Some of you may have heard about the unsolved geometry problem called "Inscribed Square Problem" or "Toeplitz' conjecture". It states that in any closed curve, there exists 4 points on the curve that form a square. It sounds like a really simple problem, yet mathematicians have failed to prove it.
In this note, we will attack a similar problem: finding three points on a curve that are the vertices of an equilateral triangle. Unlike the square problem, this one is solvable.
First try it on your own. Can you devise a fool-proof way to find three points on a curve that make an equilateral triangle? Don't read on until you give it a try.
The key insight we need to think of in order to find an equilateral triangle inscribed in this curve is rotations. Here's how:
First, we take our curve. Although I will use a polygon in my examples, the idea is the same.
Now we pick any point on our polygon. It doesn't matter which; we can always find an equilateral triangle inscribed in the curve with this point as a vertex.
Using this point as the center of revolution, turn the polygon . It doesn't matter which direction.
Now we can see that we have some points of intersection. Mark one of these points of intersection. Then, rotate it back in the opposite direction of how you rotated the polygon.
We can get rid of our rotated polygon now, and connect these two marked points with our point of rotation:
And we're done! We have find an equilateral triangle inscribed in the curve.
In fact, this is only one of the many inscribed equilateral triangles in this curve. We could have picked any of the intersections and created a whole new equilateral triangle.
In addition, we could have picked any point on the entire curve to begin with, and found equilateral triangles. Thus, there is an infinite number of equilateral triangles that can be inscribed in a random closed curve.
Now that we've found a way to inscribe equilateral triangles, can we do the same thing for other triangles? Yes, we can! The method of inscribing any arbitrary isosceles triangle is similar to the equilateral triangle; just change the number of degrees you rotate the curve by. For example, rotating the curve by will give you triangles inscribed in the curve. As an example, here is one inscribed in the same curve we used:
This is cool and all, but here's the finale: we can actually inscribe any arbitrary triangle in any arbitrary closed curve! However, you'll have to wait until the next note in order to find out how. Try to figure it out yourself in the meantime.