If you press ANY one of the islands, a bridge comes out behind you, to that building in the middle of the lake. When you go into that building, you can click on the buttons and rotate them, to find the coordinates of where the spinning dome is on each island. Remember that the spinning dome on Boiler Island is underground, and when you are there and look up, there is a hole. It took me a long time, and several trips back there, to even know what I was looking for. Spike Island has answers to the Marble Puzzle, and I think you have to actually see the Marble Puzzle to know what you are looking for.
Some islands are larger than others, and have more squares to choose from and explore and rotate.
When you are through with one island, and want another, go back to the balcony and press down another island's grid.