Vorlesungen ueber Zahlentheorie. Mit Anhang vom Dedekind by Dirichlet P.G.L.

By Dirichlet P.G.L.

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These are seen to follow the sequence of Fibonacci numbers and the recursion relations as derived above for the number of rabbit pairs can be shown to be applicable to the bee problem as long as it is assumed that bees, like rabbits, are immortal. Fibonacci numbers also appear in the field of optics. A system is constructed from two plane sheets of glass with slightly different indices of refraction. Rays of light which are incident on one piece of glass will undergo various numbers of internal reflections before emerging.

If, on the other hand, each edge of an o c ~ e ~ r is o nd i ~ into d ~ two segments with relative lengths in the ratio of 1 : r then these points do form the vertices of an icosahedron. Some care is required in locating these vertices. Four edges form each vertex of the octahedron. W o opposite edges are divided so that the longer edge segment is adjacent to the vertex while the other two opposite edges are divided so that the shorter edge segment is adjacent to the vertex. Each vertex may be treated in thts manner.

40 The GoidenRatio andFibonacciNtrmbers 3, = Fn*2 * As a final example of the occurrence of Fibonacci numbers, a somewhat more mathematical problem will be considered here. A staircase consists of n stairs. This is climbed by taking either one step or two steps at a time and the number of different ways of climbing the stairs, S,,, is to be determined. E n is 1 then the solution is simple, S,, = 1. e. 1 + 1, or 2. For n = 3 there are three different ways; 1 + 2, 2 + 1 or 1 + 1 + 1. Thus the number of possibilities for n stairs is equal to the mun of Sn-i and S+2.

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