Double Maths First Thing is part of Colin’s fight against the forces of tedium.
Hello, and welcome to Double Maths First Thing! My name is Colin and I am a mathematician, on a mission to spread joy and delight in maths.
More from me
I promise not to make this whole thing about me, but if I’ve got a blog post about something I find delightful, it would be rude not to share it. Here’s a link that took me a long time to make about the relationship between the binomial expansion and the binomial distribution. The clue’s in the name, right?
New Largest Known Prime!(?)
I am decidedly ambivalent about finding larger and larger Mersenne primes. I feel like some of those involved in the hunt are in it for the money, the mersennaries. Even if it’s been six years since the last one, the announcement that there’s a new one is not one that thrills me. I think throwing more compute at the same problem is of limited use. However, it has reminded me about the Lucas-Lehmer test, which is a very nice piece of maths that happens to coincide with the structure of computers, making it efficient (although still lengthy) to calculate.
Somewhere deep in the list of tabs that seemed like a good idea to open, I found instructions for making a giant windball. It uses some sort of construction kit called makedo, but I’d be surprised if you couldn’t find some butterfly pins and spare cardboard.
I was surprised by a result, which is always a nice feeling: if you’re thinking about balls (settle down back there), you’d expect to see \( \pi \) show up. Finding \( e \) was not on my bingo card.
Stretching the theme still further, I hadn’t heard of Pappus’s centroid theorem(s), which you could use to work out the volume of a sphere (see! There is a link!) — they’re reasonably obvious once you think about them a little, but it’s still a nice way to approach surfaces and volumes of revolution.
Other nice things!
From Reddit, probably to be filed under “absurd but also very impressive”: a computer cuber broke a world record. Not just any world record, but the record for a 121-by-121-by-121 cube. By 69 hours. My understanding is that a 121-cube is just like a 5-cube, only more so — but still, the concentration and dedication you’d need to do that… chapeau! Oh, and they say this is the fifth-largest cube ever solved by a human.
Over on the platform-still-referred-to-as-Twitter-by-everyone-sensible, David K Butler has an interesting way to look at addition and multiplication using parallel and intersecting lines (respectively). I’m always up for a new thing to add to my mental models!
In podcast news, I am given to believe that Sam Hansen is at it again. I’m not sure they ever stopped, honestly; Sam and Sadie Witkowski now co-host Carry The Two, recently with a theme of elections and representation. It’s almost enough to get me to the gym so I can listen to it in peace. Almost.
And — if you’re quick about it — you might be able to subscribe to the Finite Group in time for their first anniversary livestream.
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something I should be aware of.
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.
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
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 Pascaline. An original is displayed in The Musée des Arts et Métiers in Paris, France
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.
A rare copy of the Essay pour les coniques which is kept in the National Library of France
In 1647Pascal 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.
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.
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.
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. Paul Getty museum in Los Angeles, California.
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’:
The first seven rows of Pascal’s triangle, showing some pairs and their sums below (highlighted)
