The kaleidoscope is a tube of mirrors containing loose coloured beads or pebbles, or other small coloured objects. The viewer looks in one end and light enters the other end, reflecting off the mirrors. Typically there are two rectangular lengthways mirrors. Setting of the mirrors at 45° creates eight duplicate images of the objects, six at 60°, and four at 90°. As the tube is rotated, the tumbling of the coloured objects presents the viewer with varying colours and patterns. Any arbitrary pattern of objects shows up as a beautiful symmetric pattern because of the reflections in the mirrors. A two-mirror model yields a pattern or patterns isolated against a solid black background, while a three-mirror (closed triangle) model yields a pattern that fills the entire field. For a 2D symmetry group a kaleidoscopic point is a point of intersection of two or more lines of reflection symmetry. In the case of a discrete group the angle be

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