Yes I wondered how they lathe an Octagon without ripping up their backing plate.Not that this makes any kind of definitive answer, but I think about the cymbal rotating on a lathe. It spins around an axis, so that the cutting edge scribes a circle or spiral as the plate turns. As long as the whole thing turns around a central axis, a circle is the natural result.
I have a bit of University physics under my belt from long ago, but I did my Masters and PhD in Statistics. But you could try me on the proper equations and modeling if you know of some. Complex nonlinear models with "ridiculous numbers of variables" are something I'm used to. All I take from what you have said is "it's complicated" but no specific prediction about a sonic shift. These guys, on the other hand, have a computational model which would probably do the trickThey play with the flow and collisions of the waves to achieve particular sonic properties. A cut on the edge dissipates the wave motion while the sharp angles cause wave collisions ... etc etc etc.
Again, a simple review of waves and the basics of movement through a medium will enlighten the subject.
Imagine all the different ways you can have five skillets (or garbage cans or tin buckets or even brass rings that would represent waves) collide and the varied sounds you can get from them. There is a staggering number of ways they could collide and with each there would be a slightly different sound perceived.
Again, this is Elementary explanations but it should be clear enough ... without having to deal with the ridiculous number of variables associated with cymbal design.
And yet...despite what people are suggesting Octagons don't sound dry and lifeless and aren't a problem on the stand. Spider Rondinelli using an Octagon as his main ride on a jazz gig. He also has a smaller Octagon crash on his right and it sounds a little trashy to me.The only 2 shapes that would easily balance on a stand are the circle and square. And sure is a lot easier to reach for a circular cymbal (even without looking) at any time and know where it is. A square, or other shape not so easy. Those corners on 4-6-8 sided cymbals are a pain. A square cymbal would effectively act like a round cymbal with chunks cut out of it. Would probably be pretty dry and somewhat lifeless. As mentioned by Mcjnic it's nice to be able to follow the tonal grooves to flow the pressure waves of sound. A square would continually drop some off the edge of the cymbal (like a chunk is missing). And those areas will sound deader.