Not Being Square

I meant to post today a sequel to the circular triangles from last week, but I got carried away looking at right angled quadrilaterals bounded by circular arcs. Like the pillows, but more general. Like so:

Circulon 2

The question arises for what choices of four points we can find a right angled quadrilaterals bounded by circular arcs?

By the way, how do we call these? I thought about circulons (taken) and horny squares (oops). For now, I call them circulions (like centurions), to avoid a lawsuit about trade marks. Above you see a solution that is not a square but where the vertices are at the corners of a square. There are more like these, in fact a 1-parameter family.

Circulon 1

Below you can see the entire family at once, you just have to follow all dots with the same color.

Squaregradient

Can we do that for any choice of four points? Not so, but: Möbius allows us to move three points anywhere we like (and he will send circulions to circulions), so we can ask: where are we allowed to place a fourth point so that there is a circulion through all of them?

Möbius also tells us that this is easy if we place all four points on a circle (by sending that circle to a line, and then connecting the four points on the line alternatingly by segments of the real line and half circles, for instance). Here is an example where the first three points are at the corners of an equilateral triangle, and the fourth point is on the circle through them.

Circulon 3

Again, there is a 1-parameter family of such circulions through these points.

Trianglegradient

Pretty, isn’t it?

Now, surprisingly to me, for each choice of three points, there is a second circle on which the fourth point can reside: Take the circle that contains the given three points, and construct the circle orthogonal through it that passes through the two points between which we want to put the fourth vertex. You can put the fourth point anywhere on that new circle. Here is an example, with the first three points again at the corners of an equilateral triangle.

Circulon 4

Below is again an entire family, color coded and adorned with moiré.

Trianglegradientb

Next week you’ll see conformally correct squares. Promised.

Advertisements