High School Students Publish Impossible Proofs of Pythagoras Theorem

What started as a bonus question in a high school math competition has led to an astonishing 10 new proofs of the ancient Pythagorean theorem.

It has long been considered impossible to use trigonometry to demonstrate a theorem that is foundational to trigonometric principles, as this creates a logical fallacy of circular reasoning by attempting to prove a concept using the concept itself.

“There are no trigonometric proofs because all the fundamental formulas of trigonometry are based on the validity of the Pythagorean theorem,” mathematician Elisha Loomis stated in 1927.

What began as a bonus question in a high school math contest has resulted in an impressive 10 new proofs of the ancient Pythagorean theorem.

The Circular Reasoning Dilemma in Trigonometric Proofs

However, For a long time, it has been deemed impossible to use trigonometry to prove a theorem that underpins trigonometric concepts, as this leads to a logical fallacy of circular reasoning by trying to validate a concept using itself.

“There are no trigonometric proofs because all the fundamental formulas of trigonometry rely on the truth of the Pythagorean theorem,” mathematician Elisha Loomis wrote in 1927.

Pythagoras’ Theorem

Pythagoras’ theorem explains the relationship between the three sides of a right-angled triangle. This theorem is highly valuable in engineering and construction and was utilized by people centuries before it was formally associated with Pythagoras. Some argue that it may have even been applied in the construction of Stonehenge.

The theorem is a fundamental principle in trigonometry, which primarily focuses on calculating the relationships between the sides and angles of triangles. You probably remember being taught the equation (a^2 + b^2 = c^2) during your school years.

“Students might not be aware that two conflicting versions of trigonometry share the same terminology,” Jackson and Johnson explain.

“In such a situation, understanding trigonometry can feel like attempting to decipher a picture where two distinct images are superimposed.”

However, by clarifying these two related yet distinct variations, Jackson and Johnson devised new solutions using the Law of Sines, effectively avoiding direct circular reasoning.

However, Jackson and Johnson detail this method in their new paper, acknowledging that the distinction between trigonometric and non-trigonometric approaches is somewhat subjective.

New Trigonometric Proofs

They also highlight that, according to their definition, two other mathematicians, J. Zimba and N. Luzia, have also proven the theorem using trigonometry, challenging previous claims that this was impossible.

In fact, In one of their proofs, the two students pushed the definition of calculations involving triangles to the limit by filling a larger triangle with sequences of smaller triangles and employing calculus to determine the measurements of the original triangle’s sides.

“It looks like nothing I’ve ever encountered,” said Álvaro Lozano-Robledo, a mathematician at the University of Connecticut, in an interview with Nikk Ogasa at Science News.

In total, Jackson and Johnson present one proof for right triangles with two equal sides and four additional proofs for right triangles with unequal sides, leaving at least five more for “the curious reader to explore.”

“Having a paper published at such a young age is truly astonishing,” remarks Johnson, who is currently pursuing environmental engineering. Jackson is studying pharmacy.

To conclude, “Their findings highlight the potential of new perspectives from students in the field,” says Della Dumbaugh, editor-in-chief of the journal where their work is published.

Read the original article on: Science Alert

Read more: Quest for Fractionalization in Condensed Matter Physics