Beautiful Symmetry

Emeritus Professor Cheryl Praeger with Dr Tom Carruthers

Episode #12

Bring your coffee in your favourite mug and tune in to hear about the science of symmetry.

In this very special episode of Mug of Science, we had the wonderful opportunity to share a coffee with one of Australia’s most esteemed scientists, Emeritus Professor Cheryl Praeger AC FAA. 

Seated in the UWA university club, Tom and Cheryl explore the wonderful world of symmetry, and how it is present in all areas of science and life. They also take a dive into Cheryl’s extraordinary life in science that has taken her all over the world.


Cheryl has just been awarded an AC (Order of Australia) for eminent service to mathematics and as a mentor to women in STEM. She is the inaugural recipient of the Ruby Payne-Scott Medal from the Australian Academy of Science, and was the first person from WA to receive the PM's Prize for Science.

You can find out more about Cheryl at her website.




Emeritus Professor Cheryl Praeger: I mean, it had just not made sense to me that you couldn't subtract five from three. It just didn't make any sense, and then in Year Two, it was allowed. And so I remember being so relieved that this could make sense. 

Dr Tom Carruthers: G'day and welcome to Mug of Science. It's a Pint of Science initiative where we bring local scientists to a cafe or club or somewhere nearby to have a chat about their work. My name is Tom Carruthers and I'm here I've got the wonderful pleasure to be here with Emeritus Professor Cheryl Praeger, who is Emeritus Professor at the University of Western Australia, which also happens to be where we're at. We're at the University of Western Australia Uni Club, is that correct? 

Cheryl: Right, that's correct. 

Tom: Fantastic, so we're here to chat with Cheryl Praeger. She is a fantastic mathematician who does some really, really awesome stuff, but I'll leave that juicy bit for a little bit later in this in the episode. But first, to start off Mug of Science, the way we like to start these is to find out what's in your mug this morning. What coffee are you having today? 

Cheryl: I'm having a flat white. 

Tom: A flat white. 

Cheryl: Yes, I love having, you know, milky coffees when I'm out, because I can't be bothered doing that at home. 

Tom: Oh, it's just too much effort when you're at home. Can we just get like a quick snapshot of what Cheryl Praeger's life looks like, outside of her academic career? What do you like doing? 

Cheryl: Oh, well, as often as I can, I love getting into forests and walking. It's not so easy around here, but we've got a beautiful river and parks... 

Tom: I think maybe we might be able to see a bit of the river over the corner. 

Cheryl: Yes, I love walking, being in nature. I also love music and I've got lots of things which are perhaps associated with academia, but they're more service activities, which involve mainly looking and supporting kids in maths. That's the challenge activities at school, right and like this morning, we got the results of the European Girls' Mathematical Olympiad and Australia's only, our I think third or fourth time in participation. We came 12th, which is amazing. 

Tom: That is so cool. 

Cheryl: One of the four girls is from Perth and she won a gold medal, so I was extremely proud of her, very excited. 

Tom: So you're involved in a lot of these kind of things, because I remember there being a Math Olympiad when I was in school....

Cheryl: Yeah, and there wasn't when I was in school! [Laughs] 

Tom: So I imagine there is quite a bit of work that goes into trying to coordinate those things and get like community involvement, schools involved. 

Cheryl: It's a huge national program with every state involved and preparing enrichment and challenge activities with kids. It starts off with say, the Australian maths competition in schools, and then there are more challenge and enrichment activities through the Australian Maths Trust, and that happens state-wide, and then once there are kids really, really involved in the program, there are national training programs, which of course has been rather difficult with COVID. A lot of it's happened to happen online, but the girls, the European Girls' Maths Olympiad is a new activity running alongside the International Maths Olympiad, and this year because of the problems of having face-to-face training and selection of schools, we still managed to choose an extremely good team of four girls who were announced a few weeks ago. And the competition was just this week. 

Tom: Fantastic. Wonderful. So let's have just a little talk about your field of expertise, which is mathematics, specifically symmetry if I understand correctly, and looking at like different symmetry groups. Could you perhaps explain when we're talking about symmetry in groups, what that actually means, what they are? 

Cheryl: What they are, yeah. So I was looking at this square table. Yes, and if we forgot that it had slats this way, we could just turn it a quarter way around, half, three-quarters and the whole way around and we'd say that's four symmetries that this table's got. 

Tom: Because we can rotate it...

Cheryl: We can rotate it and it looks the same. 

Tom: Would there be another symmetry if it didn't have legs, then we could... 

Cheryl: If it didn't have legs we could rotate it halfway around, or we could imagine putting a mirror here and reflecting it or a mirror here and reflecting it. Or one that goes diagonally this way…

Tom: Right, and so for each one of those would be considered...a symmetry. 

Cheryl: A symmetry. And you put the symmetries together and you get the group of symmetries. 

Tom: Right, so an object or a something that's within a group like this square table, is going to have a similar group to other objects…

Cheryl: ...which are square...

Tom: …which are approximated by a square table or whatever it might be. 

Cheryl: Yeah, right and it depends what you are going to allow as your symmetry. 

Tom: Because we started by saying, assume these slats aren't here. 

Cheryl: These slats here, right. So then you could do it a quarter way around. But you know, if you want to preserve the slats, then you... 

Tom: …have to have to go all the way around in this case. But that's assuming that none of the grain in the wood's there and so on and so on. 

Cheryl: Yeah, that's right. So it depends on what you want, what you're thinking of as the squareness. So if the squareness has the slats or doesn't have them, well, squareness doesn't have slats but the table does. Yeah, so the mathematics accommodates all of the constraints you want to put on it. 

Tom: Right yeah, and so that gives us basically the language and the tools to be able to describe things...and that's our world around us, yes, but also things that perhaps we can't necessarily imagine...

Cheryl: ...can't see in three-dimensional space, but you might want to imagine them. Scientists would imagine accommodating time. So Einstein wanted to accommodate time and space and much more. 

Tom: So being able to effectively come up with a number of groups or symmetry groups that describe something like a table, you could then look at another object and do some testing of that object to recognise whether it looks like a particular group. 

Cheryl: To see whether it's really a table! 

Tom: And then you could infer that maybe it's a table, like in its properties or whatever it might be. Okay so and that's where it becomes really, really powerful, in terms of things you can't necessarily touch or see, like if I was looking at an atom or a molecule or something like that, I could start to make potentially some predictions... 

Cheryl: ...and then we might have some technology that could help us test them... 

Tom: ...and therefore it makes the model stronger... 

Cheryl: And so, it's a wonderful example, like Crick and Watson are the people that discovered DNA. They also predicted that these containers, the capsids for viruses had what they called an icosahedral symmetry that is... looks like one of these five-dimensional, I mean sort of five-fold symmetry things, and in those days there wasn't really much you could do for testing it, but there was the beginning of X-ray crystallography or measuring X-ray diffraction patterns, yes and they managed to confirm that this was the case, that there were 60 symmetries in the simplest viral capsid, and the reason for suggesting that there would be this symmetry in their terms was a biological economy, that the virus can reproduce itself and it doesn't have to remember too much biological stuff, because it can just rotate itself around 60 different times and reproduce itself... 

Tom: and then if it just clicks together, then great. 

Cheryl: Then that's great. 

Tom: Because okay, so just going back to our table for instance, if I was a virus making this, I could make this segment... 

Cheryl: This bit of this triangle here, or maybe even just that triangle there, and then you would just reflect it, reflect it and so on…

Tom: ...and then of course the smaller that part is, the less information is needed to store in the DNA or RNA of the virus, and so the idea of having multiple repeating units that go... right, yeah. 

Cheryl: And that's one of the early realisations of the mathematical symmetry in biology. I mean, it was always sort of known in geometry, in architecture and even in the spiral galaxies in space, it's sort of more obvious somehow, than in some of the biological settings. 

Tom: Yeah, because I've seen a couple of the photos…yeah, please do have a sip! I've seen a couple of photos of some of the beautiful like spirals that you've used in presentations, yeah, as well that symmetry... Oh okay, so here's me starting to go: hang on, how does this work? So that's like that spiral that goes out like this, how does symmetry even... 

Cheryl: How is that symmetry, because you can't flip it. So it's something that sort of goes in one direction, or you can reverse it, but it's not a finite thing, so it's a sort of an infinite symmetry and it's maybe more like a trajectory. We might call it a semi-group, rather than a group because you can't so much reverse it. So the big thing about groups is that you can reverse the symmetry. So in this finite case, you do the rotation a quarter of the way around, and you only do it four times and you're back where you started. 

Tom: Yeah, whereas for the spiral and there's a scaling that's happening, as well, like the thing gets larger, and that's allowed within symmetry? 

Cheryl: Yeah of course. Yeah, it's like calculus. [Laughter]

Tom: Oh, you can just do whatever you want. I'm sure that there's calculus mathematicians who are like, you know, that's not true, that you just can do whatever, no... Starting the maths wars right here on Mug of Science... [Laughs] But yeah okay, so there's scaling, but yes, I think I see what you say, what you're saying is...

Cheryl: So what you do here is reflected, and you do a bit more and a bit more, a bit bigger as you go out...

Tom: So there's no actual end point because you don't let land where you started. 

Cheryl: Except in the real examples, the shell does end, it doesn't go on forever and the plate...

Tom: Because it breaks the symmetry. 

Cheryl: Yes, that's right. So some of the really interesting things with symmetry are that it gets broken. So Penrose tiling is never exactly replicated anywhere, but it looks very beautiful and it's subtly symmetric locally, but it's not periodic. You can't see one bit repeated and repeated, it's different, and that was a very interesting discovery that was very useful in physics. And we've got this Penrose tiling in the floor in the chemistry building, the new chemistry building here at UWA. It's wonderful. 

Tom: Have you got any cool little references in buildings anywhere? Has anyone made a building about your work? 

Cheryl: To me? Well, no. They've named a lecture theatre after me. [Laughs] 

Tom: So someone who's watching, you need to build something that references Cheryl's work! They named a lecture theatre after you? That's here at UWA? 

Cheryl: Yes. 

Tom: Yeah, that's I guess that's pretty cool. Do you want to just give us a little bit of a history lesson on your little journey through how you became a mathematician? Because I imagine that it wouldn't have been as straightforward as, for instance, my path to get into chemistry. 

Cheryl: Yeah, maybe not. Yes, so both of my parents left school aged 14 or 15, because their parents were ill or had died and they needed to work and my dad worked in the bank, so we moved very often but I remember being in Year Two, six, seven years old and I discovered that negative numbers were illegal. I mean it had just not made sense to me that you couldn't subtract five from three... like it just didn't make any sense, and then in Year Two it was allowed, and so I remember being so relieved that this could make sense, and then when I started doing science, I was very sceptical of anything that was claimed, like I heard about mirror images and I thought no, there's no image, what are they talking about? And then we did this experiment, and you looked through the mirror and you did some measurements and I eventually decided I would believe in mirror images. 

Tom: So you were convinced? So you saw enough evidence to be convinced that mirror images existed. 

Cheryl: So I was convinced. And it seemed like every bit of science was explained by something in mathematics, and mathematics seemed to be the basis, the fundamental of it all, and I started to be very, very interested in it. I found it pretty scary, because you had to work really hard and you might get questions in an exam which would be not so easy to solve. I didn't find maths easy in that sense. I found it terrifying but easy in another sense, it all made sense to me. I wanted to study mathematics, but no one in my family had ever been to university. They hadn't finished school even, and that was still an issue that I'd be allowed to finish school. I was only allowed to finish school on the grounds that maybe I would go to a business college afterwards and catch up on the skills I'd missed out on, the commercial skills... Because I would have been doing maths and science, and no room for typing and shorthand and bookkeeping. But I was allowed to finish school, and then I thought I really wanted to study maths at university. So I tried to get what information we could to say, well, how do you study maths at university, and I was actually put off by a government vocational guidance officer who said that girls don't do maths. They don't pass, and there aren't any jobs anyway, all of which is false. But I found out more advice from the University at Queensland, the University of Queensland, and was able to study, which was fantastic. I loved it. I found it scary, just the change from school to university with no one to tell me it's going to be all right. But it was it was great, but then by second year, I was the only girl in my Advanced Mathematics units, and I was still doing these sort of Advanced Honours units in Physics as well, and there was a second girl there in second year, but you know, it was pretty lonely. I had to become an honorary guy. But the big watershed was getting a scholarship one summer vacation to the Australian National University in Canberra, and seeing mathematicians doing research projects and being given my own little research project, which worked. And I published my first paper from that experience, and I happened to have the opportunity of sitting in on lectures at a summer research institute of the Estonian Maths Society, which coincidentally happened during the period I was in Canberra, and I thought, this is what I want to do. I so much want to do a PhD in Mathematics, and I really want to do it in category theory, because that's going to unify all of mathematics and it's going to change the world. [Laughs] But well, I didn't work in category theory, but you know, I just became passionately sure that that's what I wanted to do next, I think for as long as I could. 

Tom: Yeah, so those kind of experiences are so important, right, like you're describing right now of going into that summer program, but you're exposed to the breadth of this world that you had only been tangentially touching on. 

Cheryl: And the community and the culture...

Tom: Yeah, it's so important, those kinds of programs, to be able to capture the passion and the interest...

Cheryl: And the vision and the imagination that something might be possible. 

Tom: Yeah, definitely. I think it's important to add into this line that it wasn't just that you were doing this summer project, and then you were the only girl in the maths class, you were also absolutely killing it in terms of doing really, really well. 

Cheryl: Yes, I got the University Medal for Mathematics, right, when I graduated my Honours. 

Tom: Which can also contribute to that sense of loneliness a little bit as well, you know, or was it that you were competing with a bunch of other people? 

Cheryl: Actually, I felt very guilty, because by the time I was doing my Honours year, there were eight of us. I was the only girl of course, and I desperately wanted to get a scholarship so I could do my PhD, and I felt it was almost unworthy that I wanted to get better than everybody else, because that was the wrong motivation for a girl. I mean, I knew I wanted to be really really good, but I knew I wanted to win the scholarship, and it seemed like it was unworthy. 

Tom: It wasn't a pure enough reason for you...

Cheryl: Yeah, but I did know that I wanted to go! 

Tom: How very "mathematician" of you to be like, you know, my motivation isn't pure enough here! [Laughter] So that's the stereotype, right, that mathematicians tell off all the other scientists for not being accurate or pure enough in their work, and this is actually your experience, critiquing yourself. That's hilarious. 

Cheryl: So I did get the scholarship to Oxford and that was wonderful and I was introduced to the mathematics of symmetry there, because I'd only done lots and lots of different areas of coursework as an undergraduate student, and it was only in Oxford that I really met groups. I've done a little bit through my Master's program before I went to Oxford, I learned something about groups, but then through my doctorate, I just, I just fell in love with symmetry. 

Tom: Symmetry is beautiful. So then you eventually made your way back to Australia? 

Cheryl: Yeah, I came back straight away I wasn't quite ready to come back, but I'd signed a form that said I would come back. Right. And I got a three-year research fellowship at the Australian National University, which was great because that had been where I would have liked to have done my PhD if I hadn't had the chance to go overseas. And two weeks after I started my research fellowship, I got offered a six-month visiting assistant lectureship at the University of Virginia and they gave me, after two weeks, they said yep, sure you can have a six-month leave without pay. So I went to the University of Virginia where they had a special group theory semester. Extremely exciting, like every week, some group theorists came in from somewhere in the States and gave a lecture, and I started working with other people. That was fun, and then I came back to my research fellowship. I met John, who was studying statistics. We got married. He was doing his PhD in Canberra and there were no jobs anywhere. 

Tom: So in a way kind of that government vocational officer…

Cheryl: He was right, maybe...[Laughs] 

Tom: Yeah, no, but there were jobs… 

Cheryl: So on our honeymoon, when we were at separate conferences as you do. 

Tom: [Laughter] Because this is how you do it. You go to your conference, honey, I'll go to mine. 

Cheryl: Someone said, there are these four jobs at the University of Western Australia, two for one year, two for two years, but unfortunately the application dates passed. So I thought, oh gosh. So I wrote this letter saying: my husband and I would like to apply for these jobs, and when we get home from our honeymoon, we will send you our CVs, which is on the records of this university! [Laughs] So in the end, we came here. I had a two-year-old... 

Tom: And they bent those rules? Cool.  

Cheryl: And John had a one-year job. And we thought...I thought we'd just be here one or two years, depending, but...

Tom: It's been a few more than one or two years... [Laughter] 

Cheryl: So I was very young, I was the second woman to be a professor at an Australian university. 

Tom: Second professor...? 

Cheryl:...of mathematics. Mathematics, yeah. And probably it only worked because we didn't have to relocate. The situation with such young kids and that's so...you know, we didn't ever plan to stay here, but we did and it's been very, very good. 

Tom: This became home. 

Cheryl: Yeah, it did. 

Tom: Thank you. Thank you for showing that journey, and there's of course more that happens after that, which is where you get recognised for the contributions that you've made to mathematics. You're elected a Fellow of the Australian Academy of Science. You got yourself the AM...

Cheryl: That was the AM, yes, and now I will get the AC in a few weeks when I travel to Canberra, yes. 

Tom: Which means you get to get rid of that and put on a new one, don't you. So yeah, congratulations for those things being recognised, earned and recognised. 

Cheryl: Thank you. It is wonderful for mathematics. I mean, mathematics is kind of invisible. It's so important in everything, but you see the sexy things and you don't see the mathematics so often, and I think it's just important, you know, these computers, these cameras, they just rely on the mathematical algorithms which make them work. 

Tom: There was also the PM's Prize for Science, as well, wasn't there? 

Cheryl: Now that was amazing. That was the end of 2019, that's right, yeah. So I was the first person working in Western Australia that ever won that, and it was 20 years of the PM's prize before that happened. So that was astounding for that to go to pure mathematics again, that was great, and it's so wonderful that it's also for women, as well as men and there's a role for women. I mean there's so many layers in which that was a lovely experience. 

Tom: So you don't you don't get awarded the PM's Prize for Science, you don't get given an AC, you don't get a Fellowship to the Australian Academy of Science if you haven't done something that has made a huge impact. I'm just wondering if you might be able to potentially find a few words to think about what your work may have done for my mum who might be watching this, or this society as a whole? How have you impacted my life without me knowing? 

Cheryl: Gosh, yes it's really hard to know with mathematics how it's been going to be used in the future, but the sort of work I've done really introduced new methods and new ways of analysing symmetric structures and then new algorithms for making computers act super-fast if they were studying symmetry, which computers do sometimes, and they are...and the work of some of my students are looking at codes for encoding, sort of adding error correction to sort of communication codes in new ways, but I've been using the finite simple group classification which classifies the building blocks of symmetry, and that was like a watershed result in the 1980s, really only pinned down in maybe 2011, when the last papers and books were published, but we were using it from then on and it changed the way mathematics in my area was conducted and I was lucky to be in the first few people who are sorting out how to use it and how to study the internal structure of these simple groups and how to then apply it, how to work out what new fundamental theory was needed in order to study and analyse networks, the sort of networks that you might use for communicating across the world. Yeah, I mean there's two different sides of this. One is the encryption, so you stop the wrong people reading it, and the other is robustness. So you send a spaceship into space and you want that spaceship to send back pictures that you can reassemble and view on earth…

Tom: …and you want to know whether or not you can access that information...? 

Cheryl: You want to add some extra error correction capability, you want to add some redundancy, so that when you get the stuff back and it's not quite what was sent, you can say aha, the only thing the only other message that's a true message from this gobbledygook that I've got is this one, so I'll correct it this way, and so it's adding that redundancy in a special way, in a symmetrical way that you are very likely to be able to correct it, so that's code that's encoding stuff... 

Tom: …which does speed up communications as well, because you don't have to resend stuff... 

Cheryl: You don't resend it, and you can correct it, yeah, and you often don't have the extra opportunity of correcting if it's coming from Mars! 

Tom: Yeah, of course. I hadn't realised that symmetry would have an impact in this space. 

Cheryl: So you sort of imagine you're in a multi-dimensional space, and there are so many things which might be a message, you might interpret as a message, but you want to sort of space out the messages, so that they're not too close. And so you imagine that each message is a string of zeros and ones because you can go on-off, on-off electronically, yeah, and so this is a message, and that's a message, and you send it back and the sunburst hits it and destroys this one, so it's sort of moving that message away from itself. So what you get back isn't a valid message, it's something else, but you say, oh it's closer to that one than anyone else, so I can correct it. So it's trying to do that in a smart way, which is the...which is the code. [Laughs] 

Tom: Yeah okay, so very very cool, and I know that that was a bit of a rough question originally, because it's really hard to say, well what's the impact of your work been, but what I'm hearing here is, not only are there like direct impacts that you've made on people's lives and work by providing or assisting with extra tools, you've effectively... you've enabled a huge world of mathematics to exist in a much more efficient way and in a more powerful way, and from there, then there's the discovery that might have a practical impact or whatever. 

Cheryl: I mean, things are starting like that. There's these power line communications, you want to be able to have zillions of different people communicating at the same time, you know, through the one...through the one cable... 

Tom: The power lines are already there, that infrastructure…Why another cable when there's already one? 

Cheryl: They’re already there, make use of them. One of my PhD students is sort of seeing the consequences of his work or our work in working out the kinds of codes which you might be able to use in that application, which is nice, yeah. 

Tom: Very cool. Do you think you can have like a punt in the dark as to what might be the kinds of things that people will use your work for in 20 years, or 50 years time? What would you like to see them doing? What's the next step? 

Cheryl: Oh, I'd like to see better service through communications for everyone in the regions not just in the cities and so I don't know whether some of these new methods would be so applicable, that it would be cheap, it would be effective, it would just work to make to make a more equitable society so that we could have people in any place to have the same abilities to access all the facilities. 

Tom: Yeah, because the copper for the electricity does run first... 

Cheryl: Or it might go through the through the satellites, or it might... Who knows. I mean there'll be technology, but the mathematics will also help with the technology and I hope the solutions will come. 

Tom: I think that's actually a really nice place to round off, because what I'm hearing through your story of your education as you came through schooling and through your university studies, before eventually having your Chair as a professor here at UWA, and your work outside of your academic work, where you're supporting programs like the Mathematics Olympiad for girls and those other things, and what you've just described of bringing access to people who don't necessarily have access, this sounds as though it's a life dedicated to equity, and actually I think that's a lovely legacy to be able to leave behind, potentially the tool to continue that work. I wish that I would be able to have that kind of impact myself, so thank you. 

Cheryl: Thank you, Tom. 

Tom: Just for the final question: to me, a scientist is someone who…dot dot dot... 

Cheryl: To me, a scientist is someone who's passionately wanting to understand the world and works towards that in their whole life. 

Tom: I wholeheartedly agree with you. So thank you once again, and thank you for joining us at Mug of Science. Keep tuned for the next episode, whenever that happens to go out, and happy Pint of Science festival!