Polyhedral kaleidoscopes
"Caleidoscopis polièdrics" (polyhedral kaleidoscopes) [1] is an exhibit on display at MMACA.
Description
The exhibit consists on a set of mirrors arranged on an inverted truncated pyramid structure, together with a set of wooden pieces that fit inside the mirrors pyramid. As a result of the reflected piece, one sees several kinds of symmetric polyhedra. MMACA displays three exhibits, each one with a different set of symmetries, of type (3,3,3,3), (3,4,3,4) and (3,5,3,5), that is, [...]
Activities
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 deeper mathematical discussion, one can mention that these groups are generated by order 2 elements, and hence are Coxeter groups, which are completely classified.
Background information on the books of M. Wenninger (Spherical models) <link> and H.S.M Coxeter (Regular polytopes) <link>
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 <link>. First public display was in 19?? at ???. Rey presented the exhibit at the JAEM conference (200?) <link> with a set of didactic activities.
Similar exhibits
- Kaleidoscopes (Erlebnisland mathematik)
- Kaleidoscopes at Matemilano (2003-2004)
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