Difference between revisions of "Polyhedral kaleidoscopes"

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Exhibit

Polyhedral Kaleidoscopes
On display at	Museu de Matemàtiques de Catalunya
Type	Hands-on
Topics	Mirrors, symmetry, polyhedra

Calidoscopis polièdrics (polyhedral kaleidoscopes) is an exhibit on display at Museu de Matemàtiques de Catalunya (MMACA) in Barcelona, Spain.

Description

The exhibit consists on a set of three spherical kaleidoscopes. For each kaleidoscope, four equal mirrors in the shape of circle sectors are arranged on an inverted rhombic pyramid structure. A set of coloured wooden pieces accompanies each kaleidoscope, so that these pieces fit inside the mirrored pyramid. As a result of the reflected piece, one sees several kinds of symmetric polyhedra. Each kaleidoscope has a different set of symmetries, depending on the dihedral angles of the mirrors. Those are (120,120,120,120), (120,90,120,90) and (120,72,120,72) degrees, giving locally at each edge dihedral symmetries of order (3,3,3,3), (3,4,3,4) and (3,5,3,5) respectively.

Activities

Photograph of a (3,4,3,4)-type kaleidoscope from MMACA. Note that the wooden piece has one blue square, two yellow pieces that are a third of hexagon, and two red pieces that are a quarter of octogon. Each yellow piece is reflected three times around the order-3 axis, and each red piece is copied four times around the 4-order axis.

The activities adapt to the age and background of the visitor. For younger visitors (5-10 years old), it is sufficient to bring them the wooden pieces and let them discover the beautiful shapes that are obtained. Next, play to identify the shapes with the list of polyhedra on the wall.

For an intermediate level, one can challenge them to tell in advance which of the polyhedra on the poster will appear when putting a wooden piece inside the kaleidoscope, and induce them to see the number of copies seen at each corner. This number of copies around the edge of the kaleidoscope depends on the angle of the two mirrors meeting at this edge. Discuss why the three different kaleidoscopes give rise to three families of polyhedra.

For advanced visitors, one can discuss the notion of group and symmetries. Each family of polyhedra share a symmetry group, which is generated by the reflections on the mirrors. This way, one see that the abstract concept of symmetry is related to the structure that provides the kaleidoscope, but the actual shape depends on the particular piece that we put inside, that can be one of the wooden pieces or any object.

For a deep mathematical discussion, one can mention that these groups are generated by order 2 elements, and hence are Coxeter groups, which are completely classified.

Background

The following books are a good source for constructing kaleidoscopes:

• M. Wenninger, Spherical models
• H.S.M Coxeter, Regular polytopes

History and museology

This exhibit has become an iconic feature of MMACA, and it is usually part of MMACA's travelling exhibitions. Although there are precedents (e.g. the Matemilano mirrors), this exhibit does not use the minimal set of three mirrors to generate the polyhedra, but sets of four mirrors and symmetric wooden pieces. This has the advantage of being stable (the pieces hold in place) and the aperture of the basket is bigger, trapping light and offering a good view of the polyhedra. It is believed to be the first exhibit with this four mirrors design.

The exhibit was designed and built by Josep Rey for MMACA. First public display was in 19?? at ???. Rey presented the exhibit at the JAEM conference (200?) with a set of didactic activities.

Similar exhibits

• Kaleidoscopes (Erlebnisland mathematik)
• Kaleidoscopes at Matemilano (2003-2004)
• Le Labosaïque

Resources

• Page at MMACA