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A Curious Inversion

September 28, 2015


The math of “The Curious Incident of the Dog in the Night-Time”

Mark Haddon wrote the book, The Curious Incident of the Dog in the Night-Time, which was Unknownpublished in 2003. It is about an autistic 15 year-old boy, who is a math savant, and who solves a mystery, in spite of his limitations in relating to people.

Today I want to comment on a minor historical inversion at the end of both the book and the current play that is based on Haddon’s book.

I had the great pleasure to see the play recently and found it an amazing experience. The story is told solely from the point of view of an autistic boy, named Christopher Boone. Amazon says:

Christopher John Francis Boone knows all the countries of the world and their capitals and every prime number up to 7,057. He relates well to animals but has no understanding of human emotions. He cannot stand to be touched. And he detests the color yellow.

I re-read the book days before seeing the play, and was unable to even imagine how the play could capture the feel of the book. But they did it. A New York Times review says:

Such a state of being is conjured with dazzling effectiveness in “The Curious Incident of the Dog in the Night-Time,” which opened on Sunday night at the Ethel Barrymore Theater. Adapted by Simon Stephens from Mark Haddon’s best-selling 2003 novel about an autistic boy’s coming-of-age, this is one of the most fully immersive works ever to wallop Broadway.

It was definitely a wallop. Both the book and the play end with a nice geometric problem. In both the answer, which is a proof, is left out of the main part. It is detailed in the book in an appendix; in the play it is delivered by Christopher after all the curtain calls. An “appendix” to a play—what a clever idea.

The Problem

So let’s start by stating the geometric problem from both the book and play.

Prove the following: A triangle with sides that can be written in the form {n^{2}+1}, {n^{2}-1}, and {2n}, where {n>1}, is a right triangle.

The Proof

The proof starts by showing that {n^{2}+1} is the longest side; this uses {n>1}. Then it proves that

\displaystyle  (2n)^{2} + (n^{2}-1)^{2} = (n^{2} + 1)^{2}.

It then states that by the Pythagorean Theorem the triangle is a right one.

But this is inverted.

Some History

The famous Pythagorean theorem states:

Theorem: Let the sides of a right triangle be {a,b,c} with {c} the largest. Then {a^{2} + b^{2} = c^{2}}.

The proof of the problem from the book uses not this theorem—this is the inversion. Rather it uses the converse: For any triangle with sides {a, b, c}, if {a^{2} + b^{2} = c^{2}}, then it is a right triangle.

Happily this converse of the Pythagorean theorem is also a theorem. Indeed Euclid already had proved it. I must admit that I was not sure it was a theorem.

At the play I heard the problem for the first time, since when I read the book I skipped the appendix. As Christopher proved the theorem for the audience, I was almost ready to raise my hand—as if it were a seminar talk—and ask: isn’t there a potential issue with the proof, since it relies on the converse not the actual Pythagorean Theorem? Then I realized this wasn’t a lecture hall, and left the theater quietly.

A Curiosity?

The proof of the converse is not hard, but it is definitely a different theorem. What’s curious, however, is that its proof uses the original Pythagrean theorem. Here is Euclid’s proof as relayed by Wikipedia from this source:

Let {ABC} be a triangle with side lengths {a}, {b}, and {c}, with {a^{2} + b^{2} = c^{2}}. Construct a second triangle with sides of length {a} and {b} containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length {c = \sqrt{a^{2} + b^{2}}}, the same as the hypotenuse of the first triangle. Since both triangles’ sides are the same lengths {a}, {b} and {c}, the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths {a} and {b} in the original triangle is a right angle.

So here we have a proof of the ({\Longleftarrow}) direction of an equivalence whose proof uses the ({\Longrightarrow}) direction. How common is that?

Open Problems

Did you know that the Pythagorean Theorem was an “if and only if theorem?” I did not. Are there other notable cases of equivalences where the proof from the “Book” of the converse direction uses the forward direction?

12 Comments leave one →
  1. September 28, 2015 9:13 am

    Hi,

    There is a proof of the converse without using Pythagoras Theorem itself. My classmate in college did it.

    Casey, Stephen; Mathematical Gazette; 2008; 92 (524); 309-312; 00255572

    https://jpmccarthymaths.files.wordpress.com/2013/10/pythagoras-converse0001.pdf

    Steve was an excellent student: top of our class. Sadly he was lost to physics for his doctorate and subsequently finance.

    Regards,
    J.P.

  2. MadHatter permalink
    September 28, 2015 9:45 am

    Feels good when you know something that Lipton doesn’t🙂

  3. Serge permalink
    September 28, 2015 10:47 am

    In architecture, the “only if” part of the Pythagorean Theorem is among the few reliable methods available for building right angles.

    • Alex Lopez-Ortiz permalink
      September 28, 2015 11:51 am

      Exactly, I learned the inverse when I was 10 years old when a bricklayer apprentice back home needed to create a right angle and proceeded to construct a 3-4-5 triangle with strings, nail and boards.

  4. Daniel permalink
    September 28, 2015 3:33 pm

    Finally I was able to understand thoroughly an article from this blog.

  5. September 28, 2015 5:11 pm

    The Pythagorean Theorem is generalized by the Law of Cosines, which makes it clear it’s a biconditional.

    • Javaid Aslam permalink
      October 3, 2015 12:55 am

      Isn’t this true that the basic Trigonometry (the definition of Sine & Cosine, Tangent, etc) is founded on the Pythagorean theorem by taking a right angled triangle?

  6. September 29, 2015 2:28 pm

    A right angle can also be constructed by inscribing a triangle in a semicircle. Also, what I really like about Euclid’s first proof of the PT are that it only uses the fact that the area of a triangle is half the area of a rectangle with same base and altitude and that it shows how to divide the square on the hypotenuse into two rectangles with areas equal to the squares on the other two sides.

  7. October 2, 2015 9:17 pm

    The converse of the Pythagorean Theorem is pretty standard college algebra material. For example, see Michael Sullivan, College Algebra, end of section R.3.

    • October 3, 2015 10:44 am

      And now that I think of it: People working in theater would be particularly knowledgeable of this. My partner would use the converse Pythagorean principle to square stages back when she worked in theatrical set design.

      • October 3, 2015 10:49 am

        Partner comment just now: “It’s probably the only piece of math that most theater people know!”

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