Pythagoras and his theorem

In this guest post, David Benjamin shares a cornucopia of concepts and stories relating to Pythagoras and his famous theorem.

I admit to mild irritation when I’m told that Pythagoras’ theorem is $a^2+b^2=c^2$. The theorem is based on area – in particular, that of squares. There are many proofs of the theorem and in this post we present a miscellany of Pythagorean Theorem curiosities, including some of my favourite proofs, the theorem’s links to algebra, geometry and number theory, an assassination of a president of the USA, an alleged murder in Greece, an infinite spiral of surds, Gauss and coordinate geometry – plus another connection between Pascal and Fibonacci.

The theorem

A square is added to each side of a right-angled triangle as shown in the above image. The sum of the the areas of the two smaller squares is equal to the area of the largest square. If the hypotenuse of the triangle has a length of $c$ and the other two sides are of length $a$ and $b$ then $a^2+b^2=c^2$

When the length of each side of the triangle is a positive integer, the three numbers make a Pythagorean triple. $(3, 4, 5)$ is the smallest triple with $3^2+4^2=5^2$. The Chinese text Chou Pei Suan Ching – original title Zhoubi – (周髀算经), (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) gives this visual proof for the $(3, 4, 5)$ triple.

$(3, 4, 5)$ is a primitive triple since $3, 4$ and $5$ are coprime – their only common divisor is $1$. $(n\times3, n\times4, n\times5), n = 2, 3, 4,…$ are part of the same ‘family’ and clearly not primitive triples. Another primitive triple is $(5, 12, 13)$ and an ordered sequence of hypotenuses for such triples are listed here. In the sequence I was surprised to see $185$ appearing twice. In fact there are exactly four distinct triples with hypotenuse $185$. With the aid of a spreadsheet, I was able to find them: $(57, 176, 185), (60, 175, 185),(104, 153, 185)$ and $(111, 148, 185)$

Generating Pythagorean triples

The sequence $\frac{1}{1},\frac{3}{2},\frac{7}{5},\frac{17}{12},\frac{41}{29},\frac{99}{70},\frac{239}{169},\frac{577}{408},\frac{1393}{985},…$ produces a Pythagorean triple from every other term beginning with $\frac{7}{5}$:

$\frac{7}{5}=\frac{3+4}{5} \rightarrow 3^2 + 4^2 = 5^2$

$\frac{41}{29}=\frac{20+21}{29} \rightarrow 20^2 + 21^2 = 29^2$

$\frac{239}{169}=\frac{119+120}{169} \rightarrow 119^2 + 120^2 = 169^2$

$\frac{1393}{985}=\frac{696+697}{985} \rightarrow 696^2 + 697^2 = 985^2$

As an added bonus, the decimal equivalent of each term of the sequence converges to $\sqrt2$, in a similar way the Fibonacci sequence converges to the golden ratio $\psi=\frac{1+\sqrt5}{2}$

Another method to find Pythagorean triples uses consecutive even numbers and the sum of their reciprocals as shown below.

Choose any pair of consecutive even numbers:

$8$ and $10$ $\rightarrow \frac{1}{8} + \frac{1}{10} = \frac{9}{40} \rightarrow 9^2 + 40^2 = 1681 = 41^2$

$12$ and $14$ $\rightarrow \frac{1}{12} + \frac{1}{14} = \frac{13}{84} \rightarrow 13^2 + 84^2 = 7225 = 85^2$

$20$ and $22$ $\rightarrow \frac{1}{20} + \frac{1}{22} = \frac{21}{220} \rightarrow 21^2 + 220^2 = 48841 = 221^2$

Euclid of Alexandria (325BC – 265BC) was a Greek mathematician who wrote a treatise, The Elements – a collection of 13 books. Books 1 to 6 are on plane geometry and books 7 to 9 on number theory. Euclid created a formula for generating Pythagorean triples from any pair of positive integers $m$ and $n$, where $m>n$:

($m^2-n^2, 2mn, m^2+n^2$) is the triple.

If $m=7$ and $n=3$, the triple is ($40, 42, 58$) $\rightarrow 40^2 + 42^2 = 3364 = 58^2$

If $m=89$ and $n=11$, the triple is ($7800, 1958, 8042$) $\rightarrow 7800^2 + 1958^2 = 64673764 = 8042^2$

A lovely link between Pascal, Fibonacci, Euclid and Pythagoras comes via any four consecutive Fibonacci numbers

Using $3, 5, 8, 13$

• Multiply the first and the last numbers $\rightarrow 3 \times{13}=39$
• Multiply then double the middle two numbers $\rightarrow 5 \times{8}\times{2}=80$
• Sum the squares of the middle two numbers $\rightarrow 5^{2}+8^{2}=89$
• The Pythagorean triple is $(39, 80, 89) \rightarrow 39^{2}+80^{2}=7921=89^{2}$

$F_{1}$ to $F_{4}$ gives the primitive triple $(3, 4, 5)$

$F_{2}$ to $F_{5}$ gives the primitive triple $(5, 12, 13)$

$F_{3}$ to $F_{6}$ gives the triple $(16, 30, 34)$

$F_{7}$ to $F_{10}$ gives the primitive triple $(715, 1428, 1597)$

In addition, for any set of four consecutive Fibonacci numbers, $F_{7}$ to $F_{10}$ for example, the following connection is true

$7+10=17$ and the $17^{th}$ Fibonacci number is $1597$, the third member of the triple, the hypotenuse of the triangle!

Amazingly, if we use $m$ and $n$ as consecutive Fibonacci numbers when using Euclid’s method, then the last number of the triple is again a Fibonacci number

A visual proof of the theorem

I first came across a visual proof of Pythagoras’ theorem for all right-angled triangles in Roger B. Nelsen’s wonderful book Proofs Without Words, Exercises in Visual Thinking. Nelsen noted the proof (author unknown, circa B.C. 200?) is adapted from the Chou Pei Suan Ching. The two images below combine to show the proof:

A proof by trapezium area and an untimely painful death

James Garfield (November 19, 1831 – September 19, 1881) was elected as the United States’ 20th President in 1880. He was assassinated after just 200 days in office after being shot on July 2, 1881, in a Washington railroad station. Garfield remained mortally wounded in the White House for many weeks where Alexander Graham Bell, inventor of the telephone, attempted to locate the bullet with an induction-balance electrical device which he had designed. Bell and physicians were unsuccessful in their attempts and Garfield died from an infection and an internal haemorrhage on September 19, 1881.

In 1876, Garfield had an elegant proof of Pythagoras’ theorem published. The proof makes use of the formulae for the areas of a triangle and a trapezium. The proof is demonstrated in the image below

A proof using the Shoelace formula

The brilliant German mathematician Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) developed a formula to calculate the area of a polygon if every vertex of the polygon lies on a known Cartesian coordinate. The formula is widely known as the Shoelace formula and makes use of the calculation to find the determinant of a 2 by 2 matrix.

John Molokach observed that the Pythagorean theorem follows from Gauss’ Shoelace Formula, as shown below

It’s not just squares

If the same regular $n$-gon, $n$ = 3, 4, 5,.. is drawn on each side of a right-angled triangle, then the sum of the areas of the two smaller $n$gons equals the area of the $n$gon on the hypotenuse. Semicircles also produce the same result. As $n\rightarrow \infty$, a regular $n$-gon approaches a circle and so circles, where the sides of the triangle act as tangents to the circles can be said to satisfy Pythagoras’ theorem. The same result can be obtained by rotating the semicircles through $180^ \circ$ and adding matching semicircles.

The spiral of Theodorus of Cyrene

When $n$ is not a square number, $\sqrt{n}$ is called a surd

Pythagoras and his followers, the Pythagoreans, believed that the universe can be explained by whole numbers and the ratio of whole numbers. Their moto – “All is number” – was carved above the entrance of their meeting place. However, a spanner was thrown in the works when Hippasus of Metapontum, one of the Pythagoreans suggested that $\sqrt2$, the length of the hypotenuse of the right-angled triangle with sides $1, 1$ and $\sqrt2$, could not be written as the ratio of two whole numbers. Such numbers are now called irrational and it was approximately 200 years before Euclid of Alexandria proved $\sqrt2$ was irrational. The Pythagoreans were sworn to secrecy and one legend suggests that Hippasus was thrown off a boat and drowned for revealing his discovery to non-Pythagoreans. Then again, some believe Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The ($1, 1$,$\sqrt2$) triangle is the start of the spiral of Theodorus. Further right-angled triangles, each containing a side of length $1$ unit, are added sequentially to produce the sequence $\sqrt2, \sqrt3, \sqrt4, \sqrt5,…$. The sequence is the length of each new hypotenuse, as shown in the diagram below. Theodorus probably stopped at $\sqrt17$ as it the length of the hypotenuse of the triangle before the triangles begin to overlap. The spiral can be expanded here.

Aperiodical News Roundup – December 2021

Here’s a roundup of news stories from December 2021 that we didn’t cover at the time.

Maths results

Firstly, some nice news of a proof of a result on the density of unit fractions – a set of integers of positive density must contain distinct $n_1,\dots,n_k$ such that $\frac{1}{n_1}+\ldots+\frac{1}{n_k}=1$. (via Thomas Bloom)

According to this post on Gil Kalai’s blog, Ringel’s circle problem has been solved. The problem states:

Consider a finite family of circles such that every point in the plane is included in at most two circles. What is the minimum number of colors needed to color the circles so that tangent circles are colored with different colors?

Turns out, you might need all the colours – the authors of a new ArXiV paper have found ways to construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number.

Plan 28, a project aiming to collect and understand Babbage’s notes about the analytical engine (and possibly finish building it) has issued a statement to the effect that they now think they understand all of the designs – an exciting step forward.

We have for the first time both an aerial view that integrates partial and seemingly unrelated developments, as well as the most detailed analysis yet of the specifics of implementation.

The group are hoping to be able to rewrite these notes into a format that can be used to build a physical implementation of the machine, as Babbage’s original notes didn’t include a design for a complete engine, and the work so far has taken five years. This is exactly the kind of unnecessary nerdery I love to see.

Prizes

Per Nalini Joshi on Twitter, Serena Dipierro has been awarded the Australian Mathematics Society medal for 2021, which is given within 15 years of the award of someone’s PhD for distinguished research in the mathematical sciences. According to the AustMS citation,

Professor Serena Dipierro (University of Western Australia) has made outstanding contributions to the area of analysis and PDEs, with a special focus on the theory of nonlocal operators and free boundary problems. She is a prolific researcher with a large international network of collaborators and has become one of the leaders of her field. In the nine years since the award of her PhD, her publications have amassed over 1100 citations in the MathSciNet database; since moving to Australia in 2016 she has averaged one publication per month, including many in journals of the highest quality.

According to a blog post by Gil Kalai, Richard Stanley has won the Leroy P. Steele prize, awarded annually by the AMS for distinguished research work and writing in mathematics. According to the announcement,

Stanley has revolutionized enumerative combinatorics, revealing deep connections with other branches of mathematics, such as commutative algebra, topology, algebraic geometry, probability, convex geometry, and representation theory. In doing so, he solved important longstanding combinatorial problems, often reinvigorating these other fields with new combinatorial methods. Through his outstanding research; excellent expository works; and many PhD students, collaborators and colleagues, he continues to influence the field of combinatorics worldwide.

In early December, the European Mathematical Society announced that Jacques Tits (pictured left) has died.

Jacques Tits was a highly influential group-theorist, proving the celebrated “Tits Alternative” (that every finitely generated linear group either has a solvable subgroup of finite index or contains a free subgroup of rank 2). Probably his most important contribution was the development of group-theoretic “Buildings”, a profound unifying idea which has subsequently had deep applications in diverse mathematical fields.

Following the publication of a fairly painful article in The Times just before Christmas entitled ‘Phwoar! Look at the vital statistics on these lads’ and listing the apparently increasingly attractive, and exclusively male, mathematicians and statisticians responsible for ‘crunching the data’ on the pandemic, the i newspaper published this excellent response pointing out the shocking news that some mathematicians who aren’t men have also been involved, and highlighting some of the top data experts who’ve been looking after us all with maths. The Times article includes a quote from “maths professor and author Hannah Fry — a woman” (that is literally actually what it says) who had correctly expressed on Twitter that mathematicians are hot – but I’m pretty sure she meant all of us and not just men.

Speaking of bad opinion pieces, what better way to sum up the year than this collection of terrible maths takes? Highlights include ‘How does misogyny impede a mathematician of doing a good job?’ [sic] and the wonderful ‘Physics is not math.’

The American Mathematical Society has cancelled this year’s Joint Mathematics Meetings, scheduled to take place in Seattle on 5-8 Jan, and will be refunding tickets and organising an online event instead. Unfortunately, they initially failed to notify attendees of this by email, and many found out via Twitter.

The AMS also announced in mid-December that they were shutting down all their blogs with two weeks’ notice. The AMS Blogs site has been replaced with an archive collecting all the past posts, but those who used it as a regular blogging outlet will have to find somewhere else to do that. (Hi!)

And finally

Dynamic geometry powerhouse Geogebra has been bought by an online tutoring company called BYJU’S, run by a group of former maths teachers from India. They’ve stated that all current employees, contracts, agreements and software licenses will remain in place, and the software and online resources will continue to be free to use. (via Geogebra on twitter)

PROMYS Europe is a programme designed to encourage mathematically ambitious secondary school students to explore the creative world of mathematics. Competitively selected pre-university students from around Europe gather at Wadham College, Oxford for six weeks of rigorous mathematical activity. This summer it will run from 10th July – 20th August, and applications open on 11th January.

Gathering 4 Gardner’s 2022 Wall Calendar is now available to download and print, and some print copies are also available. Including important dates of huge mathematical significance (my birthday, among others) and a selection of bios, sketches, photos and puzzles any maths fan would enjoy, it’s the perfect solution if you forgot to get a calendar and like maths.

The Geometry Center videos, which brought brought concepts from geometric topology to general audiences through computer-generated visualisation in the early 1990s, have been remastered and are available for free. (via Robin Houston)

If you spot something you think should be in a future Aperiodical News Roundup, send it our way!

The Sound of Proof

Marcus du Sautoy has tweeted about a mathematics and music project he’s involved in, called The Sound of Proof. Five classical proofs from Euclid’s Elements have been interpreted by composer Jamie Perera into musical pieces, and they’ve put together an app/game to see if you can work out which one corresponds to which.

They’ll be announcing the results at an event as part of Manchester Science Festival in October. The project is a collaboration with PRiSM, the research arm of the Royal Northern College of Music in Manchester.

The Sound of Proof, at RNCM PRiSM

Just how big is a big proof?

With news that a recent proof of the Boolean Pythagorean Triples Theorem is the ‘largest proof ever’, we collect and run-down some of the biggest, baddest, proofiest chunks of monster maths.

Theorem: every 5-connected non-planar graph contains a subdivision of $K_5$.

The above statement, conjectured independently by Alexander Kelmans and Paul Seymour in the 70s, is very easy to say. And the video below, starring Dawei He, Yan Wang, and Xingxing Yu, makes it look very easy to prove:

It’s like they got Wes Anderson to film an academic PR video. In the normally uninspiring world of maths press releases, it’s quite refreshing. And the written press release is pretty snappy, too. Let’s not make this a thing, though.

Not mentioned on The Aperiodical, March 2016

There’s been a lot of maths news this month, but we’ve all been too busy to keep up with it. So, in case you missed anything, here’s a summary of the biggest stories this month. We’ve got two new facts about primes, the best way of packing spheres in lots of dimensions, and the ongoing debate about the place of maths in society, as well as the place of society in maths.

A surprisingly simple pattern in the primes

Kannan Soundararajan and Robert Lemke Oliver have noticed that the last digits of adjacent prime numbers aren’t uniformly distributed – if one prime ends in a 1, for example, the next prime number is less likely to end in a 1 than another odd digit. Top maths journos Evelyn Lamb and Erica Klarreich have both written very accessible pieces about this, in the Nature blog and Quanta magazine, respectively.

Oliver and Soundararajan’s paper on the discovery is titled “Unexpected biases in the distribution of consecutive primes”.

László Babai reckons he can decide if two graphs are isomorphic in quasipolynomial time

László Babai in Chicago. Photo by Gabe Gaster, used with permission.

We’ve been slow to cover this, but if this week has taught us anything, it’s that taking your time over Important Maths News is always a good idea.

A couple of weeks ago, rumours started circulating around the cooler parts of the internet that László “Laci” Babai had come up with an algorithm to decide if two graphs are isomorphic in quasipolynomial time. A trio of mathematicians including Tim Gowers were on BBC Radio 4’s In Our Time discussing P vs NP while these rumours were circulating and made a big impression on Melvyn Bragg as they talked so excitedly about the prospect of something big being announced.

If Babai had done what the rumours were saying, this would be a huge advance – graph isomorphism is known to be an NP problem, so each step closer to a polynomial-time algorithm raises the P=NP excite-o-meter another notch.
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