1. The Golden Rectangle
2. The Fibonacci Sequence
4. A Geometric Connection
References and Links
The Algebraic Connection
De Nive Sexangula
In 1611, Johannes Kepler published an essay entitled On the Six-Cornered Snowflake,[B3] in which he attempted to account for the hexagonal symmetry of snowflakes. He notes the same hexagonal symmetry in the pomegranate and in the bee's honeycomb. These two examples are explained as solutions to the problem of packing a maximum number of objects in the least space. He then notes that many flowers have a five-fold pattern of petals and many kinds of fruit have a five-fold pattern of seeds, which raises the question: "why?" Kepler's answer is the following:
Of the two regular solids, the dodecahedron and the icosahedron, the former is made up precisely of pentagons, the latter of triangles but triangles that meet five at a point. Both of these solids, and indeed the construction of the pentagon itself, cannot be formed without the divine proportion [golden section] as modern geometers call it. It is arranged that the two lesser terms of a progressive series together constitute the third, and the two last, when added, make the immediately subsequent term and so on to infinity, as the same proportion continues unbroken. It is impossible to provide a perfect example in round numbers. However, the further we advance from the number one, the more perfect the example becomes. Let the smallest numbers be 1 and 1, which you must imagine as unequal. Add them, and the sum will be 2; add to this the greater of the 1's, result 3; add 2 to this, and get 5; add 3, get 8; 5 to 8, 13; 8 to 13, 21. As 5 is to 8, so 8 is to 13, approximately, and as 8 to 13, so 13 is to 21, approximately. It is in the likeness of this self-developing series that the faculty of propagation is, in my opinion, formed; and so in a flower the authentic flag of this faculty is flown, the pentagon.([B3], p. 21.)
Note how Kepler attempts to tie together the use of the Golden Ratio in the construction of the regular pentagon and the dodecahedron and the occurrence of the Fibonacci sequence in the propagation of living things. Although the connection between the fruit, the flower, and the Golden Ratio is dubious, this passage does seem to be the first mention of the sequence since Leonardo's Liber Abbaci in 1202 ([A1], p. 150), and therefore the first mention of the relationship between the sequence and the Golden Ratio, as Leonardo himself made no note of the relationship ([B2], p. 309). Oddly enough, it does not seem likely that Kepler was aware of Leonardo's work ([A1], p. 150).
So why does the Golden Ratio occur in the Fibonacci Sequence?
To answer this question, we must re-examine the sequence and its ratios.
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(1) At any place in the Fibonacci Sequence, three successive terms, a, b, and c, will have the relation that b + a = c. If we therefore look at the ratios of successive terms, we find a relation between successive ratios: that the next ratio is one more than the reciprocal of the preceding ratio. |
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(2) Using this relationship, we can calculate the ratios of successive terms in the sequence, just as we calculated previously. |
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(3) The nth ratio can be quite complicated as n gets larger. The mathematical entity you see in fig. 3 is known as a continued fraction. The question we must now ponder is this: does this continued fraction converge to a value as n gets larger and larger? |
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(4) This question can be re-translated into an algebraic one with a little thought. If the continued fraction converges to a value, then successive ratios will eventually be nearly equal. We only need ask the question: what is the limiting value of r, such that the n+1st ratio approaches the nth (or previous) ratio? It should now be clear why Euclid's Book IV, Definition 3 yields the Golden Ratio: we are requiring the ratio of b to a to be the same as the ratio of c to b, where b + a = c. |
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(5) In the limit, the n+1st ratio approaches the nth ratio. If n were infinitely large, the two ratios would be equal. Looking at the resulting equation, we see that we are really looking for a number which is equal to its own reciprocal plus one. Setting these equal gives us a simple quadratic equation to solve. |
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(6) Courtesy of the quadratic formula, we have two answers. As you may have suspected, one is the Golden Ratio. Where does the other answer fit in? We notice that it is one less than the Golden Ratio. Recall that we were looking for a number which is one plus its own reciprocal. Well, when you take the reciprocal of 1.618.., you get 0.618.. (sometimes known as the Golden Section). The Golden Ratio is unique in this respect: no other number is one more than its own reciprocal. Note also that the derivation of this limit did not assume anything at all about the numbers which were chosen to begin the sequence (only the specific examples of ratios given in steps (2) and (3) assumed that the sequence began with 1, 1). The Golden Ratio is therefore the limit for any sequence whose terms are defined as the sum of the previous two terms in the sequence. This should help to explain why the Golden Ratio occurred in the Lucas Sequence. |
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