Interview with Myfanwy E. Evans: Entanglements On and Models of Periodic Minimal Surfaces

Abstract

KK: Maybe we can start with a question concerning your training and profession.

KK: Maybe we can start with a question concerning your training and profession. You’re a trained mathematician and you often work in fields which are by no means fully mathematically developed. Hence, you’re doing fundamental mathematical research on the one hand. But, on the other hand, you’re often classified as an applied mathematician, since a lot of your work is relevant to the materials sciences and biology. How do you see the relationship between pure and applied mathematics? Where do you see the advantages and disadvantages of applied mathematics concerning research, in the sense that, for example, applied mathematics advances mathematics as a whole?

ME: My initial training was in mathematics, but my PhD is actually in physics and mathematics together, that is, in the Department of Applied Mathematics inside the Research School of Physics (at the Australian National University). As such, I was exposed extensively to physics (and a little chemistry and biology) throughout my PhD, even though the content of my research was based in mathematical techniques. I was always very much connected to the idea that these things should be useful in physics, but also the physics should be useful for generating new directions in mathematics. The interdisciplinary research that I do, along with many other people in my research community, is an interesting connection, because it’s an application directly from pure mathematics to the natural sciences. The typical field of mathematical physics involves a very different type of mathematics and very different type of physics to what I’m looking at. But we’ve made a connection from pure geometry and topology, subjects which always sit inside the pure mathematics department and never in the applied mathematics one. And this connection was made between these domains and the materials sciences and physics. One might say we’re doing applied pure mathematics. We’re using these pure mathematical constructions applied in a theoretical context in the natural sciences, such as in theoretical biophysics, theoretical biology, and theoretical chemistry. We’re staying on a theoretical level behind all of these different disciplines. In that sense, it’s applied from the perspective of a mathematician, but it’s not really applied from the perspective of a natural scientist. At no point are we really going into real-world applications of the ideas. We’re just stepping between the disciplines on a theoretical level.

MF: I would like to focus on this connection between mathematics and research on materials that characterizes the applied pure mathematics which you just mentioned, and to concentrate on a particular area of your research which looks into mathematical three-dimensional entanglements. The starting point for this research is, surprisingly, the structure of the skin, to which we’ll return later. But to concentrate for now on these entanglements, you’ve worked with visual models created by computer visualization programs, focusing on three-dimensional periodic entanglements (see Fig. 1a). Can you explain what these visual models mean for you? Are they illustrations? Do you understand them as geometrical models? Or do they serve as a source of inspiration?

Fig. 1

© All rights reserved

a Three-dimensional periodic entanglements Reprinted with permission from Myfanwy E. Evans and Roland Roth, “Shaping the Skin: The Interplay of Mesoscale Geometry and Corneocyte Swelling,” Physical Review Letters 112, no. 038102 (2014): 3, Fig. 3. © 2021 American Physical Society, all rights reserved. b A braid with four strings. From Emil Artin, “Theorie der Zöpfe,” Abhandlungen aus dem Mathematischen Seminar der University Hamburg 4 (1925): 47–72, here 54, Fig. 7.

ME: When it comes to a three-dimensional entanglement, and particularly a periodic one, we can’t say anything (yet) about it mathematically. If I have two different complicated weavings of filaments in space that are sitting side by side, I can’t say whether they’re the same or different. You could compare their exact geometry, you could try to lay one on top of the other, but when you look at entanglements, you want them to move around. The question you want to ask is: can I take one and deform it a little bit in space, not allowing the strings to cross each other? Can we deform one entanglement into the other? That question is possibly unanswerable. When I look at graphs (or collections of them), there’s information that I can write down—how the edges and vertices are connected, for example. When I have a filamentous structure, I’m lacking concrete characteristics that I can write down. So when I employ enumerative techniques to build three-dimensional structures, my end product is simply just the model. The models really are the research in this sense, that is, until we can find a better way to describe the entanglements mathematically. But until then, the models are it. So I think they do fulfil a very important role, as they are the research.

MF: Taking these difficulties into account, can one assume that braid theory¹ is not a sufficient tool to research these entanglements?

ME: While being careful about absolute statements, I think that it’s unlikely that braid theory (as it stands) will be sufficient for understanding these structures. Braid theory works nicely, because you can project a braid to a plane, examine the pattern of crossings on the plane (see Fig. 1b), and associate to this pattern of crossings an algebraic structure. If I have something that’s three-periodic (so truly three-dimensional), I can’t project it to a plane in a reasonable way. Two-periodic (like a fabric weaving on a flat page): that works. As soon as it’s three-dimensional, I can’t do it anymore without distorting or losing essential information. It becomes a purely three-dimensional geometry problem. Without this planar projection, the algebraic techniques of braid theory are simply no longer defined.

KK: We would like to examine a more concrete example: filaments.² One of the inspirations for your research is skin. Skin has a highly complicated structure that’s in constant exchange with the environment, and therefore undergoes transformations—for example, the swelling of skin in water. Modeling such a changing structure mathematically is a challenge. You’ve suggested modeling the filaments of the outer layer of the skin by describing the geometry of the swelling of dead skin cells with the help of triply periodic minimal surfaces (TPMSs).³ How can these surfaces model this geometry? And can you explain how this idea of modeling with these TPMSs came to your mind?

ME: One approach to analyzing the filaments in a three-periodic entangled structure is to consider them as physical structures, like a tube of a given radius (rather than simply curves in space), and try and minimize their energy as a way of representing that structure,⁴ giving it a canonical form, and as such giving us a way of comparing it with other structures. That’s something that’s done in knot theory already, and is termed ‘ideal knots.’ By finding a canonical form for the structure, you can compare it to other canonical forms as a way of comparing entangled structures. This is typically done computationally, as analytic solutions have proven difficult in most cases. I was extending these ideas of ideal knots to periodic structures, where I take a particular weaving of tubular filaments of a given tube radius and I computationally minimize the length of the filaments within the box, not allowing the tubes to intersect each other. There’s a set of filamentous structures that are constructed as decorations of TPMSs, where I found that the filaments weren’t straight in the minimized state, which was a big surprise and somewhat counterintuitive. These structures could be realized as symmetric cylinder packings in space, so I assumed that it would minimize their length when they take on the straight cylinder geometry. The actual minimizer is found when I allowed the filaments to curve, which in turn decreases the size of the periodic translations, kind of like compacting the structure down. This compaction actually decreased the length of the structure, and hence was identified as the minimizing structure. This physical property was counterintuitive, and lead to the question: when the structure is in its minimal form, is it somewhat ‘spring-loaded’? The simple process of pulling the filaments straight is going to isotropically expand the material. While working on this, I came across a paper by a researcher, Lars Norlén, in Sweden,⁵ who had been looking at the structure of skin and had proposed this particular weaving as the internal structure of corneocyte cells. I started to look at the swelling property and was wondering if it could be explained by our spring-loaded model. Our filament model has this nice deformation mechanism where it can expand and contract, keeping filaments in contact through the process. We started exploring our model in the context of the skin swelling, and it turned out to be a very nice way to model the behavior that we can see in the skin (see Fig. 2). We were coming at it from a purely mathematical perspective, exploring configurations, and every time we started to do calculations, it always matched up with the experimental data from skin. The relationship of the geometric structures to the TPMS in this case is also not coincidental, where the TPMS decoration is likely a biological formation mechanism for the intricate structures in the skin cells.

Fig. 2

© Myfanwy E. Evans, all rights reserved

The compact (left) and swollen (right) states of filaments.

KK: The history of these periodic minimal surfaces and their discovery is intertwined with the production of material models. The production of the material models of TPMSs started in the last third of the nineteenth century with Hermann Amandus Schwarz and his student Edvard Rudolf Neovius,⁶ who not only drew sketches but also built material models of these periodic minimal surfaces (see Fig. 3). This was followed in the twentieth century by Alan H. Schoen, during the 1960s, who built plastic models, but at the same time also did computer drawings in order to explore these kinds of surfaces, discovering in 1970 another type of TPMS, the gyroid.⁷ Schoen’s work is a reference in your own work. How do you consider the models of Schwarz, Neovius, and Schoen? And how are these models different—also compared to your own kind of modeling—from computer visualizations?

Fig. 3

a Neovius’s photo of his model of a triply periodic minimal surface from 1883. From Edvard Rudolf Neovius, “Bestimmung zweier spezieller periodischer Minimalflächen,” in Akad. Abhandlungen (Helsingfors: J. C. Freckell & Sohn, 1883), Plate 4. b The first physical model (1968) approximating the gyroid, made by Alan Schoen (https://schoengeometry.com/e-tpms.html, all rights reserved). It is composed of several lattice fundamental domains (source: https://schoengeometry.com/e-tpms.html, all rights reserved). c Approximation of the Schwarz P minimal surface. Graphic by Andreas Sandberg, 2012. CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/, all rights reserved).

ME: I find the spirit in which Alan H. Schoen did research very inspirational: a three-dimensional exploration. To come across the gyroid minimal surfaces, the mathematics is extremely difficult in order to get there. The mathematicians during the nineteenth century never got there, in the sense that they never found those minimal surfaces. Whereas Schoen’s more conceptual exploration of networks and partitioning of space was a different approach. He was looking at the network channels of the P and the D surfaces—two known types of TPMS already discovered by Schwarz (see e.g. Fig. 3c), which have a simple cubic network structure through the channels, or respectively a diamond network structure. Schoen’s observation was that there was a third network in a similar class. The cubic network has degree six vertices, the diamond has degree four vertices; Schoen was aware that there’s a third network that has degree three vertices. He started exploring what the surfaces would look like between these networks. This idea of exploring what you can do with three-dimensional structure materially and then diving back into what that means mathematically—that’s a really nice way of doing things that are too complicated for mathematics as it currently stands.

These exploratory techniques rely heavily on models, be they real models to hold and explore, or computer models that you can rotate and explore computationally. Observing different features and perspectives of the models brings new intuition for the structures. Virtual reality models are starting to play a bigger role in these explorations too, and to bridge the gap between the physical and computational models.

MF: If we return to your research on entanglements considered as a sort of ‘tiling’ of TPMSs (see Fig. 4), you also work with hyperbolic geometry, which is associated with this tiling and with the TPMSs. As you note with respect to your research on skin, “keratin filaments of the skin lie almost exclusively inside one of the channels of an appropriately sized Gyroid, leaving the second channel filled with water.”⁸ As is known, it’s impossible to isometrically immerse a complete hyperbolic plane—a surface which has everywhere constant negative Gaussian curvature—in a three-dimensional Euclidean space⁹; that is, a surface with constant negative Gaussian curvature embedded in the Euclidean space ${\mathbb{R}}^{3}$ doesn’t exist. But the gyroid is a minimal surface with the most uniform negative curvature and with finite average Gaussian curvature; hence, the gyroid can be considered as the ‘best’ embedding of the hyperbolic plane ${\mathbb{H}}^{2}$. In your work you choose a model of the hyperbolic plane called the Poincaré disk model. Here though, we’re not talking about a material model but we’re rather dealing with a model in another sense, in the sense of a realization of a set of axioms. Why did you choose this model? What advantages does it have?

Fig. 4

On the right, a ‘weaving’ of filaments on a basic unit of one of the TPMSs, induced from a free tiling (or packing of lines) in the hyperbolic plane ${\mathbb{H}}^{2}$, modeled via the Poincaré disk model on the left. Reproduced with permission of the International Union of Crystallography from Myfanwy E. Evans, Vanessa Robins, and Stephen Hyde, “Periodic entanglement II,” Acta Crystallographica Section A 69 (2013): 270, Fig. 14, all rights reserved. Here, the (cut) hyperbolic plane ${\mathbb{H}}^{2}$ is thought of as a covering of the basic unit of one of the TPMSs

ME: Yes, you’re correct that this is not a physical model but rather a mathematical model of the hyperbolic plane. The Poincaré disk allows me to visualize tilings of the hyperbolic plane in a nice way. In particular, this is a good model to use because it’s conformal, so it preserves the angles between lines in the plane. I also think that symmetry, reflections and rotations in this model has quite an intuitive feel. The research does require a lot of intuition. Because it’s not really clear what techniques should be used, and it may be that this research is in its infancy, you need models that display the properties that you would like to be able to explore.

MF: Can you explain in more detail the connection between the tiling of the Poincaré disk model and the tiling of the TPMSs?

ME: The hyperbolic plane is the universal cover of the minimal surfaces, which means that I can essentially wrap the hyperbolic plane over the minimal surfaces. If I have a cylinder, its universal cover is a flat Euclidean plane. When I have a flat plane (or, more precisely, a rectangular piece of it), I have two independent translation directions, X and Y. If I make one of those translations into a loop, and one of those I leave open, then I get a cylinder. In a similar manner, if I take a particular tiling of the hyperbolic plane, I can cut out a particular portion of this plane (a hyperbolic dodecagon), and then I have six independent translation directions in the hyperbolic plane—so, one can say, there’s a lot more space to move in the hyperbolic plane. Out of these six, I make three of them into loops and I leave three of them open. So three of them become closed loops on the resulting surface, and the other three become X, Y, and Z—the translations in three-dimensional space. This is the wrapping process from the hyperbolic plane to the minimal surfaces (see Fig. 4). Everything that I draw in the hyperbolic plane, I can just print onto the surface. That’s really analogous to drawing a line in the Euclidean plane, which would wrap to a helix of a given pitch on the cylinder. So a packing of lines in the Euclidean plane gives me helices on a cylinder. In the same way, a packing of lines in the hyperbolic plane—and there are infinitely many ways to construct these—gives me a set of helices on the minimal surfaces once I wrap up the hyperbolic plane.

MF: If I return to what was noted earlier about the impossibility of isometrically immersing a complete hyperbolic plane in the Euclidean space ${\mathbb{R}}^{3}$, this means that one has to have some distortion, seen e.g. in the fact that the curvature of the TPMS is not everywhere negative. How does this distortion affect the research?

ME: There’s some kind of distortion of the surface, but I haven’t been concerned with it because the resulting structures that we’re looking at are the entanglements in three-dimensional space; we’re not looking at the precise geometry of the curves (so far). If you wanted to do this more precisely, you could certainly use some approximations to follow geodesic curves on the surfaces rather than geodesic curves on the hyperbolic plane. This could be an interesting mathematical direction, which would involve some advanced differential geometry; however, in the connection to the physical systems, this doesn’t seem to be significant.

MF: My next question is a more general one. How does the mathematical theory of entanglement express the connection between geometry, topology, and mechanical and material behavior? Were new mathematical tools and theories developed for this?

ME: The way the project exists at the moment is enumerative. That’s to say, here’s a technique that we have which has been developed related to various objects that we find in the natural sciences, in particular the minimal surfaces. And so the question arises: how can we add an extra layer of complexity to start looking at more complicated three-dimensional structures using these minimal surfaces as a stepping stone? As with most enumerative techniques, the difficulty is in making sense of how we order and characterize the structures. That’s where various forms of mathematics has been—or needs to be—developed, in order to answer questions like: do all possible structures exist in the enumeration, can I uniquely describe each of these structures, and how can we have a sensible complexity ordering? In terms of a comprehensive mathematical theory of entanglement, we’re still a long way away from something like that.

MF: I want to make the question a little bit more precise concerning the mechanical behavior, focusing on whether the materiality of entanglement was taken into consideration, and also connecting this to the materiality of the models. You said the tactile models have the advantage that we can touch them. But you also work with filaments and entangled nets which have a certain thickness, that is, with the material itself. But can one say that the mathematical situation is close in some way to the real world? The filaments are not one-dimensional strings, like mathematical knots. Does this materiality have any influence on how to formulate a corresponding mathematical theory?

ME: That’s a good question. This type of question is already being tackled from the mathematical perspective in the topic of ‘ideal knots’ that I mentioned earlier, where the knots are made from a tube of a given thickness. Exact mathematical descriptions of these objects are really difficult to obtain in most cases, but approximate results are already proving useful in applications, like in properties of knotted DNA. Developing a parallel idea for the filamentous structures is what we’ve done computationally but, again, any kind of exact mathematics on these structures is elusive. So my answer would be that yes, mathematics can deal with the materiality, but we jump to the field of geometric knot theory and geometric analysis rather than the algebraic structure of braids that we saw earlier, and this makes the mathematics significantly more difficult.

KK: The focus on the real entangled object leads me back to the topic of visualization and modeling. Is there a difference for you when you visualize something in your computer or your mind and when you model something?

ME: No, I see them as the same thing. I can’t really come up with an example where I differentiate between the two. To visualize something is to model it.

KK: And what kind of programs do you use for these kinds of visualizations?

ME: I use Houdini, which is a proprietary animation software. There are a few research groups using Houdini for mathematical purposes. But with the increasing use of 3D printers, modeling software is now easy to obtain and relatively user friendly. Since we’re using specific mathematical formulations as inputs, it doesn’t matter too much what kind of visualization software you use in the end.

KK: Is there a link between your visualization programs and the programs that the materials scientists use?

ME: Not really. A materials scientist or a physicist will typically want to simulate a process, and will have specific programs to fulfil this. Basic visualization will often be incorporated into these programs already, and further visualization is rarely done.

KK: If we take a look at the beginning of the twentieth century, together with a certain decline in the production of material models, there was a strong tendency in certain areas of mathematics to distance oneself from intuition, and especially from visualization. I wonder, when I listen to you as you describe a somewhat opposite tendency, how this changed at the end of the twentieth century, as intuition and three-dimensional tangibility seems so important and such a driving force.

ME: I don’t know if it has really changed across most of mathematics. I think, if we walked around the corridor here¹⁰ and started to discuss these questions, one would get very different responses. I wonder if this is how the sciences are starting to filter back into mathematics. One may suggest that rigorous mathematics and physics can only describe things up to a certain complexity. Biology, for example, is feeding examples back into the sciences that are far too complex to even consider in that level of detail and rigor. But when you’re taking those concrete examples and are trying to say something about them, then you’re really inventing new mathematical ideas. In order to know what’s worth describing, you need to use intuition gained from the real structure and its function. In general, the typical mathematics problem is very easy to pose but very difficult to solve. But for these interdisciplinary questions it’s the other way around: they’re relatively easy to solve but extremely difficult to pose. What question do you even want to ask and what question do you want to answer? That’s the most difficult part of the whole process, and it often takes a long time for these research questions to evolve to something reasonable.

MF: So how do you see the connection between the material models and the task of asking the right questions? Since, if I paraphrase what you said earlier, the material models really are the research object.

ME: For me, the models are the catalyst for starting to formulate what kind of research question I might ask. I make a model (using some kind of mathematically based technique), then I look at what’s interesting about it: symmetry, mechanics, something curious. These observations then guide the exploration of finding good research questions to ask.

MF: And an example of that would be the filaments?

ME: Definitely. I build filamentous structures using the TPMS; then, using the construction and observation of the structures, I can begin to formulate possible research questions, possible characteristics worth exploring, similarities and differences. There are often really spectacular arrangements of the filaments that arise, that only become spectacular when you look at the model, that become important in comparing and contrasting structures. The skin structure is one such structure, with a very interesting symmetry and entanglement. So it’s not surprising at all that nature should choose such a structure.

Myfanwy E. Evans is Professor of Applied Geometry and Topology at the University of Potsdam. She works on geometric and topological problems inspired by physical and biological structures and processes, in particular relating to tangling, surfaces, and tangling on surfaces. She works around the central question of how much of the function of a structure in nature can be described by basic geometric and topological principles. This involves various computational and theoretical studies alongside the study of some really beautiful geometric objects. She studied Mathematics and Physics at the Australian National University in Canberra, also completing her PhD there in 2011 in the Department of Applied Mathematics with the dissertation “Three-dimensional entanglement: Knots, knits and nets.” Before moving to Potsdam in 2020, she led an Emmy Noether research group at the Technische Universität Berlin. She is an active member of the Cluster of Excellence “Matters of Activity. Image Space Material” at the Humboldt Universität zu Berlin and the cooperative research center “Discretization in Geometry and Dynamics” at the Technische Universität Berlin.