Here’s a tweet from @colinthemathmo:

I’m not big on origami, but if Colin thinks it’s an interesting puzzle…

(I’m going to make the unstated assumption that the bottom-left corner of the square is at (0,0) and that the side-length is 1.)

There are, as always, several ways to tackle it.

Circular geometry

The point we’re looking for is one unit away from the origin, and half a unit away from $\br{1, \frac{1}{2}}$.

I can set up equations of two circles:

$x^2 + y^2 = 1$ $(x-1)^2 + \br{y-\frac{1}{2}}^2 = \frac{1}{4}$

… and solve.

Multiplying out the second gives $x^2 - 2x +1 + y^2 - y = 0$

Substituting in the first gives $y + 2x = 2$ (this is, it turns out, the line perpendicular to the fold passing through (1,0)).

And substituting that back into the first gives $x^2 + (2-2x)^2 = 1$, or $5x^2 - 8x + 3 = 0$

That factorises as $(5x-3)(x-1)=0$, giving $x=1$ or $x=\frac{3}{5}$; we already knew about the first of those, so $\br{\frac{3}{5}, \frac{4}{5}}$ is the point in question.

Oo! It’s a 3-4-5 triangle!


The angle between the $x$-axis and the fold is $\theta = \arctan\br{\frac{1}{2}}$.

The angle between the $x$-axis and the edge of the paper is $2\theta$, and we have $\tan(2\theta) = \frac{ 2\tan(\theta) }{1 - \tan^2(\theta)} = \frac{1}{1 - \frac{1}{4}} = \frac{4}{3}$.

We know the point is one unit from the origin, and that the angle between the $x$-axis and the edge is the same as that of a 3-4-5 triangle, so again we get $\br{\frac{3}{5}, \frac{4}{5}}$.

Complex numbers

Think of the paper as an Argand diagram; the far end of the fold is at $z = 1+\frac{1}{2}i$. The fold doubles the argument, which means that $z^2$ would lie on the (extended) fold: $z^2 = \frac{3}{4} + i$.

However, the point we want is only one unit from the origin, so we need the unit complex number associated with $z^2$ - which is $\frac{4}{5}z^2 = \frac{3}{5} + \frac{4}{5}i$, as before.

Other methods?

I’m sure there are dozens of other methods of solving this - and probably one “from the book” that I’ve missed. How would you go about it?