II
PHI AND FIBONACCI
“[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language.” 1
—Galileo Galilei
The mathematical work of the Greeks was kept alive in ninth-century Baghdad, where caliph Harun al-Rashid founded a great library that became known as the House of Wisdom. Here, Muslim, Jewish, and Christian scholars met to discuss and debate subjects such as chemistry and cartography, and translated ancient texts from Greece and India into Arabic. Many incredible advances in science and mathematics were made during the ensuing Islamic Golden Age, which lasted until the thirteenth century. For example, the scholar Muhammad ibn Musa al-Khwarizmi (c. 790–c. 850) was among the first mathematicians in the world to use zero as a place holder, and his treatise Hisab al-jabr w’al-muqabala (The Compendious Book on Calculation by Completion and Balancing) introduced the word algebra from the Arabic al-jabr, which means “completion.” The word was referring to the process of reducing a quadratic equation by means of removing the negative terms, which gave birth to the field of algebra. Interestingly, in the same book he presented a quadratic equation that represented a line of length 10 divided into 2 segments with golden ratio proportions.
This 1983 Soviet stamp above bears the visage of al-Khwarizmi, an influential ninth-century mathematician and towering figure in Baghdad’s House of Wisdom, depicted at below.
THE FIBONACCI SEQUENCE
A half century after al-Khwarizmi, Abu Kamil Shuja ibn Aslam (c. 850–c. 930), an Islamic mathematician from Egypt, applied complex algebra to geometric problems, solving three non-linear equations for three different variables. He also presented equations on various ways to divide a line of length 10 and to inscribe a pentagon within a square. Abu Kamil was the first mathematician to employ irrational numbers as solutions to quadratic equations, 2 and his Kitāb fī al-jabr wa al-muqābala (Book of Algebra), which expanded on the work of al-Khwarizmi, was influential in Europe following its translation into Latin in the twelfth century.
The work of al-Khwarizmi—particularly his discussion of Hindu-Arabic numerals—later caught the attention of a young Italian boy during a visit to an Algerian port city with his father, a wealthy merchant from Pisa. The boy, Leonardo Fibonacci (c. 1175–c. 1250), would later become one of history’s most famous mathematicians after the publication in 1202 of his book Liber Abaci, which promoted the Hindu-Arabic numbering system throughout Europe.
These pages from a 1342 edition of al-Khwarizmi’s Book of Algebra display geometrical solutions to two quadratic equations.
Arab astronomers use an astrolabe and cross-staff to determine latitude in an observatory in Constantinople (present-day Istanbul, Turkey) during the Islamic Golden Age, which lasted from about the mid-eighth to mid-thirteenth century.
This page from Fibonacci’s revolutionary 1202 work Liber Abaci, which introduced Hindu-Arabic numerals to the West, shows the association between Roman numerals and different quantities.
In writing Liber Abaci, Fibonacci relied on many Arabic sources, including the problems of Abu Kamil. Drawing the connection between two of Abu Kamil’s equations for dividing a line of length 10 and the result that produces the golden ratio, Fibonacci gave the lengths of the segments as √125 – 5 and 15 – √125, 3 which can also be written as 5(√5 – 1) and 5(3 – √5). These are both expressions of the two golden ratio points on a line of length 10. Now, divide both of these expressions by 10, and you have the algebraic formulas for phi’s inverse (1/Ф, 0.61803…) and 1 – 1/Ф (0.38197…). Recall from here that phi is the only number in which its reciprocal is one less than itself, and derive the algebraic formula for phi itself by adding 1 to both sides of the equation:
1 / Ф = (√5 – 1) / 2 = Ф – 1
Ф = (√5 + 1) / 2
In his book, Fibonacci also wrote a simple numerical sequence based on a theoretical problem of growth in a population of rabbits. That sequence—the foundation for an incredible mathematical relationship behind phi—was known as early as the sixth century by Indian mathematicians, but it was Fibonacci who popularized it in the West.
Fibonacci’s sequence can be explained using the following example. Suppose we have a newly born pair of rabbits, one male and one female. Suppose rabbits are able to mate at the age of one month, so at the end of its second month a female can produce another pair of rabbits. Suppose our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month onward. The question Fibonacci posed was how many pairs will there be in one year? The answer is 144, which is found as the twelfth number in the sequence of growth below, corresponding to the twelfth month of new-born rabbits. Starting with 0 and 1, each new number in the sequence is simply the sum of the two before it:
0 + 1 = 1
1 + 1 = 2
2 + 1 = 3
3 + 2 = 5
5 + 3 = 8
8 + 5 = 13
… and so on, resulting in the following sequence, named after Fibonacci:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …
You can estimate the nth number in the Fibonacci sequence, using Φ and √5, with the equation:
f(n) = Φn / √5
For example, the twelfth number of the Fibonacci sequence can be calculated thus:
Φ12 / √5 = 321.9969 … / 2.236 … = 144.0014 …, which rounds to 144!
In the Fibonacci sequence, the ratio of each successive pair of numbers converges on phi. To visualize this phenomenon, note that each successive value of the ratio gets closer and closer to phi, as shown:
1/1 |
= |
1.000000 |
2/1 |
= |
2.000000 |
3/2 |
= |
1.500000 |
5/3 |
= |
1.666667 |
8/5 |
= |
1.600000 |
13/8 |
= |
1.625000 |
21/13 |
= |
1.615385 |
34/21 |
= |
1.619048 |
55/34 |
= |
1.617647 |
89/55 |
= |
1.618182 |
144/89 |
= |
1.617978 |
233/144 |
= |
1.618056 |
377/233 |
= |
1.618026 |
610/377 |
= |
1.618037 |
987/610 |
= |
1.618033 |
This marble statue of Fibonacci was created by Italian sculptor Giovanni Paganucci in 1863.
At the fortieth number in the sequence—102,334,155—the resulting ratio matches phi to 15 decimal places:
1.618033988749895
Despite the obvious convergence of Fibonacci’s sequence on the value of phi, the Italian mathematician did not write specifically about the golden ratio. In fact, another four hundred years elapsed before someone made an explicit connection between the two. 4 That person was Johannes Kepler (see here), who, in a letter from 1609, became the first person known to clearly mention that the ratios of successive numbers in the Fibonacci sequence approximates the golden ratio.
In 1653, French mathematician Blaise Pascal (1623–1662) developed his eponymous triangle formation, visually describing the algebraic expansion of binomial coefficients (i.e., two positive integers that form a sum). As shown below, starting with an apex of 1, every number in the triangle below is the sum of the two numbers diagonally above it to the left and the right, and the numbers on diagonals of the triangle add to the Fibonacci series. Pascal’s triangle has many unusual properties and a variety of uses, including the following:
• Horizontal rows add to powers of 2 (i.e., 1, 2, 4, 8, 16, etc.)
• The horizontal rows represent powers of 11 (1, 11, 121, 1331, 14641) for the first five rows, in which the numbers have only a single digit.
• Adding any two successive numbers in the diagonal 1-3-6-10-15-21-28 … results in a perfect square (1, 4, 9, 16, etc.)
• When the first number to the right of the 1 in any row is a prime number, all numbers in that row are divisible by that prime number.
Also, Pascal’s triangle can be used to find combinations in probability problems. If, for instance, you pick any 2 of 5 items, the number of possible combinations is 10, found by looking in the second place of the fifth row (note that you do not count the 1s in this application).
This colored engraving from 1822 portrays French mathematician Blaise Pascal, developer of Pascal’s triangle.
THE FIBONACCI SPIRAL AND OTHER CURIOSITIES
If you have poked around on the Internet on the topic of Fibonacci sequences, there’s a good chance you came across images of Fibonacci or golden spirals. There’s also a good chance that you’ve seen some of them as overlays on everything from the Parthenon to the Mona Lisa to Donald Trump’s hairline. Typically, the spiral is created with a golden rectangle at its foundation. Divide the golden rectangle at its golden ratio point and you’ll be left with a square and another smaller golden rectangle. Do the same to the smaller golden rectangle again and again to create the image below:
Now we draw a quarter circle arc in each square to create the golden spiral:
A closely related spiral is the Fibonacci spiral. Here, instead of creating a successive pattern of golden rectangles, our building blocks are squares whose side lengths are equal to the numbers of the Fibonacci sequence, as shown:
Technically speaking, none of these are spirals. They’re called volutes. The difference is almost imperceptible, but a true golden spiral is a unique, equiangular (that is, logarithmic) spiral that expands at a constant rate. In the illustration below, the green spiral is constructed with a succession of independent quarter-circle arcs within each square. The red spiral is a true logarithmic spiral that expands by the golden ratio every 90 degrees. The portions that overlap appear in yellow. Now, you’re one of the few who knows the difference between them!
There are many unusual relationships in the Fibonacci series. For example, for any three numbers in the series f(n – 1), f(n), and f(n + 1), the following relationship exists:
f(n – 1) × f(n + 1) = f(n)2 – (–1)n
3 × 8 = 52 – 1
5 × 13 = 82 + 1
8 × 21 = 132 – 1
Here’s another: Every nth Fibonacci number is a multiple of f(n), where f(n) is the nth number of the Fibonacci sequence. Given 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, note the following results:
• Every 4th number (e.g., 3, 21, 144, and 987) is a multiple of 3, which is f(4).
• Every 5th number (e.g., 5, 55, 610, and 6765) is a multiple of 5, which is f(5).
• Every 6th number (e.g., 8, 144, and 2584) is a multiple of 8, which is f(6). 7
The Fibonacci sequence also has a pattern that repeats every 24 numbers. 8 This repetitive pattern involves a simple technique called numeric reduction in which all the digits of a number are added together until only one digit remains. As an example, the numeric reduction of 256 is 4 because 2 + 5 + 6 = 13 and 1 + 3 = 4. Applying numeric reduction to the Fibonacci series produces an infinite series of 24 repeating digits:
1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9
If you take the first 12 digits, add them to the second 12 digits, and then apply numeric reduction to the result, you find that they all have a value of 9.
This colorful arrangement of rectangles represents the first 160 natural numbers as sums of Fibonacci numbers.
As discovered in 1774 by French mathematician Joseph Louis Lagrange, the last digit of the numbers in the Fibonacci sequence form a pattern that repeats after every sixtieth number. These are:
0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9,
0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1
When these sixty digits are arranged in a circle, as shown below, additional patterns emerge: 9
Joseph Louis Lagrange, another prominent French mathematician who studied the Fibonacci sequence, is pictured in this engraving.
• The zeros align with the 4 cardinal points on a compass.
• The fives align with the 8 other points of the 12 points on a clock.
• With the exception of the pairs of zeros, the numbers directly opposite each number add to 10.
CALCULATING PHI
In 1567, Kepler’s mentor Michael Maestlin (1550–1631), a German astronomer and mathematician, presented the first known approximation of the golden ratio’s reciprocal in a letter to his former student, describing the golden ratio as a decimal fraction of “about 0.6180340.” 10
Combining Fibonacci’s value for 1/Ф with the fact that 1/Ф = 1 – Ф, we were able to generate an equation for the value of Ф shown here. But there’s another way to derive that same value using basic logic. Remember from here that at the golden ratio cut, illustrated in the diagram below, the ratio of the whole to the larger segment is the same as the ratio of the larger to the smaller segments, represented by the equation A / B = B / C.
We also know that the two line segments B and C added together are equal to A, which is algebraically represented as A = B + C.
Now, if we combine these equations, we see that (B + C) / B = B / C. Moving all variables on one side of the equation and making C = 1, we arrive at this familiar equation:
B2 – B – 1 = 0
Because this equation is now in the form ax2 + bx + c = 0, we can apply the quadratic formula, which allows us to solve for x after plugging in the values for a, b, and c (1, -1, -1):
Therefore, our two possible solutions are (1 + √5) / 2 and (1 – √5) / 2. The positive solution gives the exact value of the Φ.
As we know already (see here), the ratio of successive Fibonacci numbers converges on phi, but this is not the only series in which that relationship exists. You can pick any two numbers to create the successive ratios and the result will always converge on phi. As an example, separate the digits of 1.618 into 16 and 18, and then add two numbers and take the ratio of 18 to 16, as shown below. If you then sum the next two numbers in the sequence and determine their ratio, and so on, a familiar pattern emerges:
16 + 18 = 34, and their ratio is 1.125
18 + 34 = 52, and their ratio is 1.888889 …
34 + 52 = 86, and their ratio is 1.529412 …
52 + 86 = 138, and their ratio is 1.653846 …
86 + 138 = 224, and their ratio is 1.604651 …
138 + 224 = 362, and their ratio is 1.623188 …
224 + 362 = 586, and their ratio is 1.616071 …
362 + 586 = 948, and their ratio is 1.618785 …
Now, let’s return to the other unique property of phi, described here:
Φ2 = Φ + 1
This can also be written as Φ2 = Φ1 + Φ0, leading to our next revelation: For any number n, each two successive powers of phi add to the next one, expressed mathematically as follows:
Φn+2 = Φn+1 + Φn
Another little curiosity involves raising phi to a power and then adding or subtracting its reciprocal:
• For any even integer n we find that Φn + 1/Φn is a whole number (e.g., Φ2 + 1/Φ2 = 3).
• For any odd integer n we find that Φn – 1/Φn is also a whole number (e.g., Φ3 + 1/Φ3 = 4).
Phi can also be calculated as the limit of a variety of iterative expressions of limits, including these:
Finally, as we observed with the connection to the pentagon and pentagram, phi has a special relationship with the number 5. If we rewrite the expression for Ф, (1 + √5) / 2, using decimals, we come up with this equation that can be used in Excel or coding “(^ is a symbol for exponent, or raised the power of.):11
Ф = .5^ .5* .5+.5
Here is yet another equation for phi:
Perhaps Kepler was on to something when he described the golden ratio as a “precious jewel.” After all, this is the person whose curiosity, persistence, and insight led to the discovery of the elliptical nature of the planetary orbits around the Sun, revolutionizing our understanding of the universe. In the next chapter, we’ll explore how these beautiful concepts of geometry and mathematics are expressed in the arts.