The Farey structure of the Gaussian integers
نویسنده
چکیده
Arrange three circles so that every pair is mutually tangent. Is it possible to add another, tangent to all three? The answer, as described by Apollonius of Perga in Hellenistic Greece, is yes, and, indeed, there are exactly two solutions [oP71, Problem XIV, p.12]. The four resulting circles are called a Descartes quadruple, and it is impossible to add a fifth. There is a remarkable relationship between their four curvatures (inverse radii):
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For any two consecutive Farey fractions γi = ai/qi < γi+1 = ai+1/qi+1, one has ai+1qi − aiqi+1 = 1 and qi + qi+1 > Q. Conversely, if q and q ′ are two coprime integers in {1, . . . , Q} with q + q > Q, then there are unique a ∈ {1, . . . , q} and a ∈ {1, . . . , q} for which aq − aq = 1, and a/q < a/q are consecutive Farey fractions of order Q. Therefore, the pairs of coprime integers (q, q) wi...
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