My point was, a square die at 15mm x 15mm goes up to 21mm x 21mm when you double the area. When you double the area of a GPU you can double the transistor count. If that 15mm x 15mm chip has 1 billion transistors, then the 21mm x 21mm will have 2 billion transistors.I have no idea why you think the physical length of each side not being double doesn't take up twice the space. Its a wafer, its not a long thin piece of silicon and you can increase the width as much as you want. Double the size and you'll more than half the amount of cores you get off a wafer as yield goes down at a non linear rate as core size goes up. If there were 100 potential cores at 225mm2, then at 450mm2 you'd have probably 44-47potential cores, a silicon wafer is circular, meaning squares don't fit in perfectly, the bigger the square the more space lost on the edge the bigger the cores get, in general. But thats before yields go down, if on the 225mm2, 100 potential cores, you only got 80 working cores, or a 80% yield, that yield would be substantially down if you doubled the core size, its all theoretical with the numbers I'm using because they arent' real, but the yield's will go down exponentially and at one point just become almost completely unworkable.
At 500mm2 (a bit more, 523 iirc) the GF100 is for all intents and purposes, unmanufacturable, it can't be made at a profit because yields are so painfully low. If AMD have gone from a 330-340mm2 core to a 400mm2, and Nvidia dropped down to a little under 400mm2, you can bet that above 400mm2 is where the yields start to drop very dramatically.
15mm sides up to a 21mm side, doesn't mean a 1/4 drop in cores per wafer, it will mean a noticeably over 50% drop in cores per wafer.
Draw a square, divide it into 15mm sided smaller squares, then divide it into 21mm sided squares, see how big the difference is
The same will happen the other way, drop the size of a 5870 by half, a 5770, and you'll get more than double the cores per wafer, because then out of 200 potential cores, the yield will be higher again, call it 90%, so you'd get 180 cores, vs 80 5870 cores, more than double.
The way you've put it suggests you think doubling a GPU's size means you multiply the sides by two, which would result in a core 30mm x 30mm, which would actually be 4x the size not 2x the size, and 4 billion transistors not 2.