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By Walker R. J.

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The cube has 24 symmetries. These can be enumerated as follows: Corner 1 can be sent to its own (front) face or to the back, and to the bottom or to the top, for a count of four; there are three further possibilities for corner 2 and two for corner 3, for a total count of 4 • 3 • 2 = 24. This mode of counting identifies the group as S4 . The dodecahedron is a bit more complicated. There are 20 corners which may be divided into four families of five, each family belonging to a single face and labeled 12345.

Then 0 contains a substitution of the form z → 1 + cz, the number c being a primitive nth root of unity, and this leads to a contradiction in every case n = 2, 3, 4, 6 (n divides 12). n = 2: c = −1 and that is impossible since, together with z → 1 − z, z → −z → 1 + z also belongs to 0 . n = 3 produces, from ±1, four additional points ±(1 + c), ±(1 − c), and this is too many. √ √ n = 4: c = ± −1 and z → ±z → 1 + −1z produces, from ±1, the √ distinct points ±1 ± −1. n = 6 produces, from 1, five additional distinct points 1+c+· · ·+cm (m ≤ 5).

15. The complex structure of S is clarified thereby: Plainly, it is compatible with that of the base except over 0 and ∞ where the branching of the radical takes place. There the cover is ramified over the base, its two sheets touching as in Fig. 16, or, more realistically, as in Fig. 17, in which you see that one revolution about 0 carries you from sheet 1 to sheet 2, and a second revolution (not shown) brings you back to sheet 1. 17 hint that the complex structure of the cover goes bad at the ramifications, but this is not so: Both cover and base are complex manifolds in themselves; it is just that their complex structures are not the same: At 0, z is local parameter downstairs √ and z is local parameter upstairs; at ∞, you must use 1/z downstairs and √ 1/ z upstairs.

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Algebraic Curves by Walker R. J.


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