Andy Norton

Department of Mathematics Education

University of Georgia


*see other examples


Fibonacci and the Golden Ratio

The relationship between the Fibonacci Sequence and the Golden Ratio is a surprising one. We have two seemingly unrelated topics producing the same exact number. Considering that this number (or Golden Ratio) is non-rational, the occurance is beyond coincidence. It calls for futher examination...


The Golden Ratio = (sqrt(5) + 1)/2 or about 1.618

The Golden Ratio is, perhaps, best visually displayed in the Golden Rectangle. This rectangle has the property that its length is in Golen Ratio with its width. As a consequence, we can divide this rectangle into a square and a smaller rectangle that is similar to the first. Let the following GSP sketch illustrate:


The Fibonacci Sequence is one where each term is defined as the sum of the two previous terms:

We can create this sequence easily in a spreadsheet, using the formula above. This has been done in the center column of the spreadsheet below:

1 1
2 1 1
3 2 2
4 3 1.5
5 5 1.66666666666667
6 8 1.6
7 13 1.625
8 21 1.61538461538462
9 34 1.61904761904762
10 55 1.61764705882353
11 89 1.61818181818182
12 144 1.61797752808989
13 233 1.61805555555556
14 377 1.61802575107296
15 610 1.61803713527851
16 987 1.61803278688525
17 1597 1.61803444782168
18 2584 1.61803381340013
19 4181 1.61803405572755
20 6765 1.61803396316671
21 10946 1.6180339985218
22 17711 1.61803398501736
23 28657 1.6180339901756
24 46368 1.61803398820533
25 75025 1.6180339889579
26 121393 1.61803398867044
27 196418 1.61803398878024
28 317811 1.6180339887383
29 514229 1.61803398875432
30 832040 1.6180339887482
31 1346269 1.61803398875054
32 2178309 1.61803398874965
33 3524578 1.61803398874999
34 5702887 1.61803398874986
35 9227465 1.61803398874991
36 14930352 1.61803398874989
37 24157817 1.6180339887499
38 39088169 1.61803398874989
39 63245986 1.6180339887499
40 102334155 1.61803398874989
41 165580141 1.6180339887499
42 267914296 1.6180339887499
43 433494437 1.6180339887499
44 701408733 1.6180339887499
45 1134903170 1.6180339887499
46 1836311903 1.6180339887499
47 2971215073 1.6180339887499
48 4807526976 1.6180339887499
49 7778742049 1.6180339887499
50 12586269025 1.6180339887499

We have also taken the ratio of every two consecutive terms, in the right column. If we take the limit of this ratio as the terms get larger... the Golden Ratio!


Understanding the Relationship

Remember, now, that the Golden Rectangle can be divided into a square and another Golden Rectangle. In fact, we can repeat this process time and again. Suppose we start with a rectangle with length Y and width X. When we divide this rectangle into a square and a new Golden Rectangle, we get length X and width Y-X for the new rectangle. Since the sides of the rectangles are in the same proportion (namely 1.618...), we get:

Now, let's return to the Fibonacci Sequence. The ratio of consecutive terms forms a sequence itself. Suppose we know that this sequence converges to some real number (we can prove this in a lesson on geometric sequences). Then, for large values of n:

But , so

And, if we replace with Y and with X, we have

The Golen Ratio!