Chapter 1

Problem 1.1

In pineapples we may find families of 3, 5, 8, 13, and 21 spirals, so that the visible opposed pairs are (3,5), (8,5), (8,13), (21,13). Similar observations can be made on cones, except that the consecutive Fibonacci numbers generally obtained are smaller, as we have seen in the text.

Problem 1.2

1. Figure 1.3(1) shows the visible opposed spiral pair (7, 11).

2. Given a system (7,11), in the family containing 7 spirals, the numbers on adjacent primordia on a spiral must differ by 7, while the numbers on two consecutive primordia on any of the 11 spirals must differ by 11. So the thing to do is to choose a primordium near the rim of the specimen, let us say the one we marked with a dot in its center, and to put the number 1 on it. This primordium is at the junction of two opposed contact spirals, one in each family. The primordium with two dots in it is primordium #8 and the one with three dots is #12. Then we have everything we need to put a number on every other primordium, by applying the Bravais–Bravais theorem. For example, the primordium adjacent to #1, #8, and #12 is #19; the one on the right of #12 is #5; the one above #5 is #16; the one on the left of #16 is #23; and on the right of #16 is #9; on the right of #9 is #2, etc.