Multi-layer user interface with flexible parallel and orthogonal movement

A user interface (UI) system calculates movements in a multi-layer graphical user interface. The UI system receives user input corresponding to gestures on a touchscreen. The UI system calculates a movement of a first layer in a first direction (e.g., a horizontal direction) at a first movement rate. The UI system calculates a movement of a second layer substantially parallel to the movement of the first layer, at a second movement rate that differs from the first movement rate. The UI system calculates a movement (e.g., a vertical movement) in a direction substantially orthogonal to the first direction, in a UI element of one of the layers.