Kaleidoscope 7: Reflection Groups

Initial notes towards a bibliography:

Reflection Groups and Invariant Theory - Richard Kane

So What Exactly is a Kaleidoscope?

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 45Â° creates eight duplicate images of the objects, six at 60Â°, and four at 90Â°. 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 between consecutive lines is 180Â°/n for an integer n>=2. At this point there are n lines of reflection symmetry, and the point is a center of n-fold rotational symmetry. See also symmetry combinations. Modern kaleidoscopes are made of brass tubes, stained glass, wood, steel, gourds and most any other material an artist can sculpt or manipulate. The part of the kaleidoscope which holds objects to be viewed is called an object chamber or cell. Object cells may contain almost any material. Sometimes the object cell is filled with liquid so the items float and move through the object cell with slight movement from the person viewing.

History

Known to the ancient Greeks (the word "kaleidoscope" is derived from the Greek kalos, eidos, and skopios; meaning beautiful, form, and view, respectively), it was reinvented by Sir David Brewster in 1816 while conducting experiments on light polarization, and it was patented in 1817. The initial design was made from a tube in which Brewster placed pairs of mirrors at one end, and pairs of translucent disks at the other end. Between the two, he placed the beads. Initially intended as a science tool, it was quickly copied as a toy. Brewster believed he would make money from his popular invention. However, a fault in his patent allowed others to copy his invention. In America, Charles Bush popularized the kaleidoscope. Today, these early products often sell for over $1,000. Cozy Baker collected kaleidoscopes and wrote books about the artists who were making them in the 1970s through 2000. Cozy is credited with energizing a renaissance in kaleidoscope making in America. [For some background on the geometry of the kaleidoscope, see Reflection groups.] .... Adapted from the wikipedia article on Kaleidoscopes.

Kaleidoscope 7

Monday, September 3, 2007

Reflection Groups - a bibliography

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So What Exactly is a Kaleidoscope?

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