Symbolism, Imagery, Allegory
It begins here. The sheet is Saleem's once upon a time, his in the beginning. After all, he wouldn't be here if it wasn't for a perforated sheet. He says so himself:
… the dream of that holey, mutilated square of linen, which is my talisman, my open-sesame, I must commence the business of remaking my life from the point at which it really began, some thirty-two years before anything as obvious, as present, as my clock-ridden, crime-stained birth. (1.1.3)
That's why he opens the story way before he is born. It's where everything starts.
So, let's start at the beginning.
The Dear Doctor
Aadam Aziz falls in love with Naseem through the sheet. It has a hole (about seven inches in diameter) cut in the middle of it, and that's how they communicate. Ghani the landowner (Naseem's dad) arranges the whole thing to trick Aziz into marrying Naseem. Every week, he sees a new part of her under the guise of a new illness, until finally he gets to see her face.
This isn't exactly normal dating, and it doesn't result in a healthy relationship. They are totally mismatched for each other, but by the time they realize it, it's too late.
Mommy Dearest
The next time we see the perforated sheet is with Amina Sinai and her husband Ahmed. The sheet has gone metaphorical this time, but it's just as bad of a way to fall in love as it was the first time.
The key difference is that Amina already knows that Ahmed is a bad match for her. But she's a determined lady. Here's how she does it:
Each day she selected one fragment of Ahmed Sinai, and concentrated her entire being upon it until it became wholly familiar; until she felt fondness rising up within her and becoming affection and, finally, love. (1.5.11)
She falls in love with him, piece by piece—like if she were looking through a perforated sheet. How romantic.
But dating through a perforated sheet does not true love make. Nadir Khan haunts Amina in her dreams.
The Singer
The sheet is back and this time it's bigger and better than ever. It's real, and it's brocaded in gold. This is the last time that we see the perforated sheet, and this time it belongs to Jamila Singer.
It's a different time, a different place, and an even smaller hole, but the perforated sheet hasn't lost any of its magic. It's still strong enough to make a prince fall in love with Jamila.
Again the sheet does its job of creating a superficial love, and again there's a problem. This time it's that Jamila Singer hates everyone who tells her they love her. It becomes a way for her to protect herself from the masses. When Saleem tells her about his incestuous love, she starts wearing the veil in front of him too.
The Void
The legacy of the perforated sheet starts even before Aziz meets Naseem. It starts when he prays and decides never to pray again, making a void in his heart. That void has to be filled with something. Aziz tries to fill it with love. Obviously it doesn't work.
The void is passed down from generation to generation. Amina tries to fill the void left by Nadir Khan, while Jamila would rather be left alone with her void. And Saleem… well, isn't it obvious? He's falling apart.