A novel quadcopter capable of changing shape mid-flight is presented, allowing for operation in four configurations with the capability of sustained hover in three. This is accomplished without requiring actuators beyond the four motors typical of a quadcopter. Morphing is achieved through freely-rotating hinges that allow the vehicle arms to fold downwards by either reducing or reversing thrust forces. Constraints placed on the control inputs of the vehicle prevent the arms from folding or unfolding unexpectedly. This allows for the use of existing quadcopter controllers and trajectory generation algorithms with only minimal added complexity. For our experimental vehicle at hover, we find that these constraints result in a 36% reduction of the maximum yaw torque the vehicle can produce, but do not result in a reduction of the maximum thrust or roll and pitch torques. Experimental results show that, for a typical maneuver, the added limits have a negligible effect on trajectory tracking performance. Finally, the ability to change configurations is shown to enable the vehicle to traverse small passages, perch on hanging wires, and perform limited grasping tasks.