Odd Angles

For a while, this will be my last post about conformal spiderwebs. Today, we will still look at circular quadrilaterals that are conformal images of squares, but allow the angles to be multiples of 90 degrees. Like so:

Let’s call this a square of type (1,1,3,3). Multiply the numbers by 90 and you get the angles at the vertices. I have again employed Möbius to place three corners at (1,0), (0,1), and (-1,0). The fourth vertex is again moving cautiously along the unit circle. Below is a square of type (1,3,3,3), and here the fourth vertex is on the x-axis, the second possible case we noticed for right angled circular quadrilaterals.

Similarly, here is a square of type (1,1,1,3), also with the fourth vertex on the x-axis.

Missing are squares of type (1,3,1,3). While there are quadrilaterals of this type, all conformally correct squares I could find were only immersed (i.e. overlapping).

Then one can also have squares of type (2,2,2,2), for instance. The circle would be an example, with artificial vertices at (1,0),(0,1),(-1,0) and (0,-1), but there are also bean shaped squares like the one below.

Finally, the square with zero angles, in its most regular form.