It turns out that you can solve for the real roots of cubic functions with origami!



While I should have been doing something more productive the other day, I discovered that thanks to a technique created by Margherita Beloch in 1936, which when used in conjunction with a technique created by Eduard Lill in 1867, it is possible to solve for the real roots of arbitrary cubic equations that have real coefficients by simply folding paper!

The techniques used include Lill's method and the construction of a Beloch square. Lill's method potentially allows a user to solve for all real and complex roots (when you use the complex extension detailed in French here) of arbitrary polynomials with real coefficients in a purely visual way. However, it requires a user to be able to find specific angles to line up various reflections, which can be difficult. This is where Margherita Beloch realised that her Beloch square was the solution to finding the angles in the cubic case, and thus the real roots of arbitrary cubics with real coefficients can be solved for by simply performing what is called the Beloch fold.

I learnt this technique and the theory from the following article by Thomas Hull: article. You do need access to the journal to see the article, however most of it is also detailed in lecture slides created by Hull that you can view for free.

Why did I do this?

I thought it was just so cool that it was possible to solve for the real roots of arbitrary cubics with real coefficients by just folding paper. No calculators, no compass, no protractor, just a ruler and some grid paper and you're away!

I hope that many people will try to learn this technique even if it's just for the fun of the whole thing.

Extra Notes

This was undertaken in late 2015.

I learnt that:

-It is hard to draw perfectly perpendicular lines

-Paper can have a mind of its own when you're trying to fold it in careful ways

-There's some really cool maths out there that seems largely unknown

Supplementary Video

I received some questions about this technique via email, so here's an extra video to hopefully answer some of them.