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A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct groups.

Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups (also called space groups). Wallpaper (also desktop picture and desktop background) is an image used as a background of a graphical user interface on a computer screen or mobile communications device. On a computer it is usually for the desktop, while for a mobile phone it is usually the background for the 'home' or 'idle' screen. Though most devices comes with a default picture, users can usually change it to files of their choosing.

"Wallpaper" is the term used in Microsoft Windows before Windows Vista (where it is called the Desktop "Background"), while Mac OS X calls it a "desktop picture" (previously, the term desktop pattern was used to refer to a small pattern that was repeated to fill the screen).A 'live wallpaper' is a type of application that works on a mobile device using the Android operating system. The application works as a wallpaper – providing the background image for the home screen—but also works as a conventional application since it can provide user-interaction with the touch screen (allowing the image to change dynamically, for example) and access other hardware and software features within the device (accelerometer, GPS, network access, etc.).

Similar functionality could be found in the Active Desktop feature of Windows 98 and later versions. There is also third party software that provides this feature for various operating systems.An autostereogram is a single-image stereogram (SIS), designed to create the visual illusion of a three-dimensional (3D) scene within the human brain from an external two-dimensional image. In order to perceive 3D shapes in these autostereograms, one must overcome the normally automatic coordination between accommodation (focus) and vergence (angle of one's eyes).

The simplest type of autostereogram consists of horizontally repeating patterns (often separate images) and is known as a wallpaper autostereogram. When viewed with proper vergence, the repeating patterns appear to float above or below the background. The well-known Magic Eye books feature another type of autostereogram called a random dot autostereogram. One such autostereogram is illustrated above right. In this type of autostereogram, every pixel in the image is computed from a pattern strip and a depth map. A hidden 3D scene emerges when the image is viewed with the correct vergence.