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On display at National Museum of Mathematics
Type Hands-on, art
Topics Tilings


An artist’s studio with a mash-up of physical furnishings (easel, paint cans, canvas) and electronic assets, providing an experience of painting (electronically) in a delightful manner combining artistic freedom and mathematical symmetries.

Activities and user interaction

The user chooses among 17 symmetry groups from an electronic easel and dips a (dry) brush into (empty) cans of paint. When a paint is selected, the can glows in the chosen color and the user can then paint the canvas with the selected color. The canvas is divided into an invisible set of cells, such that a single stroke is repeated in each cell, but transformed appropriately according to the symmetry group selected.

Mathematical background

The mathematics behind PolyPaint is the notion of symmetry. Patterns and objects are symmetric if moving them in various ways leaves them looking the same. All of the PolyPaint designs have translational symmetry---there are at least two directions in which you can slide (or translate) the design without changing it. Some of them are also symmetric with respect to rotations by 90 degrees of some other angle (rotational symmetry). The fourth kind of symmetry is called glide reflection symmetry and is a combination of a reflection and a translation. Every symmetry of a 2-dimensional figure is a combination of these four types. There are 17 plane symmetry groups, all accessible in the exhibit.

History and museology

While the core experience of PolyPaint is a software program that could be experienced with a computer alone, the addition of a physical design, themed as an artist’s studio, creates a memorable physical interaction and a forum for shared experience.