I am designing a disc brake. I want to know the theory behind pattern of holes in the rotor in disc brakes.
A fully comprehensive answer on this subject is probably a bit beyond the scope of SE, that said I'll see if I can at least get you started.
First it's important to understand why some designs have holes drilled in them in the first place.
Historically this was for four reasons:
1. Heat dissipation
When the brakes are applied the friction between the pad and disc surface generates a great deal of heat, if they get too hot all sorts of problems can occur - from the spectacular in the form of the pads catching fire to the potentially lethal boiling of the brake fluid (anyone who has ever gone for a big stop when out for a spirited drive only to be confronted with brake fade will agree that this is NOT fun!). Drilling holes in the brake disc can improve the cooling to the disc and therefore helps prevent the temperatures from getting excessive.
Some manufacturers arrange the holes in a sort of outward spiral pattern - supposedly to help with the heat dissipation by working with the rotation of the disc in the airflow. Similarly to how directional tires clear water. Honestly I'm yet to see any hard evidence that it makes any difference whatsoever but perhaps someone far more versed in aerodynamics and heat transfer could explain how it does.
2. Gas venting
This isn't really an issue any more - but in the past some brake pad materials would outgas under the heat, this gas could potentially get trapped between the pad and disc, leading to reduced braking performance. Having holes drilled in the disc allowed this to escape and prevented the issue.
Cars don't operate in 100% dry environments, and when driving in heavy rain or through puddles some of this water can end up on the brake disc, to the potential detriment of braking performance. Having holes drilled gives an easy route for this water to escape and prevents it from impeding the brakes doing their job. In high performance applications where the brakes are operating at high temperatures for more of the time the holes give the steam an escape route (like the gas venting in #2)
Brake discs, especially those made of steel are heavy - and to paraphrase Colin Chapman "adding lightness" is a time honored way of making a car faster. Especially so in terms of brake discs as they are unspring weight which in a performance application is very much something you want to reduce where possible. You won't save much with the drilling of holes, but you will save some - and particularly in motorsport applications that can be enough.
So what about the pattern?
Well for each of the reasons listed above you are going to want an even distribution of the holes.
Heat - if you cluster all the holes in one quarter of the disc surface then that quarter is going to cool better than the other three quarters. Which leads to temperature differentials across the disc surface, which in turn increases the stress on the metal and can lead to premature failure.
Gas Water/steam venting - these are going to build up all across the disc, so you need to have the venting all across the disc.
Weight - weight distribution across a rotating mass is complicated (and probably more one for physics than us) but suffice it to say having it be distributed haphazardly is not good.
There's other factors as well:
Pad wear - the "empty bits" of the holes don't have the same friction against the pad, therefore a pad won't get worn as fast against a hole as it will against the disc surface. So you want to make sure that each rotation of the disc gives the whole of the pad surface the same amount of "hole" - to ensure that the pad wear remains even across the whole surface.
Structural integrity - by now you probably know why holes can be a good thing, so the more the better right? Well, no. Drilling holes in a disc reduces it's structural integrity, have too many, too close together or too large and you are going to seriously weaken the disc, which can lead to premature failure.
Braking performance - as mentioned above pads pressing against holes aren't generating as much friction in those areas, and friction is what produces the braking force in the first place. Each hole is reducing your braking surface compared to an undrilled disc, and that means you need a bigger disc with bigger pads to get the same amount of braking surface back. Which is expensive, and adds weight.
Honestly.. there's probably a book's worth in this topic but hopefully this gives you some things to consider.