a parabola, which is the reverse of the usual logic when students learn conic sections.
  • Prove the reflective property of parabolas.
  • Accomplish all of the above without using calculus.

While I’m generally pleased with the final form of the article, the necessity of publication constraints somewhat abbreviated the original goal of this project: determining a pedagogically sound way of convincing a bright Algebra I student that string art unexpectedly produces a parabola. While all the necessary mathematics is in the article, I think the article is somewhat lacking on how to sell the idea to students. So, in this series of posts, I’d like to expand on the article with some pedagogical thoughts about connecting string art to parabolas.

As discussed previous posts, we begin our explorations with string art connecting evenly spaced points on line segments \overline{AB} and \overline{BC} with endpoints A(0,8), B(8,0), and C(16,8). We will call these colored line segments “strings.” We then found the string with the largest y-coordinate at x = 2, 4, 6, \dots, 14, resulting in the following picture:

In previous posts, we discussed three different ways of establishing that the colored points lie on the parabola y = \displaystyle \frac{x^2}{16} - x + 8.

Unfortunately, checking that a statement is true for a few points (in our case, x= 0, 2, 4, \dots, 14, 16) does not constitute a complete proof for all points. Furthermore, it’s conceivable that “fuller” string art with additional strings, like the picture below, may identify a new string with a higher y-coordinate than a colored point.

To prove that the string art indeed traces a parabola, we study an arbitrary string \overline{PQ} depicted below. For brevity, this string will be called “string s,” matching the (possibly non-integer) x-coordinate of its left endpoint P. Since P is s units to the right of A, the right endpoint Q must correspondingly be s units to the right of B. Therefore, the x-coordinate of Q is s + 8.

Since the equations of \overline{AB} and \overline{BC} are y=-x+8 and y=x-8, respectively, the y-coordinates of P and Q are -s+8 and (s+8)-8 = s, respectively. For example, if s = 5, the coordinates of P are (s,8-s)=(5,3) and the coordinates of Q are (s + 8, s) = (13, 5), matching the endpoints of the blue string in the first figure.
We now use standard algebraic techniques to find the equation of string s. Its slope is

m = \displaystyle \frac{ s - (8-s)}{(s+8)-s} = \frac{2s-8}{8} = \frac{s-4}{4}.

The coordinates of either P or Q can now be used to find the equation of string s via the point-slope formula. As it turns out, the coordinates of P are simpler to use:

y-y_1 = m(x-x_1)

y-(8-s) = \displaystyle \frac{s-4}{4}(x-s)

y = \displaystyle \frac{(s-4)(x-s)}{4} + (8-s)

y = \displaystyle \frac{xs-s^2-4x+4s}{4} + 8-s

y = \displaystyle \frac{xs}{4} - \frac{s^2}{4} - x + s + 8 - s

to finally arrive at the equation of string s:

y = -\displaystyle \frac{s^2}{4} + \frac{xs}{4} - x + 8

This has the appearance of a quadratic equation, but it’s actually a linear equation in x for a fixed value of s. For example, if s = 5, we find that the equation of string 5 is

y = -\displaystyle \frac{25}{4} + \frac{5x}{4} - x +8 = 0.25x+1.75,

matching the equation of the blue string we found in a previous post in this series.

We are now almost in position to prove that the string art traces a parabola. We demonstrate this in the next post.

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