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Fibonacci, Lucas and the Golden Ratio in Pascal’s Triangle

This is the third in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.

Leonardo Pisano (1170-1250), now universally known as Fibonacci, was born in Pisa, Italy, where he was also living at the time of his death. He was educated in north Africa as his father worked there, representing the merchants of the Republic of Pisa when they were trading in Bugia, now called Béjaïa, a Mediterranean port in Algeria.

Photograph of a Fibonacci statue. The statue stands on a cuboid of marble and depicts a man in a cape holding a piece of paper.
The Fibonacci statue by Giovanni Paganucci preserved in the monumental Cemetery of Pisa

Fibonacci returned to Pisa in about 1200 where he wrote a number of important books. His book Liber abaci introduced the Hindu-Arabic place-valued decimal system and the Arabic numerals we now use. Books and any copies had to be handwritten, as it predated the printing press. Fibonacci is now mostly remembered for introducing the Fibonacci numbers and sequence which appeared in the third section of Liber abaci as a problem about rabbits:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The resulting sequence is $1, 1, 2, 3, 5, 8, 13, 21, 34, 55…$ (although Fibonacci did not include the first term in the book).

A diagram showing the growth of a rabbit population - one pair of rabbits in the first row, the same (mature) pair in the second row, and successive generations with 2, 3 and 5 pairs of rabbits.
Fibonacci’s rabbits

The ratio of successive terms converges on the Golden Ratio, $\phi$.

$\phi = \displaystyle\frac{1 + \sqrt5}{2} \approx 1.618033988749. . .$

$\phi$ is an irrational number and is the positive solution of the quadratic equation $x^2 – x – 1 = 0$ Hence, since $\phi$ is the root of an integer polynomial, it is not transcendental, unlike $\pi$.

\[ \frac{1}{1} = 1 \qquad \frac{2}{1} = 2 \qquad \frac{3}{2} = 1.5 \qquad \frac{5}{3} = 1.666 \ldots \qquad \frac{8}{5} = 1.6\]

\[ \frac{13}{8} = 1.625 \qquad \frac{21}{13} \approx 1.615384 \qquad \frac{34}{21} \approx 1.619047 \qquad \frac{55}{34} \approx 1.617647 \qquad \ldots \]

Indeed, convergence to $\phi$ remains true if we start with any pair of Natural numbers and follow the same pattern where any term after the second is the sum of the previous two terms.

Convergence when the first term is smaller than the second term
Convergence when the first term is larger than the second term
This is called the Lucas Sequence.

In Liber abaci, Fibonacci included other numeracy problems – on perfect numbers, the Chinese remainder theorem and on the sum of arithmetic and geometric series. He wrote a book on geometry, Practica geometriae, and perhaps his most impressive work was Liber quadratorum in which he included methods for finding Pythagorean triples. But it is for his sequence for which he is mainly remembered.

The Fibonacci Sequence in Pascal’s triangle

Finding out that the Fibonacci sequence can be found in Pascal’s triangle was a delight for me and I find it hard to think it is just a coincidence. To view Fibonacci’s sequence we can display the triangle as a right-angled triangle.

Pascal's triangle with columns left aligned, with arrows pointing diagonally up and to the right showing that these lines of numbers each sum to the Fibonacci numbers
Fibonacci’s sequence is hidden in the triangle

The Golden ratio in art, music and architecture

My interest in mathematics began when the film Donald Duck in Mathmagic Land was shown to our class in my first year at secondary school in Burnage, Manchester, England and as a teacher of mathematics I showed it in the lesson before Christmas to many year 7 groups.

The film illustrates how the Golden Rectangle has been used by artists and architects throughout history as well as connections between the golden ratio and music. The film mimics some of the novel Alice in Wonderland by Lewis Carroll, the pseudonym of the mathematician Charles Lutwidge Dodgson.

Further connections between the golden ratio and music can be found here and between the ratio and a Stradivarius violin here:

Image of a violin with sections labelled - the whole violin's height split at the top of the body (labelled a1 and a2), the height from the top of the next to the bottom of the side curve, split at the top of the body (labelled b1 and b2) and the distance from the top of the body to the bottom of the side curve, split at the top of the side curve (labelled c1 and c2).

The Lady Blunt shown above shows the measurements connected to the golden ratio:

\[ \frac{a_1 +a_2}{a_2}=\frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{b_2}{c_2}=\frac{c_2}{c_1}=\phi \]

Below is a geometric interpretation of the golden ratio and the golden rectangle:

An interval split into two parts in the Golden ratio, labelled 'a' and 'b', with the equation a/b = (a+b)/a = phi
A rectangle split into two parts - a square piece with sides length a, and a rectangle b by a (a and b are in the Golden ratio)

The Lucas numbers in Pascal’s triangle

Black and white photograph of Edouard Lucas, a very French and rectangular looking white man with short dark hair and an excellent moustache, wearing a white shirt and coat
François Édouard Anatole Lucas

The French mathematician François Édouard Anatole Lucas (1842-1891) served as an artillery officer in the Franco-Prussian War, and subsequently became professor of mathematics at the Lycée Saint Louis and then professor of mathematics at the Lycée Charlemagne, both in Paris. Lucas did a lot of work on number theory and was particularly interested in the Fibonacci sequence and devised the test for Mersenne primes which is still used today.

Lucas died of erysipelas (a bacterial skin infection) a few days after a freak accident. He was at a banquet when a fragment of a dropped plate flew up and cut his cheek.

His sequence, the Lucas sequence, begins with the pair of numbers $2$ and $1$ and its terms are generated in the same way as for the Fibonacci sequence.

$2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521…$

There are a number of connections between the Fibonacci sequence and the Lucas sequence. The $3^{rd}$ Lucas number is the sum of the $1^{st}$ and $3^{rd}$ Fibonacci number, the $4^{th}$ is the sum of the $2^{nd}$ and $4^{th}$, the $5^{th}$ is the sum of the $3^{rd}$ and $5^{th}$, the $6^{th}$ is the sum of the $4^{th}$ and $6^{th}$,…

Division of the Fibonacci terms $2n$ and $n$ beginning with the $2^{nd}$ term yields the Lucas terms

$2^{nd} \div 1^{st} = 1 \div 1 = 1$

$4^{th} \div 2^{nd} = 3 \div 1 = 3$

$6^{th} \div 3^{rd} = 8 \div 2 = 4$

$8^{th} \div 4^{th} = 21 \div 3 = 7$

$10^{th} \div 5^{th} = 55 \div 5 = 11$,..

With some manipulation of Pascal’s triangle and some basic arithmetic, we can find the Lucas numbers in the triangle. We begin by setting out the triangle as below and sum the columns to obtain the Fibonacci sequence

The Fibonacci numbers revealed as the column sums

We now multiply each Pascal number by its column number and divide by its row number, starting with row $1$ column $1$ and then sum the new entries in each column. The first few calculations are shown below:

The Lucas numbers revealed as the column sums

Generally, $\displaystyle\frac{\phi^n -(\frac{1}{\phi})^n}{\phi -(\frac{1}{\phi})}$ is the formula for the $n^{th}$ Fibonacci number, $\displaystyle\frac{\phi^n +(\frac{1}{\phi})^n}{\phi +(\frac{1}{\phi})}$ is the formula for the $n^{th}$ Lucas number and $\phi^n =\displaystyle \frac{L_n+ \sqrt5 \times F_n}{2}$, where $L_n$ and $F_n$ represent the $n^{th}$ Lucas and Fibonacci numbers respectively.

In the next part, we’ll consider some more connections between the triangle and particular numbers, and types of numbers.

Mobile Numbers: Truchet Tiling

In this series of posts, Katie investigates simple mathematical concepts using the Google Sheets spreadsheet app on her phone. If you have a simple maths trick, pattern or concept you’d like to see illustrated in this series, please get in touch.

Since apparently I’m now a maven for interesting fun things built using Google Sheets, someone tagged me in to suggest I might like to see this Truchet Tiling Generator, built in Google Sheets using images generated in Google Drawing.

Truchet tilings consist of square tiles which have a design that isn’t rotationally symmetrical, so each tile can occur in one of two or four visually distinct orientations. Conventionally the designs are fairly simple, geometric patterns using two colours. The design of the tile is such that when tiles are placed in a grid, the edges of the tiles match up in some way – the position of the point where the colour changes is usually at a corner or mid-way along an edge, so that the tiles create pleasing designs.

Truchet tiles were first described in a paper by Sébastien Truchet, a French Dominican priest, entitled “Mémoire sur les combinaisons” which was printed the 1704 edition of Histoire de l’Académie Royale des Sciences. Including a large number of triangle-based patterns, this was the first text to write about Truchet tilings.

In 1987, the tilings were popularised by science historian Cyril Stanley Smith, who wrote a piece for the MIT journal Leonardo (JSTOR login required) in which he described Truchet’s tilings, compared them to historical Islamic and Celtic tiling patterns, as well as discussing them in the context of combinatorics, topology and crystallography (presumably inspired by Smith’s own background as a metallurgist). The paper also included Pauline Boucher’s translation of the original text by Truchet. Smith said:

It embodies an early representation of the principles of combinatorial theory and of crystallographic symmetry including color symmetry. Simple rules of the topology of separation and junction are used to extend Truchet’s concept of directional choice and, by relaxing symmetry rules, to generate diagrams illustrating field/ground relations, the hierarchy of structural freedom and the origin and nature of structural order and disorder in general.

The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy, Cyril Stanley Smith (1987)

The good news is, you too can now explore the hierarchy of structural freedom (and make pretty pictures), using a spreadsheet! New York-based math(s) teacher Mark Kaercher has built a magically updating Google Sheet which generates randomised tiling patterns. By generating four different orientations of your chosen tile and creating cells in the spreadsheet containing those as images, you can combine them randomly to make beautiful tilings, and ticking or unticking a checkbox in one of the cells, force the spreadsheet to recalculate (generating new random numbers using the =randbetween() function) and generating a new pattern.

Mark’s sheet, which you can make your own copy of with a single click, has tabs with a variety of designs, including triangles, quarter circles, diagonal lines, Smith curves (as introduced by Smith in the 1987 paper) and a couple of different types of hexagonal pattern. And yes, it does work on a phone!

If you’d like to read more about how the spreadsheet and tiles were created, you can read Mark’s writeup in a Google Doc.

The Mathematics of Spirograph

Travel Spirograph kit, with blue and red pens

If you’re the kind of person who’s interested in doodling and/or fun toys, you might have encountered the fun doodling toy Spirograph, or some unbranded equivalent. It sits somewhere on the continuum between an artistic drawing tool and a neat mathematical gadget.

Sequences in the triangle and the fourth dimension

This is the second in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.

Sequences in the diagonals

There are many sequences of numbers to be found in Pascal’s triangle. The Natural numbers occur in the second diagonal, running in either direction, and the next two diagonals after that contain other important sequences:

Sequences in the diagonals

There are many sequences of numbers to be found in Pascal’s triangle. The Natural numbers occur in the second diagonal, running in either direction, and the next two diagonals after that contain other important sequences:

Mathematically Gifted

Between the three Aperiodical editors (myself, Christian Lawson-Perfect and Peter Rowlett), there’s a developing tradition of excellent mathematical gift-giving. This year, Christian has excelled himself by designing and creating a brilliant mathematical hoodie, which features a meme about an in-joke (and who can resist either a meme or an in-joke?)

Pascal’s Triangle and its Secrets – Introduction

This is the first in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.

The triangle of Natural numbers below contains the first seven rows of what is called Pascal’s triangle. Each row begins and ends with the number 1, and each of the remaining numbers, from the third row onwards, is the sum of the two numbers ‘above’:

Pascal's triangle, with three triangles of numbers (two in one line, that sum to the one between them on the line below) highlighted in colour
The first seven rows of Pascal’s triangle, showing some pairs and their sums below (highlighted)
Animation showing numbers in Pascal's triangle being generated by adding the pair above
Generating the triangle