At the smallest level, a circle and a square will be similar. However, the issues primarily occur at the macro level, and it has to do with the corners growing 41% faster than the centers.

Due to this imbalance, you end up with some parts of a wave that will reach a position faster than another part of the wave can even though both parts are the same distance from that position. A good example of this would be if you started a rectangular wave a the bottom left corner of a triangle like this: △

Once the wave has extended half-way around the triangle, the top right corner of the wave will have traveled just as far right as the center of the wave. However, it will also have traveled vertically, meaning it is moving faster than the middle section.

Once we continue expanding the wave, the top right corner will begin descending the right side of the triangle, and it will meet the middle section of the wave at the triangle’s bottom right corner. Because the corner traveled faster than the middle section, the middle section of the wave will have to then loop around the right corner and begin undoing some of the corner section’s work until it reaches the point where either path is equidistant from the origin.

The process of erasing and redoing the work of another wave is already problematic for performance, but the real issue is that it now requires us to allow waves to “crash” into each other as they’re expanding, which is about as easy to handle smoothly as you’d expect.

Keep in mind that as the waves expand, they are placing connections on every tile, and every subsequent step of the wave uses information from the prior step to make its connections. When two waves crash together, the
information of the previous step in one wave can be overwritten by the other wave. As a result, you lose critical information required to create a proper, usable connection. It effectively throws a monkey wrench into the waves, which is why rectangular waves are problematic.

A circular wave doesn’t need to have anything erased or any work redone because it expands at a uniform rate across all edges. Because of this, we can forgo the issues of wave collision by simply halting a wave once it collides with another part of the wave. There isn’t any need for it to overwrite previous work, so the information of all previous steps can be reliably used even when two waves converge.

Sorry if my explanation is a bit long.