Parabolas from String Art (Part 1)

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.

To begin, we use graph paper to sketch to draw coordinate axes, the point A(0,8), the point B(8,0), the point C(16,8), line segment \overline{AB}, and line segment \overline{BC}.

Along \overline{AB}, we mark the evenly spaced points (1,7), (2,6), (3,5), (4,4), (5,3), (6,2), and (7,1).

Along \overline{BC}, we mark the evenly spaced points (9,1), (10,2), (11,3), (12,4), (13,5), (14,6), and (15,7).

Next, we draw line segments of different colors to connect:

  • (1,7) and (9,1)
  • (2,6) and (10,2)
  • (3,5) and (11,3)
  • (4,4) and (12,4)
  • (5,3) and (13,5)
  • (6,2) and (14,6)
  • (7,1) and (15,7)

The result should look something like the picture below:

It looks like the string art is tracing a parabola. In this series of posts, I’ll discuss one way that talented algebra students can convince themselves that the curve is indeed a parabola.

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