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Algebraic Geometry

By Deligne P.

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Again, you will notice these are Fibonacci numbers. Tree (Species) Norway spruce Douglas fir (spruce) larch pme Number of spirals in one direction 13 3 5 5 Number of spirals in the other direction 8 5 3 8 More details can be found in an article by Brother Alfred 4 Brouseau. The following designations of spiral patterns do not make any pretense of completeness. Actual1y bracts can be lined up into sequences in many ways. The fol1owing are simply some of the more obvious patterns that have been observed.

1. M. Huber, U. Manz, and H. Walser, Anniiherung an den Goldenen Schnitt (Approaching the Golden Section) ETH ZUrich, Bericht No. 93-01, 1993, p. 57. 62 THE FABULOUS FIBONACCI NUMBERS Distribution of a Rumor (x passes the rumor on to xa and xb) Day 1 1 Day22 1, la =2 Day 3 1,2,lb=3,2a=4 Day 4 1,2,3,4, 2b = 5, 3a = 6, 4a = 7 DayS 1,2,3,4,5,6, 7,3b=8,4b=9,5a= 10,6a= 11, 7a= 12 Day 6 1,2,3,4,5,6,7,8,9,10,11,12, 5b = 13, 6b = 14, 7b 8a= 16,9a= 17, 10a= 18, 11a= 19, 12a=20 = 15, Day 7 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18, 19,20,8b = 21, 9b = 22, lOb = 23, lIb = 24, 12b = 25, 13a = 26, 14a = 27, 15a = 28, 16a = 29, 17a = 30, 18a= 31, 19a = 32, 20a = 33 Daysn Personsp Difference 1 1 (1) 2 2 1 3 4 2 4 7 3 5 6 7 12 20 33 5 8 13 Figure 2-2 8 54 21 9 88 34 10 11 12 143 232 376 55 89 144 We find Pn = F n- 2 - 1.

A composite number is one that is not prime, that is, it is divisible by numbers other than itself and 1. 22. A prime number is a number (other than 0 or ±1) that is divisible only by itself and 1. 36 THE FABULOUS FIBONACCI NUMBERS They are: F2 = 1, not a prime 233 = 1,597 FI3 = FI7 F3 =2 F I9 =4,181 =37·113 F 23 = 28,657 F 29 = 514,229 ~ =5 F7 = 13 ~1=89 However, you will notice that the 2nd and the 19th Fibonacci numbers are not primes, and therefore this analog situation does not hold true. Nothing further to prove.

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