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LG patents a mobile UI that uses a bendable display
LG received a US patent (#8,271,047, applied to in 2008) that describes a mobile device interface that uses a bendable screen. LG's patented device uses two display - the top one is bendable while the bottom one is a rigid touch display. The user can bend or fold the top display to react with the device.
This seems to be a rather weird design. LG Display has plans to start mass producing flexible OLED panels in the second half of 2013, so such a device may see the light of day in the near future (although to be fair, these first flexible OLEDs will probably not be bendable).
Back in October 2011 Nokia showed a working prototype device called the Kinect - which was a bendable device that used the flexible display as part of its UI - for example bending can be used to zoom an image or select an option. You can see it live in the video below: