Parabolas from String Art (Part 7)

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.

Our explorations of string art led us to consider 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.

In the previous post, we established that the equation for string s is

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.

To prove that the strings trace a parabola, we now determine which string s maximizes the value of y = -\displaystyle \frac{1}{4}s^2 + \frac{x}{4}s - x + 8 for a given value of x. Algebra students can determine this maximum by recalling that a quadratic function y = as^2 + bs + c is maximized (for negative a) when s = \displaystyle -\frac{b}{2a}. Therefore, the string with largest y-coordinate for a given value of x is

s = \displaystyle -\frac{x/4}{2 \cdot (-1/4)} = \frac{x}{2}.

For example, if x = 4, then string s = 4 / 2 = 2 has the largest y-coordinate, matching our previous observations.
To complete the proof that the strings above trace a parabola, we substitute s = \displaystyle \frac{x}{2} into y = -\displaystyle \frac{s^2}{4} + \frac{xs}{4} - x + 8 to find the value of this largest y-coordinate:

y = -\displaystyle \frac{(x/2)^2}{4} + \frac{x(x/2)}{4} - x + 8

= -\displaystyle \frac{x^2}{16} + \frac{x^2}{8} - x + 8

= \displaystyle \frac{x^2}{16} - x + 8,

matching the result that we found earlier in this series.

There’s also a bonus result. We further note that, for every x, there is only one string s = \frac{x}{2} that intersects the parabola y = \displaystyle \frac{x^2}{16} - x + 8. Since each x is associated with a unique string s and vice versa, we conclude that each string intersects the parabola at exactly one point. In other words, string s is tangent to the parabola y = \displaystyle \frac{x^2}{16} - x + 8 when x=2s.

We note that all of the above calculations were entirely elementary, in the sense that calculus was not used and that only techniques from algebra were employed. That said, the word “elementary” in mathematics can be a bit loaded — this means that it is based on simple ideas that are perhaps used in a profound and surprising way. Perhaps my favorite quote along these lines was this understated gem from the book Three Pearls of Number Theory after the conclusion of a very complicated multi-page proof in Chapter 1:

You see how complicated an entirely elementary construction can sometimes be. And yet this is not an extreme case; in the next chapter you will encounter just as elementary a construction which is considerably more complicated.

In the next post, we take a second look at this derivation using techniques from calculus.

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