Pascal’s triangle | The Aperiodical

You're reading: Posts Tagged: Pascal’s triangle

Probability, statistics and Pascal’s other contributions

By David Benjamin. Posted March 4, 2022

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

Probability and combinations

In Part 1 of this series we stated that Pascal is credited with being the founder of probability theory – but credit also needs to be given to other mathematicians, in particular the Italian polymath Girolamo Cardano.

The connection between probability and the numbers in Pascal’s triangle can be shown by looking at the outcomes when one or more coins are tossed. The table below, from row two, lists the outcomes for one, two and three unbiased coins.

				$1$
			$1$ H		$1$ T
		$1$ HH		$2$ HT, TH		$1$ TT
	$1$ HHH		$3$ HHT, HTH, THH		$3$ HTT, THT, TTH		$1$ TTT
$1$		$4$		$6$		$4$		$1$

Reading from the left: all possible outcomes, heads decreasing by one moving to the right.

For four coins there is $1$ outcome for four heads, $4$ outcomes for three heads and one tail, $6$ outcomes for two heads and two tails, $4$ outcomes for one head and three tails and one outcome for $4$ tails.

Row four shows us that when three unbiased coins are tossed, the probability they will land showing two heads and one tail in any order is $\frac{3}{1+3+3+1}=\frac{3}{8}$.

As the sum of the $n^{th}$ row is $2^{n}$, the number of possible outcomes for four coins is $2^4=16$, $32$ for five coins, $64$ for six coins, …

Quincunx

A Quincunx, or Galton Board, is named after the English explorer and anthropologist Francis Galton (1822-1911) – although this name is now less popular, because of Galton’s views on eugenics and racist attitudes.

The board is a triangular array of pegs. Balls are dropped onto the top peg and then bounce their way down to the bottom where they are collected in containers. Each time a ball hits one of the pegs, it bounces either left or right with an equal probability of $\frac{1}{2}$ and the balls collect in the containers to form the classic bell-shaped curve of the normal distribution.

Animation of a board of pegs with balls falling and bouncing to the left or right — Generated from Math Is Fun’s page on the Quincunx

The Quincunx is like Pascal’s triangle with pegs instead of numbers. The number on each peg represents the number of different paths a ball can take to reach that peg. If there are $10$ rows and the last row contains the containers, then the probability of landing in the third container from the right can be calculated by using the formula for the Binomial distribution.

The probability of landing in the third bin from the right is $120\times(\frac{1}{2})^3\times(\frac{1}{2})^7=\frac{15}{128}=0.1171875$, where $120$ is the number of different paths to that bin.

Statistics and permutations

The link between statistics and the triangle can be demonstrated using combinations. Consider these 5 mathematicians Euler, Pascal, Ramanujan, Hilbert and Conway and the possible teams for a three-legged race.

There are $10$ different teams of $3$:

EPR EPH EPC ERH ERC EHC PRH PRC PHC RHC

The formula to calculate the number of combinations is $_n{C}_r =\frac{n!}{r!(n-r)!}$ where $n$ represents the total we are choosing from, $r$ the number in the team and

\[ n!=n\times(n-1)\times(n-2)\times(n-3)\times…\times1\]

In our example $n=5$, $r=3$ and $\frac{5!}{3!(5-3)!}=\frac{120}{6\times2}=10$

$_n{C}_r$ can be used to calculate the rows of Pascal’s triangle as shown below for row $6$, where in the calculation of $_5{C}_0$, $0!=1$

$_5{C}_0$	$_5{C}_1$	$_5{C}_2$	$_5{C}_3$	$_5{C}_4$	$_5{C}_5$
$1$	$5$	$10$	$10$	$5$	$1$

The animation film Of Dice and Men by John Weldon is a lovely way to introduce students to probability and statistics.

Pascal the polymath: mathematics, inventor, science and religion

Pascal’s father was a tax collector and in 1642 Blaise invented a mechanical calculator to assist his father. It was called the Pascaline and had a wheel with eight movable parts for dialing. Each part corresponded to a particular digit in a number. Numbers could be added by turning the wheels located along the bottom of the machine. Subtraction was carried out by exploiting a method called nines’ complement representation, the use of which allows subtraction to be reduced to addition. Each digit in the answer was displayed in a separate window. The workings of the Pascaline are demonstrated here.

A photo of a gold metal box, with a set of dials on the front and numbers showing through holes across the top. — A Pascaline. An original is displayed in The Musée des Arts et Métiers in Paris, France

Close-up photo of the dials on the box, which have the digits 1-9 engraved and some French script text above — A close-up of the dials which are rotated by inserting a spoke

The Musée des Arts et Métiers in Paris has one of the original Pascalines. The invention was not a commercial success – it was very expensive and often only purchased as a novelty rather than for use. Essentially, it was an adding machine. Subtraction was turned into a form of addition, as was multiplication. Division was done by repeated subtraction. Nines’ complement representation is still used in modern digital computers by a similar technique called ones’ complement which is used to represent negative numbers and hence perform subtraction in the same way as addition. Pascal did not discover this method but his calculator is the earliest known device to employ it. He continued to make improvements to his design until 1652.

Conic sections – normally just called conics – are obtained when a mathematical cone is sliced by a plane. Depending on the angle of the slice, the intersections create a circle, an ellipse, a parabola and a hyperbola. Conics have many applications including the wheel of course, ophthalmic, parabolic mirrors and reflectors, telescopes, searchlights and projectile motion.

Pascal wrote a short treatise, Essai pour les coniques (Essay on Conics) when only 16. In it he included what is known as Pascal’s Theorem which states that if a hexagon is inscribed in a conic section then the three intersection points of opposite sides lie on a straight line – the Pascal line. The theorem [also referred to as Pascal’s Hexagrammum Mysticum Theorem] was his first important mathematical discovery and a breakthrough in the field of projective geometry.

Photo of a page of the essay, with some geometric diagrams at the top and the title 'Essay Pour Les Coniques' above French text — A rare copy of the *Essay pour les coniques* which is kept in the National Library of France

In 1647 Pascal expanded on the work of the Italian physicist Evangelista Torricelli, the inventor of the barometer by writing Experiences nouvelles touchant le vide (New experiments with the vacuum) in which Pascal gave detailed rules to describe to what degree various liquids could be supported by air pressure. In 1971 the SI unit for pressure [equal to one newton per square metre] was named the pascal.

Photograph of a circular gauge with values from 0 to 4000 psi and 0 to 28000 kPa, with an arrow that rotates and the text 'U.S. Gauge / MADE IN USA' across the bottom. Text in the centre reads 'USE NO OIL'. — A pressure gauge reading in psi (red scale) and kPa (black scale)

Also in 1647 he discovered Pascal’s Law of hydrostatics allowing for the development of the hydraulic press. Pascal himself used the principle to invent the syringe.

A diagram of a car on a wide platform, joined via a hydraulic setup to a hand plunger, which has a much narrower area of fluid to push down on — Pascal’s Law is the principle behind hydraulic lifting and pressing devices

Pascal wrote an extremely influential theological work which was unfinished at the time of his death. It was posthumously called Pensées (Thoughts) and contained a detailed and coherent examination and defence of the Christian faith.

Image of a book page in French; it's a title page that reads PENSÉES DE M.PASCAL SUR LA RELIGION ET SUR QUELQUES AUTRES SUJETS, with publication details below and a fancy decorative flourish in the middle — Pascal – Pensées, édition de Port-Royal, 1670

In 1655 Pascal was trying to invent a perpetual motion machine, a machine that continues to operate without drawing energy from an external source. The laws of physics now say this is impossible. Naturally he failed but he ended up inventing a basic roulette wheel, now upgraded and used in casinos as a game of chance.

The Swiss computer scientist Niklaus Emil Wirth, born in 1934, named one of his programming languages Pascal in honour of Blaise. Wirth along with Helmut Weber also designed the programming language named after another mathematician, Euler. [Recommended read: Euler: The Master of Us All ]

Pascal died in extreme pain at the young age of 39. He had a malignant growth in his stomach which had spread to his brain. Like many others, such as Évariste Galois and Franz Schubert, we are left wondering what else Pascal could have achieved had he lived longer. His work with Fermat into the calculus of probabilities helped the German mathematician Gottfried Leibniz [1646-1716] develop the infinitesimal calculus. Pascal is buried in the Saint-Étienne-du-Mont church in Paris and his death mask is held at the J. Pau l Getty museum in Los Angeles, California.

Old photograph of a death mask, which looks like a plaster case of a sleeping man's face, seen from the side — The death mask of Blaise Pascal

Numbers and number patterns in Pascal’s triangle

By David Benjamin. Posted February 4, 2022

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

Triangles and fractals

If we highlight the multiples of any of the Natural numbers $\geq 2$ in Pascal’s triangle then they create a pattern of inverted triangles.

Diagrams showing Pascal's triangle, with numbers in hexagonal cells, with cells coloured as per their factors; in each of six diagrams, the multiples of 2, 3, 6, 7, 10 and 11 are shaded, and each gives a different pattern - all containing smaller triangles shaded the same colour

The images above are evocative of the Sierpinski sieve (also known as the Sierpinski gasket or Sierpinski’s triangle), a fractal described in 1915 by the Polish mathematician Waclaw Sierpiński (1882-1969).

Image of the Sierpinski sieve/triangle, in blue on a white background — The Sierpiński Sieve

Medal of Waclaw Sierpiński in the Yale University art gallery

Fractals are beautiful geometric shapes. Small, even down to (theoretically) infinitesimal areas of a fractal are identical to the entire shape. The Koch snowflake, generated geometrically by successive iterations on an equilateral triangle, is an example of a fractal. Julia sets and Mandelbrot sets are examples of fractals generated using recursion on complex functions. Many examples of fractals appear in nature, and the Polish-born French-American polymath Benoit Mandelbrot (1924-2010) suggested that fully developed turbulent flows are fractals.

It is a lovely surprise to discover that a simple fractal can be found inside Pascal’s triangle. It is achieved by considering all the numbers in the triangle modulo 2 – equivalent to colouring in only the multiples of 2, as in the first diagram at the top of the post. In this version, every odd number becomes $1$ and every even number becomes $0$, and by considering sufficiently many lines of the triangle, the Sierpinski pattern emerges.

A red Sierpinski triangle with the gaps filled in black — The areas containing the odd numbers have been shaded red and the areas containing the even numbers, black.

Pascal's triangle with numbers modulo 2; cells containing 1s are red and 0s are black, and the pattern matches that of the Sierpinski triangle — Pascal’s triangle modulo 2. The entries when concatenated can be read as binary numbers which are palindromic.

Number patterns in the triangle

If we consider the first 32 rows of the mod$(2)$ version of the triangle as binary numbers: $1, 11, 101, 1111, 10001,…$ and convert them into decimal numbers, we obtain the sequence:

\[1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, \]

\[196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, \]

\[50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295\]

Interestingly, all members of this sequence are factors of the final term, $4294967295 = 2^{32} – 1$. Since this is one less than a power of two, it’s a Mersenne number. Why the first $31$ terms are all factors of the 32nd term is difficult to summarise here but there is a thread on StackExchange discussing what happens to the pattern after the $32nd$ term.

$4294967295$ has prime factorisation $3 \times 5 \times 17 \times 257 \times 65537$. These five prime factors are Fermat numbers – numbers of the form $2^{2^{n}}+1$ – in this case with $n = 0, 1, 2, 3$ and $4$. As of the time of writing these are the only known Fermat numbers which are also prime.

These patterns in the rows of the triangle are intriguing, and my own efforts to understand them have uncovered a few other interesting discoveries – notably, that while the 32nd term is not divisible by the 33rd, the 34th term is exactly 3 times the 33rd. The pairs of terms after that seem to alternate, as they do from the start of the sequence, between a non-integer ratio and a ratio of exactly 3, which I conjecture is a pattern that will continue.

Two welcome appearances

$e$ and $\pi$ are two of the most used transcendental numbers. The Swiss mathematician Leonhard Euler (1707-1783) connected them with the most beautiful equation, called Euler’s identity:

\[e^{i\pi}+1=0\]

There are many approximations connecting $e$, $\pi$ and other irrational numbers to be found here.

In 2012 Harlan J. Brothers proved that

\[\displaystyle\lim_{n\to \infty} \frac{\ \displaystyle\frac{s_{n+1}}{s_n}\ }{\displaystyle\frac{s_n}{s_{n-1}}}=e\]

where $s_n$ is the product of the numbers on row $n$ of Pascal’s triangle. The proof can be found on Cut the Knot, part of the wonderful website of Dr Ron Knott.

In 2007 Jonas Castillo Toloza discovered a connection between $\pi$ and the reciprocals of the triangular numbers (which can be found on one of the diagonals of Pascal’s triangle) by proving

\[\pi= 2 + \frac{1}{1} + \frac{1}{3} – \frac{1}{6} – \frac{1}{10} + \frac{1}{15} + \frac{1}{21} – \frac{1}{28} – \frac{1}{36} + \frac{1}{45} + \frac{1}{55} – \ldots\]

Three proofs are given on Cut the Knot.

Harmony in the triangle

The infinite sum of the reciprocals of the Natural numbers is called the harmonic series, $H_n$, where

$H_n = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} \ldots$

The series is divergent, but it crawls its way towards infinity, and takes $15092688622113788323693563264538101449859497$ terms just to pass a total of $100$.

The harmonic series can be used to create a version of Pascal’s triangle – the series itself is placed along the two leading diagonals, and the entries are then related by each being the difference of the fraction to its left, and the one diagonally above it and to its left. For example, $\frac{1}{30} = \frac{1}{5}-\frac{1}{6}$.

A triangle like Pascal's but with fractions in place of numbers; the rows read 1/1; 1/2 1/2; 1/3 1/6 1/3; 1/4 1/12 1/12 1/4; 1/5 1/20 1/30 1/20 1/5; 1/6 1/30 1/60 1/60 1/30 1/6

Dividing the first term in the $n^{th}$ row by every other term in that row creates the $n^{th}$ row of Pascal’s triangle. The table below shows the calculations for the $5^{th}$ row:

$\frac{1}{5}$	$\frac{1}{20}$	$\frac{1}{30}$	$\frac{1}{20}$	$\frac{1}{5}$
$\frac{1}{5}\div \frac{1}{5} =1$	$\frac{1}{5}\div \frac{1}{20} =4$	$\frac{1}{5}\div \frac{1}{30} =6$	$\frac{1}{5}\div \frac{1}{20} =4$	$\frac{1}{5}\div \frac{1}{5} =1$

In our next post, we’ll talk about probability and statistics in Pascal’s triangle, and consider some of Pascal’s other contributions.

Sequences in the triangle and the fourth dimension

By David Benjamin. Posted December 19, 2021

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

Second place in a single-elimination tournament

By Peter Rowlett. Posted July 16, 2018

I made a silly joke, and it made me think.

You may be aware that our own Christian Lawson-Perfect is running the Big Internet Math-Off here at the Aperiodical, a single-elimination tournament with sixteen competitors. I was knocked out in round one by the brilliant Alison Kiddle. I joked that if Alison went on to win, then I’d be joint second.

Much as I like and respect @ch_nira, I’ll be rooting for @ajk_44. If she goes on to win the #BigMathOff final and is crowned The World’s Most Interesting Mathematician, then I’m joint-second, right? https://t.co/8Jt37gHFif

— Peter Rowlett (@peterrowlett) July 10, 2018

I’ve been mulling this over and I felt there was something there in thinking about the placement of the non-winners in such a tournament, so I had a play.

The 12 Days of Christmas and Pascal’s Triangle

By Christian Lawson-Perfect. Posted December 26, 2015

Reader Marc Chamberlain sent this video in a bit too late to get in our advent calendar, but it’s about the 12 days of Christmas so we’re still cool, right?

Open Season – Singmaster’s Conjecture

By Katie Steckles. Posted January 29, 2013

Science and maths are all about finding things out. Mathematics in particular is about making statements, and then determining their truth (or falsity). Finding a proof, or disproof, of a mathematical theory can be as simple as finding a counterexample, or it can take hundreds of authors tens of thousands of pages.

In this short series of articles, I’m going to write about some mathematical questions we don’t know the answer to – which haven’t yet been proven or disproven. Hopefully you will find it interesting, and maybe someone will even be inspired to delve deeper and find the answers themselves.