Episode 71 - Emily Howard

My Favorite Theorem

Evelyn Lamb: Hello and welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance math and science writer based in Salt Lake City but currently podcasting from my parents’ house in Dallas, which is actually not any warmer than Salt Lake City right now, unfortunately. This is your other host.
Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I'm in my faculty office. I'm usually — I'm the chair of the department. But I'm hiding out in the faculty office today. Actually, I was looking for better wireless. And it seems to be working a little better in here. But it's so weird because I have nothing in this office, like nothing. It's very strange. So anyway, how are things going for you?
EL: Not too bad. Yeah, seeing my family, which is nice, and very excited about today's episode. So let's get right to it! We're happy today — both of us are music lovers, and we're very happy to introduce our guest, Emily Howard, a composer. Emily, do you want to tell us a little bit about yourself?
Emily Howard: Yeah. So I'm based in the UK, in Manchester. I'm originally from Liverpool. And I'm a composer. I love writing for large ensembles, large acoustic ensembles, such as the orchestra. I also write vocal music, choral music, and also chamber music. So a lots of different areas. And I suppose probably the reason that you've got me on here is that I've got a real interest in mathematics. And actually, I have a degree in mathematics and computer science, my undergraduate is in mathematics and computer science. And I suppose that, you know, definitely it's one of the key influences on my work.
EL: Yeah, I was listening to — it might have been the BBC Proms a few years ago — I was listening and saw this piece that I think was called Torus. And I thought, “You don't accidentally name something Torus.” So I decided to try to find out more about this person who had named her composition Torus. And so yeah, I found out that you had a math background, and thought it would be just really fun to talk to you on the podcast. So yeah, can you talk a little bit about the — I know, you've done some collaborations with mathematicians, you know, written pieces, like kind of in conversation with mathematicians in the composition process, and I would love to hear about that.
EH: Yeah, so I mean, I suppose actually, in 2015, I had I think it was a Leverhulme fellowship at the University of Liverpool, working within the mathematics department. I had been invited by Lasse Rempe-Gillen. He's a professor in dynamical systems. And I think he'd been in touch because he had himself played the violin in an amateur orchestra in Liverpool, and they had performed a piece of mine called Mesmerism. Actually, it was after Ada Lovelace. Ada Lovelace used to dabble in all types of things, including mesmerism, sort of a form of early hypnosis. And the piece, I mean, that piece was a piece for solo piano and orchestra, and he was playing in it. And I think he thought it will be great to invite me as someone with a mathematics background back — after 15 years away in the music world— actually back into the maths department. And, I'm so glad that this happened, because going back in and speaking to lots of different mathematicians in a different way, rather than, I don't know, having to take exams and study, I was more being an observer, having amazing conversations about people's research. Regularly I'd speak to someone in one area of mathematics and another, and I suppose I felt — what I realized was that mathematicians often don't understand what each other are speaking about. So I didn't feel so bad about it, that I could sort of dip in and I suppose it helped me to take a more global approach, or to take in more general ideas, because I think that's one thing, perhaps, you lose if you don't practice mathematics regularly, you know, you lose this very detailed approach, and that's kind of annoying. It’s really annoying in many ways, but I also find it annoying when you can’t completely understand something because you'd have to spend quite a few years really thinking about it in depth. But then something is also gained from sitting back and taking a look at everything and absorbing it in a very different way.
So anyway, we worked together on a set of chamber pieces. You can find them on my website. One’s called Leviathan, another is called, well, these are all pieces, exploring ideas from lattice research in dynamical systems. And I suppose they're all based on thinking about perturbations, and thinking about — I’ve got some etudes as well, they’re called Etudes in Dynamical Systems — just trying out very particular things with datasets, but also conceptual ideas about reaching stasis and loops from something. I also spoke a lot with colleagues in mathematical biology and also in topology. And I remember, I think this is where, kind of, talking about Torus, it was kind of born, the idea, because this period of time gave me more confidence to actually do something with mathematics. Just to go back back a bit, after my undergraduate in mathematics, I'd spent a couple of years doing a Masters at the Royal Northern College of Music in composition, and then more years at Manchester University doing a PhD in composition as well. And I suppose, when I was doing this, I really wanted to use these ideas or concepts from mathematics, but I didn't really have a musical technique to do that. I suppose that grew. And at this point, it sort of all merged, and I was able to perhaps achieve something that I wanted to achieve with these mathematical ideas through music. And so yeah, just to go on, you mentioned Torus. And I'm immensely proud of Torus. It's a big orchestral work, it's like 23 minutes long, based on the mathematical shape, a torus, a doughnut.
And this is when I had actually just met a mathematician, Marcus du Sautoy, who I’ve worked with quite quite a lot. He's based at Oxford, and we met because of this thing. And when we spoke first, actually, he was writing about, no, he’d written a play about a torus. And I was sort of saying, “Well, actually my piece is about a torus,” so we really bonded on this. And one of the things that interested me most was, you know, the, the BBC Proms are held at the Royal Albert Hall in London. And the first thing he said was, “Well, that is a torus.” And I just thought, How funny. It really is, because you can run round and round it, you know, it's just just really nice to, you know, hear this.
KK: So I'm especially fascinated here, because my son is a composer. Well, he just finished his undergraduate degree, and now he's in graduate school for composition. And so I'm always — first, I'm encouraged that composers can actually make a living, right, because whenever I would tell a mathematician, one of my friends, “Oh, my son's studying composition,” they say, “Well, how’s he going to live?” And I say, “Well, I don't know,” but clearly it’s possible to figure it out. So I'm super encouraged. Yeah, so Evelyn, you had a real question, though.
EL: Oh, well, I was just saying this is an existence proof then for your son.
KK: Yeah, that's right. That's right.
EL: Yeah, well, I was — you know, music is a very non representational art form, unlike a canvas where you could — I mean, it's still hard to represent mathematical ideas on a painting. But can you say anything about how you do use the form of music to represent mathematical ideas?
EH: I suppose — it's really interesting that you say that, because I completely agree with you actually, that it is very difficult to do that. And I wonder if, even if I, especially in abstract music, if we take aside — you know, if it has text in it, that's a very different thing — but in an abstract form, actually, I wonder if I ever tried to absolutely represent something from mathematics within music, whether in fact, anyone would know. And I suspect, actually, there are cases when I've worked very closely with people and they really know what I'm doing. I think they can certainly tell, but actually, I don't think that that's necessarily happens. And also, it's definitely not what I'm trying to do. Absolutely not. So I suppose my aim is not to represent mathematics. My aim is to — I mean, I love hearing about mathematics, and I'm completely inspired by processes and systems and patterns. And I suppose what I'm doing is taking them, and that’s a catalyst for my creative process. And so I do think something comes through, but I think it's more that I couldn't make what I'm making without doing this. And it's more that something new is occurring that came from thinking about these things. But it's definitely not a representational thing. I mean, that's definitely the case for maybe the more large-scale pieces we could talk about, like Torus.
But I think also there have been a couple of pieces when I have tried my very best to represent ideas, and one of them would be this set of dynamical system etudes. That was one of them. And then another one would be the music of Proof, which is, I wrote a string quartet because Marcus and I were discussing proof and different ways you can prove things: proof by contradiction, by induction. He wanted to, to put those to me, and then I would create responses to different kinds of proofs. And so I suppose that’s as representational as I've been, and actually, it was really useful, because in doing that, then I gained a whole set of like, I thought about — I've never tried to write music before as though I'm solving a mathematical proof. But actually, in doing that, that led me to new places. So that's kind of what I'm trying to do, find new ways to do things and make new sounds.
EL: Yeah, maybe respond more than represent you?
EH: Definitely.
KK: I mean, certainly, some music is sort of, you can tell, it's not deliberately mathematical, I don't think. I think of like, like Steve Reich's minimalism percussion pieces, right?
EH: Clapping Music.
KK: Yeah, they’re so cyclical. And you know, you can kind of, if you think about it as a mathematician, you can kind of imagine, well, if I saw this on the page, it would almost look like — our listeners can't see my hands doing this — but you can imagine sort of, you know, intersecting sine waves and things like that. So I can see how you could do that.
EH: And to take that further, I wonder if all music in some way can be — well, I say reduced, and I don't like to reduce — but you could certainly represent ideas in very complex music, I think mathematically probably. Not that anyone necessarily has. But perhaps the secret to lots of things is in really complex music represented. I mean, I don't know.
KK: Well, there’s a whole journal of mathematics and music. I mean, you could you could certainly, I sat on a PhD committee for composer on campus here, who really was trying to do these very strange time signatures that were sort of approximations of pi and things like that.
EH: Wow.
KK: And I put the question to him. I said, “Okay, I mean, you can make a machine do these. But, you know, can a human do this?” And he said, “Well, no, not really.” But it was interesting.
EL: Can humans do math anyway?
KK: This is not a philosophy podcast. I don't know.
EL: Yeah. Thankfully,.
EH: Just to say on that on that subject. I mean, I've got certain things — there are certain things you do put in notation that are absolutely beautiful mathematically, you know, but in fact, yes, they're not really possible. But they they do make a performer think in a certain way, and it will give you certain results. So I mean, I think there can be, I think they're beautiful, and they can be there. But you just can't expect perfection, perhaps.
EL: Yeah. All right. So we always like to ask our guests, what is your favorite theorem?
EH: I mean, we've kind of been speaking about it. That's, that's the main problem with this. I mean, actually, I would say, rather than a theorem, it's definitely these shapes. So I mean, let's take the torus, but I've got this fascination, I'd say, with thinking about mathematical shapes and thinking about, you know, them being far away, and also thinking about being on them. And I suppose, I mean, so you've got a torus. I mean, so, if you think about the difference between, say, flat geometry and a sphere with the spherical geometry, and then, I mean, there's a pseudosphere, and then I would call, like a negatively curved geometry, I've got a piece actually called Antisphere because I've worked out that an antisphere, there's a word antisphere, which is the same as the pseudosphere, and the word antisphere is a lovely word, and I don't like the word pseudosphere. So I called the piece Antisphere. And I suppose I've just got this — because, you know, my music is usually notated. So I find it very interesting as a process to be in my mind wandering around these shapes. And I suppose, I also think it’s very useful for music because they don't necessarily need to be 3d, I mean music, potentially, you could perhaps hear really high-dimensional mathematics. Because, you know, you could be traveling around and you're not necessarily visualizing it.
So my answer to your question is that I like these mathematical shapes. And I've been thinking about them and really studying them and using them to influence these large-scale orchestral work. So that's Torus, Sphere, and Antisphere. And going forward, I'm now looking at, well, I'm trying to look at Thurston’s eight geometries and pushing myself in this direction with the aid of a number of very kind mathematical friends, because they're very difficult. And yeah, so that's kind of the direction where I’m going.
And I find, you know, I suppose that's what we were saying earlier about the representation thing. So there are a number of stages. So I'm thinking about this tours, I'm thinking about traveling around it. Shall I give an example of the compositional process for writing Torus?
KK: Sure, please.
EL: Yeah.
EH: Okay so Torus is a piece, yes, it was for the BBC Proms. It was performed first in 2016, by the Royal Liverpool Philharmonic Orchestra with Vasily Petrenko. Now I’d say, I wrote it over at least a year or so. And I'd also been thinking a lot about this just previously, thinking about traveling on this torus. And what I was thinking about when I wrote it was the idea that you're traveling round and round one direction of the torus, you know, and I took these very consonant chords, see, I've got major sixths. If you hear it, they’re a very constant chord. And I start with this major sixth, and I travel up one and down one and up another — they're going up in semitones, and down, and they reach a point and they come back again. So that the harmony is very much also shaped like a torus all the way through. And it's skewed, when you hear it, it's more complex than just listening to it timed because I've got a real combined fascination with exponential functions and the idea of really big things becoming absolutely minuscule, but also kind of being related. So, you know, this kind of journey around the torus, the first loop in this orchestral work is about five minutes long. And if you do listen, you'll hear it in the strings. And to my mind, I've got this sort of ever-expanding enclosing torus idea in the strings, but, but maybe on the surface of it, as you go round, you hear different things each time you go round. And that's perhaps in the wind and in the percussion, and in the brass. So it’s almost like, you're on a landscape and it's changing.
Now, the piece works by this — suppose the radius of the doughnut-shaped Torus, sort of shrinking. So you go around it, and it gets quicker and quicker. And you'll hear — I think it's around about maybe 17 minutes in, there’s a viola solo. Because it's really big, and it sort of shrinks down, I think there are seven, sort of around this this way of the torus, and you get this viola solo that encapsulates these major sixth idea because you can just play all of that on one solo viola. And then there's this sort of huge shift in thinking in the piece. And, again, that almost came from thinking about dynamical systems and just completely changing the goalposts, and then you're traveling very fast. And in my mind, we've flipped to that we’re now thinking about going on the other direction of a torus forever and ever. Do you know what I mean?
EL: Okay.
EH: You’ll hear that. It’s a huge moment. And that music seems to completely change. And I don't think anyone listening to it will hear a torus shape, because you can’t, because you're hearing my ideas about journeying around it. Do you see what I mean?
EL: Right.
EH: But actually, it really helped me to think about this thing and you know, it is absolutely about this sort of journeying around on it.
EL: Yeah, I don't know, I'm almost imagining like you could, maybe — I wish I had like a, some kind of inner tube or something. But like, maybe the first loop around is on the top where it's further apart, and then you're kind of almost falling into the middle where the You know, the two sides of the torus, or the hole of the tours is kind of small. And then you you kind of flip the other way and start. The next time I listen to it, I'll have to imagine that kind of journey.
EH: The thing is, I’m sure now I've told you that, you probably will hear it, you know? And I think as well, I mean, it's not, it’s never so obvious, because I'm also doing a couple of other things as well. So each journey around this torus — so I said the first one's about five minutes — you’ll also hear these major sixths alternate in the string, so it’s almost like there's one side and there's another side. So you hear this sort of, they appear to travel like this, and then on the other way, they're sort of all, everything is together, so it's rhythmic unison. And it and there's a sort of written-out rall, so it gets slower and slower. And then you'll hear, it comes around again. So I mean, there’s that and there, and then sometimes as well, one side of the torus. That is going on in the background in my mind, and that's definitely happening. But in fact, there's something else on the surface, like there are these very loud things going on sometimes. And they obliterate this thing, but it's always there. And a bit like, say, a painter might have a layer of something to start the piece, this is my layer, this sort of journey. And then it builds up.
KK: And you totally, you had Evelyn at viola solo.
EL: Yeah, yeah, I do play viola.
EH: Great. You’ll hear that.
EL: Yeah, you're pushing all our buttons. Kevin’s son’s composer thing, my viola thing. You’re just excellent. Yeah. What I was wondering is, do you know, why were you drawn to the torus specifically? Do you know?
EH: It’s a really good question. No! I’m not sure I do.
EL: Well, you sort of mentioned the Sphere and Antisphere and Torus are kind of this set. And those can be the three different two-dimensional geometries. But I'm wondering whether you kind of already liked the torus and just were excited that you could use it here, or if you kind of sought it out because it is this model of flat geometry.
EH: That's very interesting, because I think I was trying — when I was writing Torus, I was already thinking about writing Sphere as well. And, you know, in many ways, there's something too perfect about a sphere.
EL: Yeah. It constrains you. It’s very limiting.
EH: It's a problem. And the other thing is, I was having to think of a title. And actually, the title Torus is just the most beautiful word. And there, you know, I think it's the fact that the sphere was too perfect. And there was a way to sort of have two very different things on this torus. But I think it's also true to say that I hadn't thought until later on to do the set. It emerged, you know. So I was sort of playing around and suddenly, I thought, when I'd written Torus, I thought, “Well, actually, that's quite a really wonderful way to think about things.” And then then came Sphere. And that's a shorter piece. It’s a five-minute piece for chamber orchestra. I suppose, I was thinking more about spherical geometry. As I said, I found that one more difficult because it had to come straight after I was writing Torus. However, a couple of years later, that's when I decided to write this Antisphere. And that's for the London Symphony Orchestra with Simon Rattle. And it was just performed a few months before lockdown. So I was very lucky to have that happen. I'm really grateful. And I'd say that, I mean, it's interesting, because somehow, I feel that my link to the maths in it is stronger with Antisphere. And I think that's just because I've gained experience and maybe confidence. I suppose with Antisphere, I was thinking about harmony. I like using quarter tones. They’re present in a lot of my music over the last, say, 10 years. We’re used to maybe the 12 semitones of the scale, and the quarter tones are just in the middle. And I suppose I was thinking about, you know, the classic angles in a triangle, which add up to 180. And then on the sphere it’s more, and then on the negative curves, it’s less. And actually in Antisphere, I use that so that you get these chords that we might all recognize, like a major chord, but actually it's been shrunken in some way.
EL: Oh!
EH: And so it's weird. And it sounds weird. And also, there's a section in there, it’s a very fast section of a kind of circle of fifths idea. But actually, I think it's a circle of 4 3/4. And they're great, because they last longer. Obviously, you know, these players are amazing, because they can do this in an orchestra ensemble.
EL: I was going to say how did you get — I think I would struggle to play quarter tones on purpose. I’m sure I’ve played some by accident.
EH: I mean, I was really blessed with this string section, but it is an amazing experience. And I think you do feel because you do feel this — I mean, the association of a major chord, I mean, I don’t think I’m using major chords, but whatever I’m using — you can feel it, but then you can certainly feel that something's happened to it. And I really liked that. And another thing I did was the rhythmic thing. We've mentioned Steve Reich, and I like the thought that you've got a regular pulse in our world. But if you're somewhere else, or looking in on a different kind of — well, the thought is that you're the pulse might, if it was like 1-1-1. But in fact, it could go 1-2-4-8. So actually, therefore our perception of this piece of music is not one of a regular pulse. In fact, I have created it as though it is, but it isn't. So we hear a short section and a longer one. And then and by the time you get to here, it's so long, you perceive it as something completely different. And I love that as a compositional process. So I did that. In Antisphere, I've actually taken a chord sequence that I used in Mesmerism, this this early Piano Concerto I wrote in 2011. And that’s, quite regularly, you know, there are a few chords, at the opening, you hear solo piano playing these chords, and they're quite regularly spaced. But in Antisphere, what I've done is I've taken them as the basis, these piano chords, as the basis of these sort of chords that start off like a bit and actually end up as three minute sections. And loads of weird things going on in between you like this with the percussion, there's loads of weird metals and sort of resonant sounds. And so, yeah, as I said, it gives a completely different sort of perception of what's going on.
EL: Yeah, I don't think I have listened to Antisphere yet. But I am now going to seek it out because I really want to hear this — especially if I've got in my mind this idea of, like, hyperbolic triangles with their, you know, curved in — or they look to me like they're curved in. If I actually believed enough, they would look like straight lines to me, because I would really embody the hyperbolic metric.
EH: But and this so this is happening in pitch, but it's also happening in sort of rhythmically and timbre in lots of ways as well. I did write an article, Orchestral Geometries, which I will post, you know, we could put there. And actually, I've been lucky enough to have wonderful recordings of the three pieces, and also the scores as well alongside so yeah.
EL: Oh, wonderful. So yeah, another thing we do in this podcast, which we kind of already done, is we ask our guests to pair their theorem, or mathematical object, with something. And so yes, I kind of assume that you would be pairing it with your compositions based on these, on the torus and these other shapes. But yeah, do you want to add anything else about that, or any other pairings? You know, if it's just a nice cup of tea or something?
EH: Or a doughnut? I was going to pair them with that. And actually, I was going to pair them perhaps with, you know, we could we could play a little clip from one of them. So this clip is a little bit from the orchestral work Torus, and I think, it’s sort of about two thirds through where everything changes. So before then you've been rotating round, the kind of doughnut shape of the sphere, no sorry, the doughnut shape of this torus. And then you go into this viola solo and then you hear the huge perturbation, and that's when you change as though you're rotating. At least when I was writing, I was thinking about rotating around the other part of the torus.
[music clip]
KK: So I really, now I'm sort of curious to know how you’re — I mean, I know you haven't thought about it yet. But pursuing the sort of 3D geometries, that's going to be weird. Have you seen these, people have tried to visualize these things using VR. Have you seen any of these explorers?
EH: Yes, yeah, I have. And actually, if you have any more, I would love to have them because it's very helpful to see them. We've been looking online, and there are quite a few. You know, there are some amazing websites and people, actually programs for doing it. And we were kind of exploring it. But I am interested. And it's a new pursuit, you know, so I'm really interested to, well, think about it more. I think it'll take some time as well.
I'm currently writing a piece. And I was thinking that it will be very nice, now having written these orchestral geometries, to embody sort of a process of moving between the different geometries a bit. So that you could, I mean, no one need necessarily know this, but musically, there might be a difference between this spherical and the — but they could, so I'm really interested in that. So I'll do that. But I also I'm interested in this, is it H3 [hyperbolic 3-space]?
EL: Yeah.
EH: So yeah, I'm interested in that. And it's kind of frustrating, because I'd really like to understand why. And it's quite difficult, I think. I'm not sure. I think it's probably well beyond me. But, you know, the thought of this dodecahedral space and moving around and the twist, I'm really interested in the fact that it would you know, that the spherical that these twists, change what dimension, it’s really interesting.
EL: Yeah, yeah. And I'm wondering how you can use like, the specific — what am I trying to say, like, the opportunities that an orchestra gives you to sort of, you know, can you, like, pass off ideas from one section to another, can that give you a twist? Or, you know, something like that, how you could, I guess, use the tools that you have at hand to kind of explore it in a different way than you might if you were trying to draw it on a piece of paper?
EH: Yeah, I think, I mean, it’s probably better doing that than drawing on a piece of paper actually. Like, I feel there must be something about the way I think that I like — these thoughts, and these kinds of systems and shapes immediately present musical ideas to me. So, absolutely, you know, what I would be interested in is a process that you could audibly hear becomes something different when it's spherically curved, and becomes something different when it's, you know, hyperbolic, and making that more and more extreme. So the math underlies that, and perhaps, maybe it's a process of me imagining going through this, these these dodecahedron and things. But in fact, that will be a layer, and then after that, perhaps then, I do something more musical, as in this has given me this data, but I accentuate it in ways, and perhaps also go against it and things. Because I do think as well with art, usually, you're asking a question rather than answering anything, which is a real difference, I think. Because mathematicians, you're so interested in truth, or mathematical truth, and you’re really bound by it; it’s really important. But I always feel that I'm more interested in what your process is in these wonders, and then I'm just using that to sort of leap somewhere unknown.
KK: Well, so are we. We're leaping into the unknown all the time, we just, then want we want the answer once we get there, right?
EL: I do think there's a similarity in that, you know, the way mathematicians approach things a lot is, you kind of set up these axioms, and this is the rule system I'm going to be working within. And I think that forms of art do that as well, you know, say, this is the aesthetic system that I'm working in, or this is, and maybe they're not as rule-bound as mathematics is, because when you say like, these are the axioms, or these are, you know, whatever, I'm doing, then you're just really stuck with them. And with art, a lot of the times it's about breaking the rules. But I do think, you know, you kind of set up these, sometimes composers can set up compositional rules where, like, Okay, well, I'm writing a fugue, which means that I have, you know, I have this kind of structure. And the allowed things to do are like transposing it or flipping it backwards, or things like this, and say, like, well, I'm working within this form, in this way. So, I mean, I did a lot of music in college, and I was kind of torn between going into math and music. And I think the way that I thought about them kind of tickled the same part of my brain, that’s why I was interested in both things, and ended up, you know, in math instead of music professionally, but I do think we've made these aesthetic or form kind of rules in music or art. And now we're going to work within them, just the way mathematicians do with axioms.
EH: Yeah, I mean, I completely agree with you on that. And I also would say as well that I agree, and actually working closely with professional mathematicians has really kind of opened my eyes to how much they are going, because I don't think you learn that, you know, these are the answers, actually it is a lot of guesswork. And it's really, yeah, leaping, as you say.
EL: Yeah, well, this, this has been a lot of fun to talk with you about this. I really hope our listeners will go find your pieces. And we'll definitely link to your website and the article you talked about and everything. Is there anything else that you want to share, you know, things you want to suggest that they look into or read or any concerts coming up that people could actually attend or livestream?
EH: Yes. So actually, next week, the BBC Philharmonic is playing sphere in Manchester. And that should be on Radio 3, as well. I’m based at the Royal Northern College of Music in Manchester. I’m a professor of composition, and I run, I direct, the PRiSM lab. PRiSM is a research center for Practice and Research in Science and Music. And I'm lucky enough to work there also with colleagues. Marcus du Sautoy joins us and David De Roure, as well. And I suppose we're interested in mathematics meets music meets science meets AI, and there are lots of different types of composers. We have a blog, and that's where this orchestral geometries blog will be. And lots of very exciting, very different things going on there. So I just wanted to mention that and maybe I could also give you a link for that as well.
KK: Sure.
EL: Yeah, that would be great.
KK: We will include it. All right. This has been fantastic. Thanks for joining us, Emily. It's really been great.
EH: Thank you so much for having me. It's been a pleasure.
[outro]

On this episode of the podcast, we were delighted to talk to composer Emily Howard, who uses her mathematics background in her compositions, about her favorite mathematical object, the torus, and the orchestral work it inspired. Below are some links you may enjoy after you listen to (or read) the episode.
Emily Howard's website
Her page about the composition Torus, including a recording by the BBC Radio Orchestra
Her article Orchestra Geometries
The November 11, 2021 BBC Radio concert featuring Howard's composition Sphere
PRiSM, the Royal Northern College of Music Centre for Practice and Research in Science and Musicthat Howard directs
A website visualizing the eight Thurston geometries for 3-dimensional space
An article by Evelyn about the pseudosphere (or antisphere)
Our episode with Emily Riehl, who is relevant to this episode because she is both an Emily and a violist