Until now!

Each match will pit two interesting maths things against each other. The mathematician who gives the most interesting things in each group, as decided by you, goes on to share another fun maths thing in the next round. In order to make the whole thing hang together, we’re going to call the person who wins ** The World’s Most Interesting Mathematician (2024)***.

** of the 16 people I contacted, who were available in July, and wanted to take part.*

I’ve asked the competitors to come up with maths topics they find interesting. I don’t need new things, or things that they came up with – just the kind of thing that you’d tell a fun maths friend about when you bump into them.

The tournament will start on the 1st of July. Each match will be a post here on The Aperiodical, where the two competitors will each make a pitch for something they find interesting. At the end there’ll be a poll where you can vote for the thing you found most interesting. Each poll will be open for 24 hours, and then the person with the most votes will be victorious in that match and continue to the next round.

Without further ado, here are the charming people who will be trying to out-maths each other to victory this year, in random order:

**Angela Tabiri** is a mathematician and youth mentoring in STEM expert from Ghana. She is the founder of Femafricmaths, a non profit organisation that promotes female African mathematicians to highlight the diversity in careers after a degree in mathematics. You can follow Femafricmaths on YouTube, Instagram, Facebook and X.

**Sam Kay** is a maths student at Durham University where he hosts the Chalkboard Ultra podcast and spends too much time thinking about spinors. Outside of maths, Sam runs one of the university’s jazz bands. You can follow him on X, and Chalkboard Ultra on X and Instagram.

**K.P. Hart** is a mathematician, general topologist, and occasional set theorist; he also writes for the Dutch math journal Pythagoras. You can follow him on Mathstodon, Bluesky, YouTube, or look at his homepage.

**Tom Edgar** is a math professor and the outgoing editor of Math Horizons. He enjoys thinking about and animating so-called “proofs without words.” You can follow him on YouTube and Instagram, or look at his homepage.

**Katie Steckles** is a mathematician based in Manchester, who gives talks and workshops and writes about mathematics. She finished her PhD in 2011, and since then has talked about maths at universities, schools events, festivals, on BBC radio and TV, in books and on the internet. You can find her on Mathstodon, Instagram, and as part of The Finite Group (and here on The Aperiodical).

**Howie Hua** teaches math to future elementary school teachers at Fresno State. He also likes to make math explainer videos and math memes. You can find all his socials on his linktree.

**Mats Vermeeren** is a Research Fellow and Lecturer at Loughborough University, UK. He is a Dutch-speaking Belgian. Dutch is one of the few languages in which the word for “maths” does not derive from the Greek “máthèma”, so his parents had no idea what they predestined him for. You can follow him on YouTube and Mathstodon, or look at his homepage.

**Dave Richeson** is a professor of mathematics and the John J. & Ann Curley Faculty Chair in the Liberal Arts at Dickinson College in Carlisle, Pennsylvania, USA, and is the author of Euler’s Gem (Princeton University Press, 2008) and Tales of Impossibility (Princeton University Press, 2019). You can follow him on Mathstodon and X, or look at his homepage.

**Max Hughes** is the coordinator of MathsCity Leeds, who spends their free time playing table-top roleplaying games and reading comic books, whilst being engaged with fun mathsy projects on the side. You can follow them on Instagram.

**Fran Herr** is a PhD student in mathematics at the University of Chicago. She studies low dimensional topology and geometric group theory. You can follow her on YouTube and X.

**Ayliean** (noun): Mathsy, arty, crochet crafty, origami, activist, zine author known for making badges and trouble. You can follow them on YouTube, as @Ayliean on all social media, or look at their homepage.

**Kit Yates** is an author, communicator and academic mathematical biologist who is interested in sharing stories about the places where maths can impact our lives without us even realising it. You can follow him on Mastodon and X, or look at his homepage.

**Benjamin Dickman** is a mathematics teacher at a girls’ day school in New York City and a two-time Fulbrighter. His doctoral dissertation was on creativity and problem posing with the times table. He created the word game FiddleBrix! You can find him on X and Bluesky.

**Matt Peperell** is a London-based recreational mathematician living a double-life as a software developer. Outside of work and when not doing maths he likes bellringing and playing board games. You can follow him on Mathstodon.

**Matt Enlow** teaches mathematics at the Dana Hall School in Wellesley, MA. You can follow him on X, BlueSky and Mathstodon.

**Fran Watson** is a teacher and communicator of mathematics originally hailing from Cornwall but now living in Cambridgeshire (by way of Cardiff in between – locations today brought to you by the letter C!) She loves puzzles, origami, games and musical theatre and will endeavour to weave these passions into her pitches.

You’ve got $2 \times (2^4-1) = 30$ bits of fun maths to look forward to over the next month. I’m sure there’ll be some old favourites, and plenty of stuff you’ve never heard of – looking at the list of things that are going to come up, there were a fair few things I’d never seen before.

Of course, no Summer knock-out tournament would be complete without a wall-chart to print out and follow along at home, so I’ve made one:

Here’s the full tournament schedule in text form:

Date | Match | Mathematician 1 | Mathematician 2 |
---|---|---|---|

2024-07-01 | Match 1 | Katie Steckles | Benjamin Dickman |

2024-07-02 | Match 2 | Angela Tabiri | Max Hughes |

2024-07-03 | Match 3 | Matt Enlow | Sam Kay |

2024-07-04 | Match 4 | Mats Vermeeren | Howie Hua |

2024-07-05 | Match 5 | Matt Peperell | Fran Watson |

2024-07-06 | Match 6 | Ayliean MacDonald | Klaas Pieter Hart |

2024-07-07 | Match 7 | Fran Herr | Tom Edgar |

2024-07-08 | Match 8 | Dave Richeson | Kit Yates |

2024-07-10 | Quarter-final 1 | ||

2024-07-11 | Quarter-final 2 | ||

2024-07-12 | Quarter-final 3 | ||

2024-07-13 | Quarter-final 4 | ||

2024-07-17 | Semi-final 1 | ||

2024-07-18 | Semi-final 2 | ||

2024-07-23 | Final |

The mathematicians are relying on your support to carry them all the way to the title of World’s Most Interesting Mathematican (2024, of the 16 people I contacted who were available in July and wanted to take part).

Follow along on social media – we’ll be tooting at @aperiodical@mathstodon.xyz and the competitors will be posting on their own channels. And please post your ideas for interesting bits of maths with the hashtag #BigMathOff.

]]>*Love Triangle* is available wherever good books etc. are, from June 20th – and signed preorders are available from Maths Gear

Yeah yeah yeah. Let **ABC** be a triangle with area \( \Delta \), side lengths \( a\), \(b\) and \(c\), and semiperimeter \(s = \frac{a+b+c}{2}\).

Of Matt’s many complaints about our poor first-century-CE friend’s contribution to the literature, one did strike me as valid: there isn’t a nice proof of it. You can make like Hero himself and do some jiggerypokery with a degenerate cyclic quadrilateral, you can follow Sir Isaac Newton’s lead and wrangle the algebra, or you can trigonometry yourself to death.

This surprised me. Surely a nice symmetric formula should have a nice symmetric proof? According to the Wikipedia citations, I’m not the only person to think that. It was also something that interested John Horton Conway — a clue that led me to a lovely proof.

Once upon a time, back in the heady days when Christian had an occasional few minutes of free time, this esteemed organ ran a Big Math[s]-off. One of my few successful submissions was a proof without words of Conway’s Circle Theorem. It was a very nice proof without words; however, I now have a better and clearer one.

Conway’s Circle Theorem states:

Given triangle

ABC, extend sidesABandCBthroughBby a length equal to sideAC. Similarly, extend sidesBAandCAand throughAand sidesACandBCthroughCto define the six Conway points. Then the ends of these sides lie on a circle (the Conway Circle) which is concentric with the incircle ofABC.

I mean, what?! I’ve proved this several ways and it still seems absurdly neat.

The simplest proof I have looks like this:

The radii of the incircle that reach the edges, and the segments connecting the incentre to the vertices, naturally divide triangle **ABC** into six triangles in three congruent pairs. These can be rearranged to make a rectangle as shown; the most distant point on the rectangle is a Conway point.

You can do the same thing either way from any of the three inradii, giving you the six Conway points; all of the rectangles are congruent, so their diagonals have the same length. In other words, all six Conway points are the same distance from the incentre, so they lie on a circle that’s concentric with it \( \blacksquare \)

That’s tidy and all, but what’s *really* interesting is that the rectangle shown has the same area as triangle **ABC** does — it’s made up of the same six triangles. One of the rectangle’s edges is equal to **ABC**‘s inradius and the other is equal to its semiperimeter. And the only other place I’ve ever seen a semiperimeter mentioned is in Heron’s formula.

We’re currently simultaneously close but some way off: we can see that \( \Delta = rs\) — but there’s no immediate sign of \( (s-a) \) and its friends. They’re there if you look closely, though: each of the six triangles has a leg of the right form. For example, the red triangle has one leg of length \( r \) and the other of length \( s-c \).

Unfortunately, multiplying them together is non-trivial and is going to take a little work.

You’ll also see a lot of dry, technical alphabet soup that I’ve tried to avoid here — think of this as a somewhat coherent argument rather than a proof.

For example, the first time I noticed the semiperimeter thing, I thought “I just need to prove that \( r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}} \).” I then realised how much heavy lifting that “just” was doing and set it aside until spurred back into action by a Parker diatribe.

I spotted there was another way: if I could show that the base of the rectangle was equal to \( \frac{(s-a)(s-b)(s-c)}{r} \), then I could multiply the two areas together to get \( \Delta^2 \). So that’s what I did.

We’re going to need some letters, I’m afraid.

This is the same triangle **ABC** as before, with the same triangles, but it’s helpful to label things. Call the inradius \( r \). The half-angles at **A**, **B** and **C** are \( T \), \( U \) and \( V \) (respectively), and the segments linking each to the incircle have lengths \( rt \), \( ru \) and \( rv \) — it makes the algebra simpler later to keep a factor of \( r \) in there. Lastly, the segment linking **A** to the incentre has length \( rt’ \).

The main idea of this proof is that you can multiply lengths together by scaling triangles.

Start by building a triangle similar to a blue triangle (the ones containing point **A**) and match its “\( r \)” leg to the “\( rv \)” leg of a pink triangle (containing **C**).

This scales the blue triangle by a factor of \( v \), so the new blue triangle’s other leg has length \( rtv \). Its hypotenuse has length \( rt’v \), but we won’t need that until later on.

Now I’m going to scale a yellow triangle (containing **B**) so its “\( r \)” leg matches the “\(rtv\)” leg of the new blue triangle. Because it’s scaled by a factor of \( tv \), its other leg has length \( rtuv \).

A couple of claims here: firstly and easily, the angle between the blue and yellow hypotenuses is \( V \) — this must be the case because \( 2T + 2U + 2V = \pi \) and the non-right-angles of our yellow monster add up to \( \frac{\pi}{2} \). Secondly, I claim that the apex of the yellow triangle (marked **P**) is a Conway point. That takes a bit more justification, and two more triangles.

If I drop a perpendicular to the blue hypotenuse at **C**, I get a triangle that’s similar to the pink triangle. Its scale factor is \( \frac{t’}{v} \), and the new leg has length \( rt’ \).

What about the *other* triangle, with **CP** as an edge? That has the same angles as the triangle connecting **A** and **B** with the incentre — and it has a pair of corresponding sides the same length, so the two are congruent. In particular, it means that **CP** has length \( rt + ru = c \). A point on **BC** extended, a distance of \( c \) away? That’s a Conway point.

The rectangle from Conway’s Circle Theorem earlier had area \( rs \), so we can write \( \Delta = r^2(t+u+v) \)

But we also just worked out that the yellow triangle’s leg — which corresponds to the “\( s \)” edge of the rectangle — has length \( rtuv \), so we can write the rectangle area as \( \Delta = r^2 tuv \).

Multiplying the two \( \Delta \) equations together gives something I’m going to arrange as \( \Delta^2 = r(t+u+v)(rt)(ru)(rv) \).

Meanwhile, you can remember from earlier that \( rt = s-a \), \( ru = s-b \) and \( rv = s-c \) and out jumps \( \Delta^2 = s(s-a)(s-b)(s-c) \), which is Heron’s formula \( \blacksquare \)

Just because I wondered and looked it up: Euclid flourished about as long before Hero did as Newton did before me. Assuming that writing for the Aperiodical is what counts as flourishing.

I find it a little hard to believe that nobody in the last couple of millennia has stumbled on something akin to this, but in any case, my approach to figuring things out is similar to the fabulous Moose Allain‘s approach to jokes (I paraphrase, and I speak only for me): it doesn’t *matter* if someone else thought of it first, you’re still allowed to take joy from thinking it up yourself, and you’re still allowed to share it.

This proof brought me great joy. And maybe it’ll help Matt see another side of Heron’s formula.

]]>Thomas Hales and Koundinya Vajjha have claimed a proof of Mahler’s first conjecture, that the most unpackable centrally symmetric convex disk in the plane is a smoothed polygon. *(via Greg Egan)*

There’s also a been a proof of the geometric Langlands conjecture published, as outlined in this New Scientist article.

Zhouli Xu has claimed a proof of the Kervaire invariant one problem in dimension 126. *(via Kyle Ormsby)*

And finally, Hidetoshi Mino has counted all the magic squares of order 6. Up to rotations and reflections, there are 17,753,889,197,660,635,632. *(via Walter Trump)*

The inaugural Jean-Pierre Demailly Prize for Open Science in Mathematics has been awarded to zbMath Open, “for its broad scope, recent policy changes, and commitment to accessibility and sustainability”. *(via the European Mathematical Society)*

It’s been announced that the first President of the newly-formed Academy for the Mathematical Sciences (AcadMathSci) will be Professor Alison Etheridge OBE FRS, a professor in Probability at the University of Oxford, and a world expert on stochastic processes and their applications. She will take up the role on 17 June 2024.

The Shaw Prize in Mathematical Sciences 2024 has been awarded to Peter Sarnak, “for his development of the arithmetic theory of thin groups and the affine sieve, by bringing together number theory, analysis, combinatorics, dynamics, geometry and spectral theory.” *(via Paysages Mathématiques)*

“Des chiffres et des lettres”, the French gameshow on which Countdown is based, has been cancelled after more than 50 years. *(via Sarah Dal)*

The UK Government has issued a call for £6m funding to set up a National Academy focused on Mathematical Sciences (NAM). Confusingly, this isn’t the same thing as the fledgling Academy for the Mathematical Sciences (AcadMathSci), though AcadMathSci may well bid to become the NAM. Clear?

And sadly, award-winning mathematician and co-founder of the Simons Foundation Jim Simons has died. *(via Alberto Ramos)*

- Philippa Bonay, Director, Operations, Office for National Statistics. Appointed OBE for Public and Charitable Services.
- Anne Davis, Professor of Mathematical Physics, University of Cambridge. Appointed OBE for services to Higher Education and to Scientific Research.
- Paul Fannon, Fellow, Christ’s College, Cambridge, and Volunteer, United Kingdom Mathematics Trust. Appointed OBE for services to Education.
- Ian Hall, Professor of Mathematical Epidemiology and Statistics, University of Manchester and Senior Principal Modeller, UK Health Security Agency. Appointed OBE for services to Public Health, to Epidemiology and to Adult Social Care, particularly during Covid-19.
- David Marshall, Lately Director of Census, Northern Ireland Statistics and Research Agency (now Northern Ireland chief electoral officer). Appointed OBE for services to Official Statistics and Census-taking in Northern Ireland.
- Bruno Reddy, Founder and Chief Executive Officer, Maths Circle, Ampthill, Bedfordshire, and creator of Times Tables Rock Stars. Appointed OBE for services to Education.
- Sam Rose, Deputy Director, Data and Analysis Division, Department for Transport. Appointed OBE for services to Advanced Analytics
- Matthew Woollard, Professor of Data Policy and Governance, UK Data Archive, University of Essex. Appointed OBE for services to Data Science
- George McMath, Lately Deputy Principal, Northern Ireland Statistics and Research Agency, Northern Ireland Civil Service. Appointed MBE for services to the Northern Ireland Census.

Get the full list from gov.uk. Spot anyone we’ve missed? Let us know in the comments.

]]>The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>\[ f(x) = x^2 + x + 41\text{.} \]

Using this, \(f(0)=41\), which is prime. \(f(1)=43\), which is also prime. \(f(2)=47\) is another prime. In fact this sequence of primes continues for an incredible forty integer inputs until \(f(40)=41^2\). It might generate more primes for higher inputs, but what’s interesting here is the uninterrupted sequence of forty primes.

This got me wondering. Clearly \(f(0)\) is prime because 41 is prime, so that much will work for any function

\[ f(x) = x^2 + x + p \]

for prime \(p\), since \(f(0)=0^2+0+p=p\). Are there other values of \(p\) that generate a sequence of primes? Are there any values of \(p\) that generate longer sequences of primes?

I wrote some code to investigate this. Lately, I’ve taken to writing C++ when I need a bit of code, for practice, so I wrote this in C++.

I figured the cases where \(f(0)\) is prime but \(f(1)\) isn’t weren’t that interesting, since \(f(0)\) is trivially prime. In fact, \(f(x)=x g(x)+p=p\) when \(x=0\) for any prime \(p\), but saying so doesn’t seem worth the effort.

So I kept track of the primes \(p\) whose functions \(f(x)=x^2+x+p\) generate more than one prime, and the lengths of the sequences of primes generated by each of these. This produced a pair of integer sequences.

I put the primes that work into the OEIS and saw that I had generated a list of the smaller twin in each pair of twin primes. I was momentarily spooked by this, until I realised it was obvious. Since \(f(0)=p\) and \(f(1)=1^2+1+p=p+2\), any prime this works for will generate at least a twin prime pair \(p,p+2\).

What about the lengths of the sequences of consecutive primes generated? The table below shows the sequences of consecutive primes generated for small values of \(p\). Most primes that generate a sequence produce just two, and \(p=41\) definitely stands out by generating forty.

\(p\) | \(f(x)\) | Primes generated | Number of consecutive primes generated |

3 | \(x^2+x+3\) | 3, 5 | 2 |

5 | \(x^2+x+5\) | 5, 7, 11, 17 | 4 |

11 | \(x^2+x+11\) | 11, 13, 17, 23, 31, 41, 53, 67, 83, 101 | 10 |

17 | \(x^2+x+17\) | 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257 | 16 |

29 | \(x^2+x+29\) | 29, 31 | 2 |

I was pleased to see this sequence of lengths of primes generated was not in the OEIS. So I submitted it, and it is now, along with the code I wrote. (I discovered along the way that the version where sequences of length one are included was already in the database.)

Anyway, I amused myself by having some C++ code published, and by citing Euler in a mathematical work. Enjoy: A371896.

]]>The 2024 Abel Prize has been awarded to Michel Talagrand, “for his groundbreaking contributions to probability theory and functional analysis, with outstanding applications in mathematical physics and statistics.”

This year’s Turing Award has been given to Avi Wigderson, “for foundational contributions to the theory of computation, including reshaping our understanding of the role of randomness in computation, and for his decades of intellectual leadership in theoretical computer science.” Widgerson is a previous recipient of the Abel Prize.

Nature magazine reports the discovery of a natural metabolic enzyme capable of forming Sierpiński triangles. Fractals are everywhere!

Quine’s New Foundations for set theory, in which the axiom of choice is false, has been formally proved in Lean to be consistent (PDF).

Another unreasonably effective application of maths: knot theory can be used to reveal points where spacecraft can switch between intersecting orbits using minimal fuel.

And finally, there have unfortunately been two deaths in maths education. First, maths education stalwart and generally lovely person Sue de Pomerai has died. Sue worked at MEI, FMSP and AMSP, and made a huge contribution to maths promotion in the UK. Also Hugh Burkhardt, pioneering mathematics education researcher and former Director of the Shell Centre for Mathematical Education.

]]>The next issue of the Carnival of Mathematics, rounding up blog posts from the month of April 2024, is now online at Ioanna Georgiou’s blog. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>I remember the days well when I first learnt about loci and constructions – what a wonderful thing. Granted, I love doing them now; to be able to appreciate how Euclid developed his incredible proofs on geometry.

In school, it was a slightly different story. Whilst I was meant to be constructing triangles and drawing a locus of a point, the school-supplied compasses had other ideas – slipping around unhelpfully, making them useless for the task. Understandably, I’d often put down the compass, chat with my friends, and sneakily eat crisps when the teacher wasn’t looking.

A lot has changed since then; I now have a working compass. But more than that, I have become a mathematician, mathematics teacher and all round mathematics nerd! I don’t think a day goes by where I have not been involved in some mathematical activity. My favourite on the weekend is working out how long it will be till my takeaway arrives! But I digress. Writing is a passion of mine as it allows me to share the facts and curiosities of this truly wonderful subject.

In this article I aim to share some key ideas that allowed me to develop my interest in geometry, and to appreciate the wonders hidden in plain, or rather, plane sight. Enjoy some of my favourite facts about shapes and lines, while I finish my packet of salt and vinegar crisps.

One of the first things I learnt at school was that angles in a triangle sum to 180 degrees. But I never knew why, or saw a proof. Proof is vital in mathematics as proof allows us to confirm theories and conjectures that can help us progress our mathematical knowledge. If something is proved, then we can always assume it to be true!

I first came across this proof when I was researching different ways to teach angles in parallel lines. We will therefore be using proven facts on angles in parallel lines in this proof.

The first proof I saw of the angle sum was one using parallel lines. We start off with a triangle whose interior angles are all different (called a **scalene triangle**). In order to write an equation including all three angles, we first draw two lines, both parallel to the base of the triangle:

The lines \(a\) and \(b\), which are two sides of the triangle, touch both parallel lines – we say they are **transversal lines**. We can use some existing results about angles and parallel lines to make some deductions:

\(\angle A = \angle \alpha\) (‘alternate angles are equal’).

\(\angle B = \angle \beta \) (‘alternate angles are equal’).

\( \angle \alpha + \angle C + \angle \beta = 180^{\circ} \) (‘angles on straight lines add up to 180 degrees’)

\( \Rightarrow \angle A + \angle B + \angle C = 180^{\circ} \) QED

Here, we were able to use the fact that alternate angles are equal, as this is a proven fact. This can save time when we’re proving something that can build on existing mathematical theories.

Here we have a triangle with two sides of the same length. We want to show that if this is the case, then the two base angles are equal. This proof is relatively simple, and relies on a powerful tool in geometry!

We bisect the angle at \(C\), which, in this case, will intersect at the midpoint of \(AB\) (which we denote \(c\)), as the sides \(a\) and \(b\) are the same length. It follows then, using one other existing result, that:

\( \Delta BCc \simeq \Delta ACc\) (by ‘side-angle-side’)

\( \Rightarrow \angle B = \angle A\) QED.

Congruent triangles make an appearance in a lot of geometrical proofs as they allow us to confirm certain angles or sides are equal, allowing us to draw conclusions. A related idea is that of similar triangles, which Thales used to measure the height of the Great Pyramid! But how did he do that?

Although the Great Pyramid, and Thales, are both three-dimensional, we can model this problem by focusing on the 2D plane cutting through the pyramid, as shown in the diagram.

Thales wanted to know the height of the Great Pyramid, \(d\) and to do this is placed a vertical pole \(BC\), of height \(a\), in front of the Great Pyramid. He then measured the length of the shadow cast by the pole, \(c\) and the Great Pyramid, \(f\).

Assuming the sun’s rays are parallel, Thales drew the conclusion that the triangles formed by the tall objects and their shadows must therefore be similar, and so the height of the pole and the Great Pyramid must be in the same proportion. That is to say:

\[ \frac{d}{a}=\frac{f}{c}\]

Thales will know the lengths of \(a\), \(c\) and \(f\) so, with some rearranging, he can find the height of the Great Pyramid:

\[ d = \frac{af}{c}\]

There are over 350 proofs of the Pythagorean Theorem! So many ways to prove such a simple yet powerful result. I have not seen or worked out all of them (I took a break when I got to Euclid’s proof), but the one pictured above* *is so far, my favourite, because it uses a lot of nice algebra.

The proof goes as follows: we start by enclosing a square with four congruent triangles, as seen above. We then want to work out the total area of the congruent triangles:

Each blue triangle has area \( \frac{1}{2}ab \), so the total area of the four blue triangles is \(2ab\).

We can also find the area of the triangles in another way. Because the four triangles are congruent, the sides form a larger square of length \( (a+b) \), and the area of the pink square will be \(c^2\). From this, we can work out the total area of the triangles a different way:

Total area of blue triangles \(= (a+b)^2 – c^2\)

We now have two ways to write the area of the blue triangles, so we can equate these two expressions:

\[ (a+b)^2 – c^2 = 2ab\]

\[ a^2 + 2ab + b^2 – c^2 = 2ab \]

\[ a^2 + b^2 – c^2 = 0 \qquad \textrm{(Subtracted }2ab\textrm{ from both sides)}\]

\[ a^2 + b^2 = c^2 \qquad \textrm{(Added }c^2\textrm{ to both sides)}\]

QED.

The proof is complete!

Mathematics started to become a passion for me when I first learned about expanding quadratics – it was the first topic I revised when preparing for my GCSEs. I remember spending ages on this topic, because I kept making mistakes when multiplying negatives and positives, but I kept persevering. I even rushed my tea to go and continue to expand brackets! But don’t worry, I took some crisps with me.

The obsession stemmed from the fact that it was extremely fun! I knew I had an end goal and I had to work towards it – double checking to make sure each step I took was correct. Algebra became one of my favourite pastimes, and increased my love for mathematics. So, when I heard there was a link between expanding brackets and geometry, I was excited to learn more.

If we expanded \((a+b)^2\), we would get \(a^2 +2ab + b^2\). But why? It may not initially seem that obvious. We can prove this using the distributive property – but that’s not what this post is about… so let’s use geometry!

We start by drawing a square of side length \((a+b)\). We then divide the square up into different sections: we can make a square of length \(a\)(blue square), then cut out 2 congruent rectangles with dimensions \(a\) by \(b\) (green rectangles). We are then left with a pink square that has side length \(b\).

If we work out the areas of each of these 4 shapes, the sum of these areas will equal the total area of the initial square.

The sum of the areas:

\[ \textcolor{blue}{a^2 \textrm{ (blue) } } + \textcolor{green}{ab \textrm{ (green)}} + \textcolor{green}{ab \textrm{ (green)}} + \textcolor{magenta}{ b^2 \textrm{ (pink) }} \]

Therefore, this will be equal to the area of the initial square, which measures \((a+b)\) on each side:

\[ (a+b)^2 = a^2 + 2ab + b^2 \]

I do quite like this result as it links to my first ever real enjoyment of learning mathematics!

I would like to end with a little puzzle for you. Can you work out the area of the green section, in this triangle with circle arcs centred at each corner?

The solution is below.

I am now reaching the end of my crisps and so, like at school, it is time to call it a day. Geometry is fascinating. There is no denying that. The theorems, proofs, applications are truly something to behold, and we can see the connections between solving equations and drawing shapes! But this article barely scratches the surface of the wonder that is geometry so, when I get a new packet of crisps, I will be sure to share more of the fascinations geometry has to offer.

The triangle is equilateral, since all the sides are the same length. Using Pythagoras to work out the height, \(h\), of the triangle:

\[h^2 + 4^2 = 8^2\]

\[h^2 + 16 = 64\]

\[h^2 = 48 \]

\[ h = 6.928\ (4 s.f.)\]

Area of triangle:

\[ \frac{1}{2}\cdot 8 \cdot 6.928 = 27.71\ (4 s.f.)\]

Area of sectors:

\[ \frac{60^{\circ}}{360^{\circ}}\cdot \pi(4^2)\]

\[ \frac{16\pi}{6} = \frac{8\pi}{3}\]

As there are 3 congruent sectors, total area of sectors:

\[ 3 \cdot \big( \frac{8\pi}{3} \big) = 8\pi\]

Therefore, area of the green section:

\[ A = 27.71 – 8\pi\]

\[A = 2.587\ (4 s.f.)\]

]]>We spoke to Marcello Seri and Marit van Straaten from the Bernoulli Institute at the University of Groningen, about their podcast, *It’s Not Just Numbers*.

**Podcast** **title**: It’s Not Just Numbers**Website**: podcasters.spotify.com/pod/show/not-just-numbers**Links**: Apple Podcasts, Spotify, Google Podcasts**Average episode length**: 1 hour**Recommended episodes**: Intro Episode, Teaching Mathematics (S1E03)

The idea for the podcast was born during the pandemic. In a break between online lectures we discussed with some students how hard and time consuming it can be to solve the homework exercises, and this somehow led us to talk about their perceived idea of their lecturers. It was quite mind-blowing how far from the truth this was, and how much the stereotypes from mathematical movies were shaping their impressions. We think that part of this came from the fact that they always see the “lecturer/supervisor” side of us and rarely have a chance to see the rest.

To counter this, It’s Not Just Numbers aims to address some common misconceptions and stereotypes around mathematics by showing the human side of it. In each episode we invite two mathematicians to talk about their drives, their aspirations, and the other hobbies they have outside mathematics. We use the discussion as an opportunity to get to know our guests and also present different sides of what being a professional mathematician entails. Moreover, we discuss a topic that is related to Mathematics, for instance what is involved when teaching Mathematics.

The podcast is hosted by us: Marcello Seri, an associate professor in mathematics at the Bernoulli Institute of the University of Groningen, and Marit van Straaten, a final year Master student at our institute. We record with the invited guests using facilities provided either by our faculty, or by FSE Radio, a group of student podcasters at our Faculty with their own recording room. For the editing and the planning we take turns among each other depending on how busy we are. The podcast is hosted on Spotify for Podcasters (formerly Anchor) and available on all of the major podcast platforms.

Broadly speaking, everybody that is curious about mathematics and mathematicians and what they are like. Looking at our stats, it seems that so far we are reaching young people interested in knowing what being a mathematician is like, be they just curious, potentially interested in enrolling in a mathematics track or already studying some scientific subject and curious to know what mathematics lecturers are

like and what they do.

The episodes are released on a monthly basis, the first Sunday or Monday of the month. They are structured in two parts: in the first half we interview our two guests to get to know them. We look into how they ended up becoming mathematicians, what motivates them and what they do besides mathematics. In the second part we look into some mathematics-related themes. This can vary a lot, from covering different aspects of our jobs as mathematicians to exploring what you could do outside academia as a mathematician.

The episodes are planned less as an interview and more as a discussion among the four of us, so while we have some initial idea and we stick to the two-part structure, the discussion flows freely wherever our collective dialogue brings it. The idea is that this gives a more spontaneous and truthful picture of who and what we are like.

In this first season we focused more on mathematics: teaching and studying it, engaging with people outside our university, and looking briefly at some fields of mathematics and how they relate to each other. In the second season, which will start some time in the fall, we will shift more towards the “aftermath”, what working as a mathematician can be like. Many of our listeners were curious to hear about the experience of doing a PhD, what is the academic career path like for mathematicians and, in general, what people do with mathematics outside academia.

Keeping the format that we have now, we plan to get a bit more into these topics, with the plan to also start looking also outside our department, and perhaps outside our university. We would also like to explore what you can do with mathematics “outside mathematics”: for example, we are trying to plan an episode about philosophy of mathematics for next year.

Our podcast is less about specific mathematical topics and more about the mathematicians and the surroundings of being a mathematician. This makes the podcast accessible to a broad audience, as a mathematical background is not required.

We aspire to show the people behind the trade and how varied and different their experiences are, hopefully demystifying some usual misconceptions and stereotypes and providing a glimpse to many aspects of being a mathematician that are rarely discussed. So far I have not seen other podcasts going in this direction in the same way as we do here, although I can see some analogy at least in our objective of bridging the gap between people and mathematicians with the Chalkboard Ultra podcast previously featured here.

Recently, we made two episodes about applied mathematics and theoretical mathematics, discussing the boundary between the two fields. It was a good opportunity to show that the difference between mathematics and applied mathematics is not as vast as students sometimes perceive, and that it is quite common that people switch from one field to another.

Besides this, many other insights can be taken from every episode. Each person brings with them a whole different view, making each episode special in its own way. Being able to hear the experiences of your lecturers as a student is extremely valuable. It really helped to put my (Marit) own struggles into perspective, and we have heard from other students that the insights they gained from our podcast have significantly helped them in their academic related challenges. The ability to help fellow students is the biggest highlight of all!

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