How about a smartphone that has a screen that folds out to increase its size? Taiwan’s Industrial Technology Research Institute has developed one that opens to 5 inches. They’ve posted a slideshow of a mock-up to give you an idea of how it works.
It takes the shape of a rectangle, as most smartphones do. It then opens like a book, not a flip-up phone. The bottom half the phone is a keyboard; the top half the screen. Slide the keyboard down and you see the rest of the screen. Only a small section of about 1 centimeter actually folds; the rest of the screen is flat. (The line you can see in the middle of the screen in the image at the left is a task bar, not a break in the screen.)