The Waffle Cone and a new Proof of Pythagoras’ Theorem

Jackson an’ Johnson / Murphy an’ Bronson /
One by one dey come / An’ one by one to dreamland dey go.
[From Carmen Jones.  Lyrics: Oscar Hammerstein]

Two young high-school students from New Orleans, Ne’Kiya Jackson and Calcea Johnson, recently presented a new proof of the Pythagorean theorem at a meeting of the American Mathematical Society in Georgia. It has been widely believed that no proof based on trigonometry was possible but we now know that to be false.

Pythagoras’ Theorem

The Theorem of Pythagoras has been described as the most important theorem in the whole of mathematics. It has a history going back thousands of years. It was known to the Babylonians although a proof came much later. We have no evidence that Pythagoras proved it, although tradition holds to this view. The theorem is the culmination of Book I of Euclid’s Elements, appearing as Proposition 47.

There are, of course, many proofs of the Pythagorean theorem. Children learn the theorem, often written in algebraic form as a² + b² = c² , in their geometry classes. The usual proof required the construction of several lines in non-intuitive ways. The new proof is essentially trigonometric but, of course, it avoids using the Pythagorean identity, sin² x + cos² x = 1, which would have made it circular.

Proof must not use the Pythagorean Identity.

The new proof uses the Law of Sines, a clever construction of what Johnson and Jackson call a “waffle cone”, an infinite set of similar triangles reducing in size, and the formula for the sum of a geometric series. A link to their abstract is given below, and a link to a YouTube video explaining the proof in detail can also be found there.

A classical book by Elisha Loomis, The Pythagorean Proposition, contains some 370 different proofs of Pythagoras’ Theorem. The book has a section entitled “Why No Trigonometric, Analytic Geometry Nor Calculus Proof Possible”, but this was published in 1940 and was shown to be wrong when, in 2009, Jason Zimba published a proof using trigonometry. He relied only on the formulas for the sine and cosine of the difference of two angles. The proof is on the website Cut the Knot (link below).

Zimba’s trigonometric proof is much simpler than that of Jackson & Johnson. However, given the ages of the two girls, and their relatively elementary knowledge of mathematics, their achievement is quite remarkable.

Sources

• Calcea Johnson and Ne’Kiya Jackson, 2023: An Impossible Proof Of Pythagoras. Abstract.
• Polymathematic, 2023: Pythagoras Would Be Proud: High School Students’ New Proof of the Pythagorean Theorem.  YouTube video.
• Jason Zimba’s proof of Pythagoras using trigonometry on Cut the Knot.
• Maor, Eli, 2007: The Pythagorean Theorem. Princeton Univ. Press. 259pp. ISBN: 978-0-6911-4823-6.