What shape are the golden arches?

Every day for lunch I eat salad (made with vegetables from our local farmers’ market or from our college’s organic farm) and homemade yogurt and granola. The only time I ever eat fast food is on long car trips. So why, I ask you, did the question “What shape are the golden arches?” pop into my head?

I have no idea. But once it did, I just had to investigate. A quick internet search was inconclusive. Commenters on discussion forums assert that they are a pair of parabolas or a pair of catenary curves. But the credibility of the sources is questionable. So I thought I’d see what I could determine using Geogebra.

It turns out that the arches are definitely not parabolas (I didn’t think they were). The catenary is a good fit, but it still isn’t quite perfect. The best fit is an ellipse (or part of an ellipse)! Check out the applet that I made, and see for yourself.

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7 Comments

  1. Ed Bradshaw says:

    My geometry is a bit rusty but I’m thinking a single arch is the arc of a circle viewed in perspective? The other arch is a mirror of the first.

    1. Good thought—it could be different conic section—not a parabola, an ellipse.

    2. I just updated the applet to include an ellipse. Best fit yet! Thanks.

  2. Theo says:

    Ut pendet continuum flexile, sic stabit contiguum rigidum inversum.

    Thus: a catenary would be appropriate.

    1. Excellent! Hook’s 1675 anagram: “As hangs a flexible cable so, inverted, stand the touching pieces of an arch.”

      I thought about the fact that they are called the golden *arches*. Nice quote.

  3. nlw0 says:

    Have you tried fitting a cubic (Bézier curve…)?

  4. I feel quite proud of the fact that when I saw this post’s title, my immediate reaction was “the golden arches – what are they?”

    Your applet convinces me that “Ellipse” is the correct answer to within an epsilon of artistic licence.

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