Keio University researchers have solved the problem of a pair of unique triangles with the same perimeter and area. Laymen remain confused!

You might be surprised to know that math is one of the most popular subjects in Japan, but between the musical capabilities of their math tools and sympathetic, creatively minded teachers handling the subject many avid Japanese learners dedicate their academic lives to the complex world of numbers.

There are problems out there in the adult world of math that, while relatively easy to understand, have yet to be ‘proved’ by a theorem. One example is the concept of a ‘unique’ triangle, which is to say a triangle without an equivalent. While on paper this sounds relatively easy to prove (of course any given different triangle you draw is unique, come on!) mathematic rigor requires more evidence than just common sense, so there’s an equation to prove it.

▼ I defer to this YouTube math teacher on all subjects. Or at least this subject.

Third-year graduate student Yoshiyuki Hirakawa and second-year doctorate student Hideki Matsumura, both members of the Keio Institute of Pure and Applied Sciences, have successfully proven an old theory in their thesis, titled A Unique Pair of Triangles, specifically, that there exists a unique pair of rational triangles, a right-angle triangle and an isosceles triangle, which have the same perimeter and same area. This proves by contradiction the aforementioned old theory: “There exists no pair of a primitive right triangle and a primitive isosceles triangle which have the same perimeter and the same area.”

The study of geometric shapes has been of great interest to mathematicians since the time of Ancient Greece, and being able to use proven methods to evaluate area and perimeter is a great asset across fields as varied as sculpture, engineering and architecture. Greek thinkers puzzled over the seemingly insurmountable problem of determining the existence of every single right angle triangle with rational measurements, but now they can at last lie easy in the afterlife, knowing their theories have been proven!

Admittedly, modern day thinkers are impressed by Hirakawa and Matsumura’s feat, but several don’t really understand all the ins-and-out of the explanations provided by their thesis. When reading through the relatively straightforward explanations of the unique pair of triangles, it’s not surprising those unfamiliar with concepts like “Coleman’s theory of p-adic abelian integrals” and “the 2-descent argument on the Jacobian variety of a hyperelliptic curve” couldn’t quite understand why these proved anything, much less why only these triangles made a pair with the same area and perimeter.

“I… don’t get it. At all. Can you put the question in Dragon Quest terms?”
“Amazing! People who can do math are incredible!”
“I haven’t been this emotional since hackers worked out the Arseus void glitch in Pokémon.”

When one user asked “Okay, but is this going to help anybody in the real world?” one user replied in jest “It’s gonna help these guys graduate with honors for sure”. Another commenter took it very seriously, however.

“You know, if someone can even apply one out of of the hundreds of things we learn through mathematics in the real world, that is enough for us. We don’t tend to think in terms of “This is useful, so it’s good” and “This is pointless, so it’s bad”. We study mathematics hoping that it could, potentially, be of use to someone.”

▼ It may not be the real world, but presumably someone applied real math to manage this.

Congratulations to Hirakawa and Matsumura, and here’s hoping future math majors can use their work to reach even greater heights!

Source: Science Direct, University Journal Online via Kinisoku
Featured image: Pakutaso