QUESTIONS AND PROBLEMS 39them" (quoted by Mackay, 1991, p. 172). Granting that at the final point of contact
between theory and the physical world, when a human design is to be executed in
concrete and steel, every number is only an approximation, is there any value for
science and engineering in the concept of an infinitely precise real number? Or is
this concept only for idealistic, pure mathematicians? (The problems below may
influence your answer.)
2.4. In 1837 and 1839 the crystallographer Auguste Bravais (1811-1863) and his
brother Louis (1801-1843) published articles on the growth of plants.^14 In these
articles they studied the spiral patterns in which new branches grow out of the
limbs of certain trees and classified plants into several categories according to this
pattern. For one of these categories they gave the amount of rotation around the
limb between successive branches as 137° 30' 28". Now, one could hardly measure
the limb of a tree so precisely. To measure within 10° would require extraordi-
nary precision. To refine such crude measurements by averaging to the claimed
precision of 1", that is, 1/3600 of a degree, would require thousands of individual
measurements. In fact, the measurements were carried out in a more indirect way,
by counting the total number of branches after each full turn of the spiral. Many
observations convinced the brothers Bravais that normally there were slightly more
than three branches in two turns, slightly less than five in three turns, slightly more
than eight in five turns, and slightly less than thirteen in eight turns. For that rea-
son they took the actual amount of revolution between successive branches to be
the number we call 1/Φ = (\/5 — l)/2 = Φ - 1 of a. complete (360°) revolution,
since
3 8 Λ 13 5
2 < 5 <*<T<3-
Observe that 360°-=-Φ « 222.4922359° « 222° 29'32" = 360° - (137° 30'28"). Was
there scientific value in making use of this real (infinitely precise) number Φ even
though no actual plant grows exactly according to this rule?
2.5. Plate 4 shows a branch of a flowering crab apple tree from the author's garden
with the twigs cut off and the points from which they grew marked by pushpins.
The "zeroth" pin at the left is white. After that, the sequence of colors is red,
blue, yellow, green, pink, clear, so that the red pins correspond to 1, 7, and 13,
the blue to 2 and 8, the yellow to 3 and 9, the green to 4 and 10, the pink to 5
and 11, and the clear to 6 and 12. (The green pin corresponding to 4 and part of
the clear pin corresponding to 12 are underneath the branch and cannot be seen in
the picture.) Observe that when these pins are joined by string, the string follows
a helical path of nearly constant slope along the branch. Which pins fall nearest
to the intersection of this helical path with the meridian line marked along the
length of the branch? How many turns of the spiral correspond to these numbers
of twigs? On that basis, what is a good approximation to the number of twigs per
turn? Between which pin numbers do the intersections between the spiral and the
meridian line fall? For example, the fourth intersection is between pins 6 and 7,
indicating that the average number of pins per turn up to that point is between
(^14) See the article by I. Adler, D. Barabe, and R. V. Jean, "A history of the study of phyllotaxis,"
Annals of Botany, 80 (1997), 231-244, especially p. 234. The articles by Auguste and Louis
Bravais are "Essai sur la disposition generate des feuilles curviseriees," Annates des sciences na-
turelles, 7 (1837), 42-110, and "Essai sur la disposition generate des feuilles rectiseriees," Congres
scientifique de France, 6 (1839), 278-330.