A Curious Inversion
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 published 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.
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 , , and , where , is a right triangle.
The proof starts by showing that is the longest side; this uses . Then it proves that
It then states that by the Pythagorean Theorem the triangle is a right one.
But this is inverted.
The famous Pythagorean theorem states:
Theorem: Let the sides of a right triangle be with the largest. Then .
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 , if , 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.
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 be a triangle with side lengths , , and , with . Construct a second triangle with sides of length and containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length , the same as the hypotenuse of the first triangle. Since both triangles’ sides are the same lengths , and , the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths and in the original triangle is a right angle.
So here we have a proof of the () direction of an equivalence whose proof uses the () direction. How common is that?
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?