There is a continuum of polygons with two or more sides (see ‘Between a Square Rock and a Hard Pentagon’), including fractional-sided shapes. There is also a continuum of negative sided polygons with sides numbering below minus two (see ‘Thinking Outside the Box’), the ‘holes’ left by their positive counterparts. What happens between minus two and plus two sides? Do shapes with a shortage of edges exist, and if so, what do they look like? Well I think they do and I’m taking their side! Read on… Continue reading
Pi is one of the most common mathematical symbols in use today. It appears when we calculate the area of a circle, the volume of a sphere, or the surface area of a cone. But, what do we know about pi? We all probably know that it is approximately 3.14, but what else do we know? Well, there is an endless list of interesting facts about pi that you should know. Continue reading
Bertrand Russel once said that “mathematics possesses not only truth, but also supreme beauty.” Anyone who is passionate about mathematics knows that this beauty lies in concepts, not calculations. Yet somewhere along the line we have ended up with a reputation for just number crunching! Introducing Mathscon: the organisation with a mission to reshape the world’s perception of mathematics. Continue reading
It’s only at university that I started to consider mathematics as a possible fully-fledged part of my life. During my second year, in December 2013, the most widespread Ebola disease outbreak in history occurred in west Africa, killing thousands of people. As part of our national exam, my friend and I decided to study the virus propagation. For one year, we modelled the disease spread. From this mathematical model, we determined a vaccine campaign strategy to eradicate the epidemic efficiently. Continue reading
Sometimes, as scientists, we can be too focused on the details, and this clouds our vision when discussing broader topics such as diversity. The Royal Institution’s event ‘Balancing the equation‘, held on 8 October this year, avoided this pitfall and kept both eyes firmly on the big picture. The evening explored the far-reaching implications of structural racism on the past, present and future of science, and was hosted by Alex Lathbridge, a comedian, presenter and doctoral researcher in biochemistry.
Alongside Lathbridge were three guests: Lisa Kennedy, assistant curator at the Science Museum; Segun Fatumo, a senior scientist at the London School of Hygiene and Tropical Medicine; and Riham Satti, an entrepreneur who uses artificial intelligence to improve the hiring process. The speakers’ range of backgrounds ensured that the evening was varied, and presented a well-rounded view of the issue. Continue reading
Our latest edition, Issue 08, is available now. Enjoy the articles online or scroll down to view the magazine as a PDF.
- We chat to the author of the best-selling book How to Bake a Pi and pioneer of maths on YouTube
- Colin Beveridge looks at different designs for 2- and 3-dimensional tiles
- Alex Bolton plays noughts and crosses on unusual surfaces
- Read about Maxamillion Polignac's adventures in a prime-hating world
- Adam Atkinson uses maths to try to help a sculptor
- Find out more about the weird shapes on the cover of Issue 08
- Elizabeth A Williams falls off a log
- Emma Bell explains why the Renaissance mathematician Gerolamo Cardano styled himself as the "man of discoveries".
- Just what is category theory? Tai-Danae Bradley explains
- Biography of Katherine Johnson, NASA human computer and research mathematician
- A tabletop demonstration of chaos.
- Surprise yourself by making this thing
- Print it out and play against your friends!
- Win £100 of Maths Gear goodies by solving our famously fiendish crossnumber
- Are you perfect or imaginary? Are you a square?
- Village fetes, Brexit and flowers are among the topics of discussion in this issue's Dear Dirichlet advice column
- The definitive chart of the best units
- Vote for your favourite Chalkdust regular
- Tired of hearing people ask which area of maths is your favourite? Get one of these tattooed on your forehead and you'll never be asked again...
- Many mysteries about hair are now revealed with the ponytail shape equation
- Fashion is fleeting, Chalkdust regulars are not.
We meet Eugenia Cheng a couple of hours before she’s scheduled to give a talk at City University, where she’ll make another stop on her journey to “make abstract mathematics palatable” in the public consciousness. With over 10 million views on YouTube, three best-selling books in How to Bake Pi (2015), Beyond Infinity (2016) and The Art of Logic in an Illogical World (2018), and interviews ranging from the BBC to late night US television, it’s safe to say Cheng has made incredible progress on her mission.
Not ‘just’ a mathematician
In talking to Cheng, you quickly realise that she is always trying new activities, pursuing further study and pushing herself to understand more of the world around her. She read voraciously growing up, but describes mathematics as “the only subject to stand up to [her] desire for rigour”. In attempting to satisfy her “curiosity for asking why things are true”, Cheng learned of category theory, the study of relationships between similar themes and concepts in different branches of mathematics. She describes this field as particularly abstract, but remarks that it could be viewed as a prerequisite to most undergraduate courses in the way it identifies links in areas of mathematical study. In her view, category theory does for mathematics what mathematics does for the world. This concept takes a little bit of mental gymnastics, and Cheng takes it a step further in her current research in higher-dimensional category theory. She studies the relationships between the relationships themselves, adding “an extra layer of subtlety” to her understanding of mathematics.
It’s clear that Cheng’s research has influenced the way she approaches her other passions, which include food, music, and teaching. An avid baker, Cheng’s first book, How to Bake Pi, begins each chapter with a recipe for the reader to try. This method mirrors the conversations that introduce chapters in Douglas Hofstadter’s seminal work Gödel, Escher, Bach: An Eternal Golden Braid, which she credits as a “very influential book” for her approach to writing about mathematics. Cheng believes that the general public is suffering from a severe case of “maths phobia”, and that introducing mathematics in recognisable, accessible ways is far more effective than teaching times tables and long division. On her appearance on The Late Show with Stephen Colbert in 2015, Cheng introduced the concepts of exponentials through making mille-feuille live with the host. Children as young as seven have, through How to Bake Pi, gained an understanding of complicated abstract mathematics, and she recounts a story of a five-year-old calling out to her “I’m your biggest fan!” at one of her outreach talks. By giving younger and younger children an appreciation of what mathematics can do and how it is expressed in the world around us, Cheng believes that the fear of mathematics so many schoolchildren feel will erode and disappear over time.
Cheng is the founder of the Liederstube, an environment for classical musicians to come together and enjoy performances in a relaxed setting, based in the Fine Arts building in Chicago. A talented pianist, Cheng performs alongside her busy schedule writing books and giving talks. When asked about whether performing in concert halls is more nerve-wracking than giving talks at the Royal Institution, Cheng doesn’t hesitate: “Compared with playing the piano, public speaking is easy! You can say whatever you want, and you don’t have to say particular things in a particular order.”
As the scientist in residence at the Art Institute of Chicago, Cheng teaches abstract mathematics to undergraduate art majors. She enjoys teaching mathematical ways of thinking to socially conscious students, giving them the ability to “frame social issues in a mathematical sense”. Perhaps these students represent the ultimate cases of maths phobia, but according to Cheng, there is a lot more in common between those that study mathematics and the students she sees weekly. Through her courses, Cheng is learning as much from her students as the other way around. She had not realised the applications of mathematical thinking as a “framework to agreeing on the world, something badly needed in today’s public discourse”. The parallels between Cheng’s passions and her research are immediately apparent. She finds the mathematical similarities between music, food, and teaching in the same way she identifies connections between areas of mathematics in category theory.
Women in mathematics
As a prominent woman in mathematics, especially in popular culture, the question of the gender gap inevitably came up. By Cheng’s own admission, she “was initially reluctant to address the issue” at the beginning of her career. A firm believer in the meritocracy of academia, she is certain she’s “able to achieve anything a man could in mathematics”. Cheng says that “when women in academia are young and not treated with much respect, they think it’s because they are young and junior. But as they progress, they continue to notice the lack of respect, making it clearly a feminist issue.” Cheng has also realised how important it is to act as a role model, and has embraced the challenge of becoming more visible to young women in mathematics. For Cheng, the most important message to communicate to these students is that “they are good enough”. Students that are struggling are “not finding it difficult because they don’t understand. They are seeking a deeper level of understanding-exactly the kind of person needed in higher level mathematics.” Cheng sees a marked difference in how the average female student approaches applying to PhD courses compared to their male counterparts. She admits that if she had not been offered the opportunity to study for her doctorate at her first choice (Cambridge), she would have given up pursuing a career in higher level mathematics, taking the rejection as a comment on her ability. She believes that in applying for mathematical postgraduate positions, women have to take the same persistent approach that men do, applying for any opportunity that allows them to follow their passions.
Cheng also feels that too much self-confidence in one’s own abilities doesn’t make the best students. “Most people think that self-confidence is the most important part of being a mathematician, whereas I believe that self-criticism is far more important. I’d much rather work with a student that underestimated their own abilities, than the other way around.
Most people think that self-confidence is the most important part of being a mathematician… I believe that self-criticism is far more important.
In fact, Cheng believes there needs to be a reframing of the whole argument, proposing new words to replace masculine and feminine, as “we shouldn’t prescribe behaviours to genders”. For masculine, Cheng suggests `ingressive’. “Ingressive-it’s all about getting the right answer, being competitive, being first, exactly the way we teach mathematics at a young age.” Even the way we test is ingressive: “Exams are an ingressive thing too,” Cheng says. “You have to get as much done as you can, as right as you can, as quick as you can.” But research isn’t like that at all. “Research is congressive,” Cheng explains, using her replacement word for feminine. “You’re trying to discover deeper insight, you have to work collaboratively. It takes time and it takes patience.” And therein lies the problem, according to Cheng. “We’re selecting ingressively for a subject that should be very congressive in nature,” she concludes. “I suspect we’re losing a lot of talent this way.
The art of logic
Just before Cheng has to nip off for her talk, we move onto the subject of her new book, The Art of Logic in an Illogical World. “This really grew out of teaching art students because the ones I teach are so socially conscious, and want to change the world,” Cheng says. “It was a bit like when I used baking to perk up my mathematics undergraduates. If I talked about a social issue from a mathematical point of view, then they were all completely alert.” The book is a summary on the “insights mathematical thinking gives me on social and political issues.” In summary, the book is about “the nature of disagreement.”
Mathematics is a way of being clear and unambiguous, and we need that today.
We later attend one of Cheng’s talks at the Royal Institution, based on her book, and it’s fascinating to see how mathematical thinking could be applied to questions of LGBT rights, racial privilege and political disagreement. “I always read the comments below news articles,” Cheng said, prompting a sympathetic laugh from the audience, “You have to! You have to know how people see the world so you know how to talk to them.” Cheng’s belief that mathematical principles allow us to cut through overly complicated debate is infectious and so clear, you wonder how anyone could possibly disagree.
“Mathematics is a way of being clear and unambiguous, and we need that today,” Cheng concludes. Through her writing, talks and outreach work, there’s no debating the important work she’s doing for the subject, curing cases of maths phobia every day.
It is altogether too hot, it is altogether too full of people, and it is altogether too lunchtime for what feels like 8pm. Fortunately, the organisers have fed mathematicians before, and have thoughtfully provided plain paper tablecloths and pens with which to postulate, puzzle and prove while we eat. We’re in Atlanta, Georgia for the 13th Gathering 4 Gardner, or G4G: every couple of years, mathematicians, magicians, sceptics, jugglers and assorted others gather to honour the work and memory of popular science writer Martin Gardner with a week-long conference.
An idea from an earlier talk had lodged in my head. Cindy Lawrence of MoMath—New York’s Museum of Mathematics, where one can ride a square-wheeled tricycle or explore the inside of a Möbius strip, the kind of thing that Gardner would certainly have written about—had raved about Truchet tiles. What are they? Well, start with a square, coloured either black or white. Pick two diagonally-opposite corners and shade them with a quarter-circle of the other colour, as shown. Make several. Then place them however appeals to you! The way they’re set up, it’s practically impossible not to start making patterns: blobs and whorls that seem almost alive.
Of course, you don’t need the tiles themselves. You can just as easily doodle them on a convenient sheet of paper, such as a tablecloth. And you can just as easily start asking yourself questions like, what kinds of tiles can you get if you remove the restriction of squareness? What happens if you move into three dimensions? And is there any pretty maths underlying the pretty patterns?
Sébastian Truchet and his tiles
The tiles Lawrence talked about are not, strictly speaking, due to Truchet. In 1704, Sébastien Truchet published A Memoir On Combinations, in which he discusses squares split diagonally into triangles (pictured right), giving four possible orientations for each tile.
It’s an interesting read. Truchet methodically looks at the number of ways you can place two such tiles next to each other, edge-to-edge. (He says that he’s started work on three tiles, but isn’t happy with it yet. I’ve seen nothing to make me believe he ever was happy with it.)
In the paper, Truchet carefully reasons that there are:
- four possible orientations for a single tile;
- four positions to place a second tile next to a first (north, south,
east, or west); and
- four orientations for each of the second tiles
… making a total of 64 possible arrangements.
He then notes that some of the arrangements are indistinguishable from others: placing a tile in orientation A to the left of a tile in orientation B is the same as placing B to the right of A, reducing the number to 32. Furthermore, some arrangements are rotations of others—for example, arrangement AA (pictured below) is a rotation of arrangement CC (not pictured below unless you’re reading Chalkdust upside down).
In all, he reduces the 64 original possibilities to 10 (six appear in eight configurations each, and four—those with `stripes’ across the middle—appear four times apiece).
It turns out to be worthwhile to consider the overlying structure of these arrangements. Of the ten possibilities, eight appear in pairs: swapping the colours of one gives the other. The other two are self-inverse: swapping the colours gives a rotation of the original arrangement.
And that’s where Truchet tiles remained until the 1980s, when CS Smith, writing in Leonardo, took a deeper dive into the topology of Truchet tiles.Among other things, he suggests several possible changes to the design—for example, removing colour from the equation altogether and simply tiling with diagonal lines, or—rather nicely—by a pair of arcs in opposite corners. This is, in terms of symmetries, just the same as a diagonal, but placing large numbers of them together makes for much more appealing patterns—which you should totally play around with, but after you’ve finished reading this article.
Smith also suggests extending these tiles to include colours: if you make the cut-off corners one colour, and the remaining strip of the tile the other, you have exactly the blobby-cornered tiles MoMath brought to G4G.
But why limit yourself to squares?
Extending tiles to 2n-gons
Don’t get me wrong, I have nothing against squares. They’re certainly in my top ten favourite shapes. Just… even if the tessellation patterns are neat, the configurations of the tiles themselves are not all that interesting. When you go beyond four sides using blobby corners and two colours, though, fascinating things begin to emerge, without even having to put the tiles next to each other. (From here on, you can assume that all tiles are blobby-cornered and two-coloured.)
Indeed, any regular polygon with an even number of sides can be turned into blobby Truchet tiles (although hexagons are the only ones that would tile the plane alone). Odd numbers of sides don’t work with this kind of colouring, because the vertices need to alternate between two colours. How many hexagonal tiles are there? Well, it depends how you count.
I choose to count rotations of the tile as the same tile—so there exactly two square tiles, one with two isolated white corners and one with two isolated black corners.
With hexagons, it’s simple enough to do the counting: let’s start by considering the three white corners and how they connect (or don’t connect) to each other. Either each corner is on its own, all three are connected, or a pair of corners is connected and the other isolated. (Note that if any member of a group of corners connects to another corner, all members of that group must connect—we can’t have a case where the first corner connects to the second and the second to the third unless the first and third are also connected.)
These correspond to the figures to the right—three isolated white corners and three interconnected white corners, followed by—more interestingly – an isolated white corner, followed by a black band that ends halfway across the hexagon, followed by a white band and an isolated black corner.
I’ve paired them like this for a very good reason: inverting the colours on the black dots tile gives the white dots, and vice versa; inverting the colours on the split hexagon gives… a rotation of itself! I count the hexagons as having three possible tiles, one of which is self-inverse.
What possible patterns are there for an octagon?
Obviously, octagons don’t tile the plane on their own (although there’s nothing to stop you filling in the gaps with square tiles!) Independent of how we’re going to arrange them, we can still consider the viable patterns.
Again, it’s good to start by considering the four white corners and how they connect. When none of them connect, we have another white dots pattern; when all four connect, we have a corresponding black dots pattern. These two are inverses.
If two neighbouring corners are joined, and the other two left unattached, the resulting shape looks like a pair of black pants. Its inverse pattern, the pair of white pants, arises from any three white vertices being connected.
We could also connect a pair of diagonally-opposite corners to give a white stripe; the black stripe comes from connecting one pair of adjacent vertices, then connecting the remaining pair.
The octagons therefore have six possible tiles, none of which is self-inverse. (Or do they? Have I counted correctly? How do you know?)
Decagons are where (for me) it gets interesting: but perhaps you want to try working out how many decagon tiles there are yourself first. Scroll down once you’ve had a go…
There are ten decagon tilings, up to rotation—of which two are self-inverse. That’s a structure we’ve seen before: it’s the same as the structure behind Truchet’s pairs of adjacent tiles. Don’t you think that’s neat? Well, hold my coffee.
Truchet tiles in three dimensions
The following day, I pick a table with the MoMath people, who have a set of the square blobby tiles out to play with, black with white corners one side, white with black corners on the other. “How many ways can you make a cube?” asks Tom. We pick up the tiles and try to arrange them, one per face.
Let’s adopt the reasonable rule that the corners of each tile that meet a vertex must be the same colour. Each face then has two diagonally-opposite black vertices, and two diagonally-opposite white vertices, and there are only two ways to place a blobby tile to satisfy that: either the black vertices are joined by the stripe, or the white ones are. We can therefore think of each face as either black (if its black vertices are connected), or white (if the white pair is joined).
If all six faces are white, there’s only one possibility. The same goes for no white faces.
If five are white, there is again only one possibility (up to rotational symmetry); this is also true for one white face.
With four white faces, there are two possibilities: either the black squares are on opposite sides, or they are on adjacent sides. As you might expect, the same goes for two white faces.
Now, simply listing the possibilities isn’t all that interesting. However, the structure behind the list is: of the ten arrangements, eight have a natural colour inverse, and the other two are self-inverse – switching the colours gives you a rotation of the same cube.
That’s precisely the same as the structure of the decagonal tile arrangements—and Truchet’s original pairs.
It’s not clear to me whether this repeated structure is a coincidence, or some deep property of Truchet tiles. It’s not unnatural for a ten-element set to have that same structure—two elements that are their own inverses, and eight that form inverse-pairs.
Indeed, if you consider the group of integers from 0 to 9 under addition modulo 10, you also get that structure (0 and 5 are self-inverse). However, to consider the tiles or cubes as a group, we’d need a way to combine them in pairs, and if there’s a simple operator for any of the Truchet sets, it’s not obvious to me.
So, I’m opening it up to you, knowing that the readership of Chalkdust has, collectively, far more insight than I do: is this common structure a coincidence, or something deeper?
While you’re working it out, draw some Truchet tiles of your own. You’ll be glad you did. See you later, tessellator!
Tic-tac-toe also known as noughts and crosses) is a classic game known for its simplicity, and has been popular since ancient times. You and a friend (or enemy!) take turns to mark the squares of a $3 \times 3$ grid. The winner is the first to get three of their symbols in a row (horizontal, vertical or diagonal). A winning game for X is shown below right.
It’s not difficult to work out how to play optimally on a standard board, where you’re guaranteed to at least draw with your opponent. However, what if you’re not restricted to the standard two-dimensional square grid? How would you play then?
To present a fresh challenge and to make tic-tac-toe exciting again, here is a collection of puzzles where you will be swapping your standard square board with one on various topological surfaces. Have fun!
The first new board to consider is the cylinder. To form a cylinder as in the diagram below, imagine that the board is wrapped around so that the left and right edges of the board are connected to each other, like a piece of paper that has been rolled into a tube. Matching edges are denoted with a $\uparrow$.
In all the subsequent puzzles, it is X’s turn to play, and it is possible for X to win in some number of moves, even if O plays optimally.
Puzzle 1: cylinders
How does X win both of these games on cylindrical boards? It is possible to win the first game in one move.
The first game can be won in one move. A demonstration of this, along with the folded cylindrical board, is shown below.
The Möbius strip
How about a Möbius strip? Imagine that the right edge of the board is wrapped around the back of the board and given a half turn, so that it connects to the left edge of the board. The half turn means that top and bottom become flipped as we move off the left or right edge of the board. The edges with a half turn are denoted by a $\uparrow$ and a $\downarrow$. (If you want to actually construct this board, draw each grid square as a wide rectangle so that the grid is wide enough to wrap around with a half twist.)
The figure on below shows adjacent squares in the new board by making copies of the Möbius strip board, and puzzle 2 gives two puzzles on the Möbius strip board.
Puzzle 2: Möbius strips
How does X win both of these games on Möbius strip boards? It is possible to win the first game in one move.
There’s no need to limit ourselves to only connecting the left and right edges—we can also connect the top and bottom edges. If we return to the cylinder formed by wrapping the right edge of the board round to meet the left edge, we can now connect the top and bottom edges together to form a torus. Now we have two pairs of connected edges, denoted by $\rightarrow$ and $\uparrow$.
For tic-tac-toe purposes, the cylindrical and toroidal boards are identical, and this holds even if we change the game to be `make a line of length $n$ on an $n \times n$ board’ for any $n$. However, for puzzle 3, you need to find a line of length 3 on a $4 \times 4$ board.
Puzzle 3: torus puzzle
What should X do to make three in a row on a torus?
On the torus, we can think of any row (or column) as being the central one, so it’s easier to prove facts about the torus than for other boards. Have a go at the following challenges.
Puzzle 4: more torus puzzles
Can you show that making any line of length $n$ on an $n \times n$ cylindrical board is also a line of length $n$ on an $n \times n$ toroidal board and vice versa (so a game on a torus is equivalent to a game on a cylinder)?
Are any starting positions better than others on a $3 \times 3$ torus?
Is it possible for the game on a $3 \times 3$ toroidal board to end in a draw, with neither player getting 3 in a row (the previous question gives you a shortcut to solving this one)?
The Klein bottle
Now we will think about playing on a Klein bottle, a shape that cannot be constructed in three dimensions without it intersecting with itself. Fold the top edge of the board over to touch the bottom edge, and connect the left and right edges with a half twist like for the Möbius strip.
The figure to the right shows which squares are adjacent on the Klein bottle board, and puzzle 5 gives two puzzle.
Puzzle 5: Klein bottle puzzles
How does X win both of these games on Klein bottle boards? It is possible to win the first game in one move.
The projective plane
Another shape that it is not possible to construct in three dimensions without the shape intersecting itself is the projective plane. This is a shape where the top and bottom edges are connected by a half twist, as are the left and right edges. To imagine how it would be made, think about connecting the single edge of the Möbius strip to itself. (You’ll have to be very dextrous to actually create this from paper!) Puzzle 6 gives two puzzles using the projective plane. Top tip: construct an adjacency map like the one for the Klein bottle.
Puzzle 6: projective plane puzzles
How does X win both of these games on projective plane boards?
(Note: I’m assuming that a valid line comprises three distinct squares, so a single square does not appear more than once in a valid line.)
That brings us to the end of our foray into topological tic-tac-toe. I hope you enjoyed these mind-bending puzzles! The inspiration for this article came from Across The Board by John Watkins, the most complete book on chessboard and other grid puzzles. I would recommend this book for some further interesting puzzles, eg how many queens are needed so that every square on a chessboard is targeted or occupied by one of the queens? The author gives an interesting history of chess problems and shows that they have inspired important advances in maths.
Some people like to hear about mathematics being used to address real-life problems. I am going to claim that the problem I describe in this article is real-life because it arises from a conversation between two non-mathematicians.
Specifically, one of my sisters-in-law did an art degree, and as part of a project she did for this she visited Sardinia to interview the sculptor Pinuccio Sciola. At least some of his works are quite large, by which I mean maybe 3m high or more, based on things I see on the web. During their conversation, he said something about wanting to install one of his sculptures on a named mountain somewhere in the Catania area, for the benefit of the residents. I don’t know his exact words, but my sister-in-law found this remarkable enough that she reported it to me and other members of the family. I’m not naming my sister-in-law here because she is not a public figure, and feels that this article is not the way she would choose to become one.
It may be important to note that my sister-in-law lives in a small town near Catania, and this may be what prompted him to say this. It seems entirely likely that he had not spent any real time looking into this idea. Indeed, I am told that when he later visited Catania he immediately realised that his idea was probably unrealistic.
Let’s use maths, and some other disciplines, to consider his idea, pretending for the sake of discussion that he or someone else really does want to go ahead with it. You may notice that even though Sciola named the mountain, I have chosen not to do so. I shall do this later. For starters, let’s have a look at the mountain. I took this photograph from just outside Catania airport.