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5. Rotations

Other isometries of Euclidean space, such as rotation, can be effectively illustrated using animation. Consider Klein's statement “The rotations which bring one of the regular solids into coincidence with itself collectively form a group” [6, p.5]. Using a ray tracer, you can illuminate this idea in a manner not possible in the real world. You can model two copies of the solid: one stationary and the other in rotation. Figure 10 and its corresponding animation show how this approach can be used to demonstrate the five-fold rotational symmetry of the dodecahedron. As with several of the examples given above, the animation illustrates the mathematical concept with a vividness difficult to achieve in any other way.

Figure 10 with linked animation: Five-fold rotational symmetry of the dodecahedron

*Want to learn more?* The companion paper “Creating Photo-realistic Images and Animations using POV-Ray” explains the third image in detail, providing an introduction to both image and animation generation and allowing interested readers the opportunity to launch their own investigations.

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6. References:

[1] Arnold, Douglas, and Jonathan Rogness, *Mobius Transformations Revealed* (video) http://ima.umn.edu/~arnold/moebius.

[2] Leys, Jos, Etienne Ghys, and Aurélien Alvarez, *Dimensions* (video) www.dimensions-math.org.

[3] Hilbert, David, and Stephen Cohn-Vossen, *Geometry and the Imagination* (reprinted 1999: AMS), p. 90.

[4] Burn, R.P., *Groups: A Path to Geometry* (1985: Cambridge Univ. Press), p.57-59.

[5] Needham, Tristan, *Visual Complex Analysis *(1997: Oxford Univ. Press), p.140-142 http://www.usfca.edu/vca/.

[6] Klein, Felix, *Lectures on the Icosahedron *(reprinted 2003: Dover).

[7] Coxeter, H.S.M., *Regular Polytopes *(reprinted 1973: Dover).