## Abstract

Mathematically, the program [SURFER] visualizes real algebraic geometry in real-time.

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“Mathematically, the program [SURFER] visualizes real algebraic geometry in real-time. The surfaces shown are given by the zero set of a polynomial equation in the variables \(x\), \(y,\) and \(z\). All points in space that solve the equation are displayed and form the surface. [For] example[,] [by entering] \(x^2 + y^2 + z^2 - 1 = 0\), [one obtains] a sphere. […]

The great thing about SURFER is that you don’t have to understand the underlying mathematics (algebraic geometry) a priori, you can experiment, try, follow your intuition and creativity and this way learn math[ematics] and create unique artwork like pictures or animations.

SURFER is the new, Java-based version of the program SURFER2008 that was developed for the IMAGINARY exhibition in the year of Mathematics 2008 in Germany. The program is platform-independent and runs on a Windows, Linux or Mac operating system.”

^{Footnote 1}

**MF**: We would like to start with the history of the project IMAGINARY and the exhibition concept behind it. As can be seen from the quotation, taken from the IMAGINARY website, the first exhibition used a computer program called SURFER, a program which, given their equation, enables the visualization of, among other things, surfaces in the real Euclidean space \(\rm{\mathbb{R}}^3\) (see Fig. 1 for an example of such a surface). Could you tell us how IMAGINARY and SURFER were developed? How did you come to the idea of visualizing mathematical objects?

**ADM**: It all started in 2007 when it was already clear that 2008 was going to be the German Year of Mathematics. The Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics) wanted to contribute to that year. At first one had the idea of establishing a mathematics museum in Oberwolfach. This idea was floating around, and the institute was looking for people to help realize it. At that time I was in touch with the Austrian mathematician Herwig Hauser, an algebraic geometer, who was experimenting with algebraic surfaces. What he did was to develop images of algebraic surfaces that he created using POV-Ray, an open-source ray-tracing program—though it took several hours to render a single image. He prepared these images and added, one may say, an aesthetic or artistic component to them. He began this project even before 2000, I believe, and had exhibited some of his pictures—for example, at the ICM (the International Congress of Mathematicians) in Madrid in 2006. So there was already a gallery of his pictures available.

In 2007, with Hauser, I had large-sized pictures of algebraic surfaces printed. He told me that one of his colleagues, Gert-Martin Greuel, an algebraic geometer who was the director of MFO at that time (between 2002 and 2013), was looking for somebody to carry out a public outreach project in mathematics, similar to a museum. So I had my job interview at MFO, and they hired me right away. Though the idea of a mathematics museum was a bit too physical and too big, monetarily speaking, it was clear that algebraic geometry nevertheless had to be involved. We had Hauser’s pictures after all, and Gert-Martin Greuel agreed to present surfaces—it was also his topic of research—and the pictures looked attractive as well. The idea was to use touch screens. I had already experimented with big touch screens a few years before—recall that this is the pre-smartphone era; touch screens were not yet common; you really had to explain to people that they could touch the screen using their fingers. In a math and artificial intelligence exhibition I did with touch screens in 2002,^{Footnote 3} people could indeed interact in and with the exhibition using their fingers. It was then very clear for me that this real-time interactivity and feedback between the visitors and the object is a must.

My goal was to see whether we can just change the equation in real time and immediately present the image of the resulting surface. At that time it was considered impossible, as it took five hours of calculation time for one image. Hence, my first task was to find people who could implement this. At the Technische Universität Kaiserslautern there was a software called Surf, developed by Stephan Endrass and others.^{Footnote 4} It was open-source software, though without an impressive user interface—but it was able to do quite fast ray tracing of images. At the same time Henning Meyer and Christian Stussak joined the team—Christian was then doing his master thesis on real-time ray tracing using graphics cards, which was very new at that time as well.^{Footnote 5} Nowadays everybody uses graphics cards all the time to do faster operations for geometry—and not only for video games; that is, today one constantly uses the parallel processing power of CPUs. In 2007 we had another key person joining: Oliver Labs. He had just finished his PhD and was also working on algebraic geometry. He found a few very nice singular surfaces, on which I’ll elaborate later. In addition, he had developed another software called Surfex, which had added a new interface to Surf.

The key issue was, having all of that research on hand, to develop an accessible user interface on Surf. In the end we didn’t change too much. We did the Windows port, which was complicated because Surf was pure Linux software. This was how the first exhibition functioned. The user interface was comprised of just some parameters, some buttons on the touch screen—one could change parameters. We also presented a gallery of surfaces (see Fig. 2b), including some surfaces Labs found, and also surfaces with nodal singularities, surfaces which have the maximum number of singularities of this kind (see for example Fig. 2a for the Barth surface).

From that point onward we continued adding interactivity. At the very first exhibition it was simply that. At the second, however, we added a printer, giving the visitors the opportunity to take with them a print of the surface they created. One could also see the formula on the printout and leave it in a new user gallery in the exhibition. Other visitors could then retype the formulas and play with surfaces done by other visitors. After this we allowed downloading the program, adding some instructions online. We organized several competitions in which people sent us images they created with SURFER. They were online competitions—so we had the very first online exhibition competition, which was called “Kunst aus Formeln” by *ZEIT online*.^{Footnote 6}

The online competition was an open one—all of the participants could see the other entries as well before the competition ended. This created a community of people playing with SURFER, ranging from professional mathematicians to hobby mathematicians, artists, designers, architects, etc. I sometimes compare SURFER to a digital camera—in a sense, we created a digital camera with which you can take pictures of abstract mathematical objects. Changing the lens or the camera settings changes the equation. You could really see how people interacted and how they also came to discover the flaws of the camera by going into extreme settings. Of course, it’s still impossible to have a hundred percent correct visualization of surfaces, even for curves.

**KK**: This leads me to the following question. What were the challenges that you encountered when developing the program? Since, eventually, what is presented on the screen are pixels and not mathematical points.

**ADM**: Yes, we’re talking about surfaces, but these are eventually just pixels on the screen, and one has to deal with infinitely small lines or points. Hence, we constantly improved the software and even completely rewrote the core for a new software version—this was a new ray-tracing program with different algorithms which were faster and more exact. Coming back to Surf and SURFER, to this interactive component, the whole trick was to psychologically make it feel as if it was being calculated and presented in real time. The problem was that it took a lot of time. You really have to solve a high degree polynomial for each point, for each pixel—and this isn’t trivial at all. What you need is high processing power, and then you have to calculate it in parallel. Lastly, you need to make it feel as if it’s in real time. If one, for example, rotates the surface presented—I mean with a finger—for rotation one needs a certain number of frames per second in order to make it feel as if it’s happening in real time, something like 20 or 30 pictures per second on the screen, as in film. In the end I think we used 10 frames a second and we lowered the resolution. Even now, if you look at the program, it adapts, there’s a process of adaptation. Now, it actually adapts automatically, depending on your CPU. This adaptive rendering is extremely important, so it feels interactive. I can rotate it, I can change the equation, and the results are immediate. It then becomes sharper while you simply look at it, and you don’t even notice it—it happens automatically.

If I return to the technological aspect, we had those big touch screens that nobody had. This technology, which was fascinating at that time, was completely new. Nobody could imagine that we had SURFER. Everyone was shocked how quickly we could do everything. It was the algorithms and also the design—it was such a beautifully designed mathematics exhibition.

**MF**: From your description, it seems as if the very concept and meaning of ‘exhibition’ has changed. Did this happen in other ways too?

**ADM**: Indeed, one important example is worth mentioning—it happened at the very first exhibition in Munich. During the exhibition, we received an email from a teacher of a local school in Bavaria. She told me that she loved the exhibition and asked if she could have it at her school. I immediately agreed. I asked the other organizers and contributors whether I could send the teacher the digital files of the images, so she could just go ahead and print them. And indeed she printed them and organized them in a similar structure. All the contributors were perfectly fine with this—I mean everyone had a ‘sharing philosophy.’ This was the first case of what we have been doing now for more than 10 years in many, many countries. We can call it a kind of open-source remote exhibition. The different museums, institutes, universities simply copy and add their own flavor, presenting it as their exhibition, while they base their exhibition on the content that’s already there.

**KK**: How did the project and the exhibition continue?

**ADM**: The IMAGINARY project, in the first years, was always kind of jumping from one year to the next. It was planned as a one-year travelling exhibition in Germany, 12 cities,^{Footnote 7} and that was it. But then it turned out that it was kind of longer. There were requests from other institutions. There was a very big exhibition by the Newton Institute in Cambridge, and then in Austria in various places. It started expanding in the European context, and at the same time we had our first remote open-source exhibition in Kyiv, Ukraine. Remote means that we were not even present in the preparation of the exhibition. Then we received some more funding—it was interesting because IMAGINARY always raised funds. We had the initial start-up capital from the Ministry of Education and Research. The exhibition travelled quite a lot. We still have an agreement with the Spanish Royal Mathematics Society, and I think that, since 2010, they’ve done more than 20 exhibitions, more than we’ve done in Germany. This was based on the original materials, but of course they renewed the computers, they bought some new touch screens (the technology changed a lot, clearly), although they still have the same images.

These days we have a few types of exhibitions. There are images, films (and the hands-on aspects as well), which could be data for 3D printing, but also anything you build physically, and there are also the interactive programs, which are software-related exhibitions, where the software plays a major part. Of course, in this case, one would have a touch screen or some hardware attached. There are also the accompanying texts, pedagogical aspects, and the instructions to consider, but this structure generally proved to be very good and is still in place. In 2013 we launched a platform with support by the Klaus Tschira Foundation, which made it easier for us to distribute and deploy the content, but also to collect content as well. That is, people can just upload things themselves; we can curate it online; it’s kind of automatic.

**MF**: I want to go back to the program itself. In which program language was it written? You also mentioned that the algorithms changed during the years. How did that happen? The last question, which is also very much related, concerns the role of the mathematician: does one actually need mathematical knowledge of algebraic geometry for these algorithms?

**ADM**: The original SURFER was written in C++. The original user interface used the GTK library. The new core was written in Java, which was also a kind of decision taken at that time, because we hoped—something that eventually proved to be wrong—that Java could be used in a browser. So there was this kind of Java application, which worked for a couple of years. We therefore rewrote everything in Java, including the user interface. Of course, software development is a very tricky part, because you have to rely on new technologies, but at the same time the technologies change. Thus, the languages changed, the libraries changed, and they eventually stop at some point. Hence, the new user interface library, which used JavaFX 1, was discontinued. A noncompatible new version, JavaFX 2, appeared, so we had to completely redo the whole user interface in order to be able to continue working with it. At that time—and that was indeed a good decision, which was also a professional one in terms of architecture—we separated the rendering core from the user interface.

One of the latest trends at that time was also doing everything in the browser using graphics cards. There were a lot of technological decisions to be made. For example, there was the question of whether we could really use graphics cards. If we used them, we could speed up SURFER by a thousand times. But there was a problem. The code required to render certain algebraic surfaces was too complex for certain GPUs, and you would never know in advance which GPU/driver combination would fail for which surface. So it was just not practical to deploy this to the general public in 2011 when the WebGL specification had been released.

This leads me to the role the mathematician plays in this process and the algorithm behind it. Our developer is a computer scientist with a PhD in computer science, but he works with mathematicians. He has a very strong interest in and knowledge of algebraic geometry and mathematics. I think that this is a necessary condition to work on SURFER—you must also understand the mathematics behind it.

There are a lot of known algorithms—for example, you have different algorithms to solve polynomials in higher degrees. It’s as if you have good ones, exact ones, or more or less exact, faster ones—but of course we need the fastest. Then you compromise on quality or combine algorithms, and in the end the programmer uses one algorithm. The original Surf had an option where you could switch between algorithms, so there were a lot that were implemented; you could switch and then also see the results—you could then see the errors in the visualization. This is where you need mathematicians to present test cases. Oliver Labs gave us twenty equations that can’t be visualized well—then you can test them and subsequently see what’s obtained. One of them was the famous Whitney umbrella (see Fig. 3), with a singular line continuing it below.

Explicitly, the problem with the visualization of the Whitney umbrella and this ‘singular line’ continuing below it is that one isn’t able to ray-trace this line, because you can’t ‘catch’ a line. Ray-tracing works as follows: you send rays to different pixels and you have to find the pixel; but, if it’s just one point you would like to catch, you won’t be able to do that because, even if you’re a little bit next to it, it’s still not there. This is the difficulty also with singularities, i.e., with singular points. You have to apply other tricks in this case.

Labs gave us these test cases; this showed the limits or limitations of our visualization power. As mathematicians, we want this to be as accurate as possible, but there’s always a trade-off between accuracy and speed. If the computer power increases, you have more cores, so you can do more parallel processing. Such programming wouldn’t be possible without mathematicians. It would also not be possible without computer scientists who know the mathematics.

**KK**: As you noted, there are mathematical questions that evolve from trying to visualize certain surfaces. I wonder where and how these visualizations can influence in turn the realm of pure mathematics?

**ADM**: There is one example which comes to my mind of how IMAGINARY and SURFER can stimulate research in pure mathematics. This concerns what I was talking about in terms of the maximum number of singularities for a given degree of a surface. Let’s say that you have an equation of a surface with the highest exponent being six. It has been proven that the maximum number of nodal singularities of such a surface is 65. The Barth surface or the Barth sextic, having 65 nodes, which I mentioned before (see Fig. 2a), is an example of such a surface.^{Footnote 8}

**MF**: Just to be sure—obviously, the singularities could also be complex;^{Footnote 9} so how do you visualize these complex parts?

**ADM**: We don’t visualize them. We show only the real points. And then it’s a question of whether you count the complex singularities or not (i.e., in real algebraic geometry, only the real singular points are counted, as one works only over the real numbers). If one works over the complex numbers, as is usually done, you also count complex points, and in that case you need another tool, such as the software Singular, which you can use to count also such complex singularities.

To return to our discussion on surfaces, if we now consider degree seven surfaces, Oliver Labs found an equation of such a surface with 99 singularities (that is, nodes), but it’s been proven that the maximum number of such singular points which a degree seven surface can have is between 99 and 104.^{Footnote 10} So the question arises: can one construct a surface, or find its explicit equation, which has between 100 and 104 nodes? Constructing such equations is highly complex. It’s not at all trivial. But we had thousands of users—mathematicians and nonmathematicians—who were playing all of the time with the equations, trying to find such surfaces. My hope was that one would find an equation of a degree seven surface with 100, but so far that still hasn’t happened.

**MF**: You have just mentioned that a community was forming around the attempts to find such a surface, but before that you also mentioned another way of communicating this knowledge visually to people—a three-dimensional printer could also be used to print these surfaces. Can one indeed just print these surfaces so easily?

**ADM**: For example, one can’t print the Barth sextic as it looks in (real) space, since it looks as if it’s composed of several pieces (as seen in Fig. 2a). These components look as if they’re connected at only one point. If one prints them like that, they would just break. Thus, you really have to find a printer that’s so strong that one single point would hold a structure—and, in any case, you would have to smooth it in some way. So you can print, but what one obtains isn’t exact. What I mean is that you can’t see the singular points. The other problem is the thickness. These surfaces are mathematically ‘infinitely thin.’ So how can you print that? You have to make them thick, but how do you do that? You always have two layers and you fill the area between. You need to print the volume. Such problems are not trivial. Oliver Labs now has a company called MO-Labs. He does precisely this, he has his own tools, he does a lot of manual adjustments, and he has three-dimensional triangulations of most of the surfaces now, with which he is able to print these surfaces (see Fig. 4).^{Footnote 11}

**KK**: When one compares the visualized surfaces produced by SURFER with the material models obtained using a 3D printer, I wonder what kind of models we have with SURFER. The models produced by SURFER are interactive, they can be transformed and changed in real time, they’re embedded in a sort of virtual reality as much as they’re realistic and as quasi-haptic as possible. Hence, how exactly are these objects considered as models? What is their epistemic value as models?

**ADM**: IMAGINARY’s main motivation is to communicate, to enable a transfer of ideas, of knowledge. The central idea is to always spark something with the person who is confronted with that given model. It’s really about producing different types of motivations in the user. And then, having opened that door, these incentives can lead one to develop something, to make a new program, to create something, to play with it. Since this motivation, or what sparks such motivation, is very important, we can’t separate it from when we look at the model. Hence, it’s a key ingredient of the models we use. That’s the reason why I always look at the design, the aesthetic component. Even if it were the best mathematical model, if it doesn’t appear pleasing, I wouldn’t use it. There are these sorts of criteria. Also, if it’s not participatory enough, I couldn’t use it to catch people’s attention.

So what are the models? Epistemically, the models are definitely surrounded by these communication criteria, which are essential. Maybe the latter are even the key issue. A technology which allows and prompts aesthetical and participatory elements should be used, even if this means that the model would subsequently change. Were a new real-time three-dimensional printer to appear which is perfect, we would use it right away. Three-dimensional printing is a good example, because we don’t really use it, since it’s simply not up to the standards of real time that we require. Each criterion has a minimum. To wait for five hours for something to be printed is just boring. Being interactive, aesthetic, participatory—these criteria really define the desired aspects of the model.

Maybe one should also look at the model in a more abstract way. What the ideal tool or object is for us, for example, is a very interesting question. What’s the ideal object? SURFER as an exhibition was somehow ideal, because it’s attractive, it’s interactive, and it has a lot of mathematics behind it—it’s there right in front of your eyes; the mathematics isn’t hidden somewhere. Because of its aesthetics, it also attracts all types of audience. You can three-dimensionally print it, you can show it—there are quite simply a lot of possibilities, which is what makes it ideal. We’re still considering this when creating other exhibitions. It’s not as though we’re only working with algebraic geometry, and not even only with geometry; we also try to have applied mathematical topics in many types of other exhibitions, but so far we still believe that SURFER is somehow unique. The disadvantage of it is that it starts on a higher level. For example, in museums, as a stand-alone exhibition, it doesn’t work that well, since people would just see these equations, and if they don’t know what the signs by which exponents are shown mean, they won’t understand. Thus, there are some small details…

**MF**: …which can be understood only by mathematicians.

**ADM**: Yes. Thus, there’s this one problem with SURFER, which makes it unsuitable for complete stand-alone exhibitions. That said, as soon as you have somebody there to introduce it, then the audience will stay for hours. You have other exhibitions where you start right away, but after five minutes it becomes boring. SURFER doesn’t get boring. It really does have this infinite number of possibilities to present.

**MF**: You were talking about the problematics of 3D printing the virtual models: the incapability of the material to be thin enough, or that there are singularities. The problem was how to present them. Are there other problems that SURFER should tackle? This relates to a question we discussed earlier: which object is presented? If we use virtual reality on the computer screen, we can visualize points or singularities in a very precise way. The question is how you actually transfer such things to the three-dimensional world without causing a visual distortion.

**ADM**: One possible way is to use AR [augmented reality], which is a simpler approach. You have the digital projection in three dimensions in a real space. With that you have the option of seeing real singularities in space. One may try this with HoloLens, which we’re using. The object is then floating—you can look at it and walk around it. Of course, I would love to have something like what I previously mentioned, that is, real-time three-dimensional printing.

In terms of challenges, even if you can improve the algorithm, using more GPU [graphics-processing-unit] power, making it super fast, at some point SURFER will reach its limit. It’s similar to the situation with the visualization errors of algebraic surfaces: you give me an equation and I can show it to you, I can print it in two- or three-dimensions, I can have different displays, etc., but there will also be problems; at some point a limit is reached.

**KK**: Making—or, more precisely, creating—is an essential part of the exhibition, and of the entire procedure. The exhibition has something that drives curiosity, which also propels research forward and motivates a certain playful aspect; it’s an open process and it’s not functionalized in only one direction. With these visualization techniques, is there a threshold crossed or a new exchange established between mathematics and ‘play’ or even playfulness through which this creative process unfolds?

**ADM**: The philosophy is that you, as the user, should be able to create something that surprises me, that I would respond to with astonishment. Even if I’m the author, the programmer, the visitor has the possibility of doing something that creates this kind of effect of amazement on me. That should be part of every object. To give a short example: Many people make polyhedra out of paper^{Footnote 12}—you can even buy such things preprinted now; it’s merely a question of cutting and folding. Personally, I find this boring, I look at it and ask myself why I should even bother. A lot of people have fun with it, however. For those people, it’s as if they construct it themselves. For me, this isn’t the case at all—it’s simply reproducing. You have no option to surprise the author. You may make it perfectly, but it still doesn’t have this aspect of creativity. What I do is to give people a tool, a software with which polyhedra can be designed. You can then create your own cardboard and print it out, so you have something that’s your own creation. You can even change the tool and come up with new ideas.

Another important aspect of the making/creation process is that it occurs within a community, of which I’m then part. Here, one can do things together. This is the part that has to do with making. I think it’s extremely important to have the process with the object. The how-to should be there, but it should not be simply a question of me giving you the print and then you just putting it together. The how-to should go back to the source as much as possible, so that every option remains open to create something new.

**KK**: To continue with this issue of community: this also points toward a possibly new pedagogical approach. Is it being taken up in universities? For example, are mathematics departments using your approach?

**ADM**: No, generally they’re not. I’ve given some courses at universities, but not in Germany. I taught an entire three-semester course in Argentina. It was called “Interactions between science and art for mathematicians,” but we also had other scientists participating as well. We had a large IMAGINARY conference where we tried to create new products for mathematics education. One project dealt with creating a university curriculum for mathematics, “Interactive mathematics communication.” Our idea is to have an open IMAGINARY approach, an open curriculum that people could build on.

**MF**: If one examines historically how material models and mathematical education were and are interwoven,^{Footnote 13} then, at the end of the nineteenth century in Europe, and especially in Germany, there was an influential culture of production of material mathematical models, of surfaces, of curves, which were also an integral part of courses given at universities. For example, the students had to produce their own models; they had to develop the skills to produce the material models from wood, string, metal, plaster, etc. in order to get a grade.

Given IMAGINARY’s and SURFER’s interaction and interactivity aspects, which bring the visitors to be actively involved, do you see this as a new mathematical culture? Or can one consider it as a renewed continuation of older mathematical cultures?

**ADM**: in March 2017 I was at a workshop in Wuppertal called “Mathematik und ihre Öffentlichkeiten,”^{Footnote 14} where I was invited to present IMAGINARY. The talk by Ulf Hashagen, who gave his talk before me, was about the mathematical exhibitions in Europe around 1900. At the end of my own talk there was a comment pointing out that what I was doing with the IMAGINARY exhibition was the same as what had been done previously, that IMAGINARY was nothing different, nothing new. However, I responded to this by saying that IMAGINARY not only has a much more international scope but it’s also far more interactive. Nevertheless, one is probably always part of a culture. Perhaps it’s only logical that IMAGINARY was created in Germany.

I think that what’s really new is where IMAGINARY breaks with tradition. What we can do right now—and it’s a trend in many fields—is to attempt to break hierarchies. It’s not a question of experts coming to teach and to present on one side and a passive audience on the other side. You may be an expert in mathematical equations, for example, someone else is an expert in visual intuition, someone in creative ideas, algorithms, the list goes on. We’re not dealing any longer with this kind of strict segmentation of fields or disciplines. I think IMAGINARY can break the hierarchy. I think that this is what’s new; there’s a different and much vaster audience involved.

This isn’t the only new aspect here; also the aspect of interactive geometry should be stressed. If the visual technology is fast, then you may arrive at new results by just playing around, playing in a good sense, experimenting, because you can test much more. Here again, we’re dealing with the question of real time. If I can change parameters in real time, I might discover things that I couldn’t discover otherwise, since I can’t do such computations in my mind (or with a single material model). I could have intuitions, but I couldn’t test them. When I don’t have to wait days for the images, for the testing, then this real-time mathematics could have a major impact.

**Andreas Daniel Matt** is the director of IMAGINARY, a Berlin-based nonprofit organization for the communication of current research in mathematics. He has a PhD in Mathematics in the field of machine learning, and worked from 2007 until 2016 for the Mathematisches Forschungsinstitut Oberwolfach, a Leibniz Institute, where he co-initiated IMAGINARY. In 2013 he received the Media Award of the German Society of Mathematics for his contribution to the communication of mathematics. In 2020 IMAGINARY was awarded the Mariano Gago Ecsite Award for Sustainable Success in Science Engagement.

IMAGINARY is currently involved in a major international training and teaching program in AI and mathematics, and has developed a new digital and physical open-source AI exhibition (https://www.i-am.ai/) and a new exhibition on the connection between mathematics and music (https://www.imaginary.org/exhibition/la-la-lab-the-mathematics-of-music). Other current projects are an interactive picture book which combines storytelling and mathematics (https://www.mathina.eu/) and a mobile mini museum on mathematical modeling for the climate crisis (https://10mm.imaginary.org/). IMAGINARY is also coordinating the website and communication for the UNESCO International Day of Mathematics, a project by the International Mathematical Union (https://www.idm314.org/). And yes, SURFER is still in our minds and hearts: we recently presented a web ray tracer of surfaces, developed by Aaron Montag at TU Munich, see https://love.imaginary.org.^{Footnote 15}

## Notes

- 1.
This description of the software module SURFER can be found on the website of IMAGINARY. See: “program. SURFER,” online: https://www.imaginary.org/program/surfer (accessed November 23, 2020).

- 2.
“Zitrus (Citric),” website of IMAGINARY, https://www.imaginary.org/gallery/herwig-hauser-classic (accessed November 23, 2021).

- 3.
See: “Interaktive Roboter-Tage im Einkaufspark Sillpark Innsbruck, vom 13.–15. Juni 2002,” http://fabula.tv/roboter/ (no longer accessible).

- 4.
See: “What is

*surf*?,” http://surf.sourceforge.net/ (accessed September 6, 2021). The program SURFER is based on Surf. In 1997 Stephan Endrass also found what is now known as the Endrass surface: a nodal surface (i.e., a surface whose singularities are only nodes) of degree eight with 168 real nodes. Note that the maximum number of nodes that a surface of degree eight may have is 174. See: Stephan Endrass, “A projective surface of degree eight with 168 nodes,”*Journal of Algebraic Geometry*6, no. 2 (1997): 325–34. - 5.
In 2013 Christian Stussak obtained his PhD with the dissertation “On reliable visualization algorithms for real algebraic curves and surfaces” at Martin-Luther-Universität Halle-Wittenberg. In 2010 Stussak, together with Peter Schenzel, published a paper on their program RealSurf, which also discusses the IMAGINARY project. See: Christian Stussak and Peter Schenzel, “RealSurf—A Tool for the Interactive Visualization of Mathematical Models,” in

*Arts and Technology: ArtsIT 2009—Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering*, vol. 30, ed. Fay Huang, and Reen-Cheng Wang (Berlin, Heidelberg: Springer, 2010), 173–80. - 6.
N.N., “Kunst aus Formeln,”

*ZEIT online*(May 30, 2008), https://www.zeit.de/online/2008/23/bg-matheskulpturen (accessed November 6, 2021);*Endbericht IMAGINARY: Wanderausstellung des Mathematischen Forschungsinstituts Oberwolfach, Dezember 2007–Dezember 2008*, 3, website of IMAGINARY, http://data.imaginary2008.de/Endbericht-IMAGINARY-2008.pdf (accessed November 6, 2021): See also: Björn Schwentker, “Mathematik Skulpturenprojekt: Anleitungen”,*ZEIT online*(March 11, 2008), https://www.zeit.de/online/2008/04/mathe-wettbewerb-anleitungen (accessed, November 6, 2021). “From February to December 2008, mathematics-art competitions using the program SURFER were organized together with media partners (*Die Zeit*,*Spektrum der Wissenschaften*). In total, more than 8000 pictures together with their formulas were created for the competitions.” [“Gemeinsam mit Medienpartnern (Die Zeit, Spektrum der Wissenschaften) wurden von Februar bis Dezember 2008 Mathematik-Kunst-Wettbewerbe mit dem Programm SURFER veranstaltet. Insgesamt wurden über 8000 Bilder zusammen mit ihren Formeln für die Wettbewerbe erstellt.”*Endbericht IMAGINARY: Wanderausstellung des Mathematischen Forschungsinstituts Oberwolfach, Dezember 2007–Dezember 2008*, 3, website of IMAGINARY, http://data.imaginary2008.de/Endbericht-IMAGINARY-2008.pdf (accessed November 6, 2021)] (Trans. by M.F./K.K.). - 7.
Among the cities were Munich, Berlin, Stuttgart, Potsdam, Leipzig, Cologne, and Saarbrücken. There were also many special exhibitions during this year (2008). See:

*Endbericht IMAGINARY*, 2. - 8.
See: Jaffe and Ruberman, “A sextic surface cannot have 66 nodes.”

- 9.
If (

*x*,*y*,*z*) is a point on the surface*z = z*(*x*,*y*), then ‘complex’ here means that at least one of the coordinates is of the form*a + bi*, when*b ≠*0 and*i*= −1. - 10.
In 2006 Labs found a degree seven surface with 99 real nodes. See: Oliver Labs, “A septic with 99 real nodes,”

*Rendiconti del Seminario Matematico della Università di Padova*116 (2006), 299–313. - 11.
In 2019 Oliver Labs developed another method of three-dimensionally visualizing these surfaces—that is, by lasering them in a glass cube, which enables a more accurate presentation of the singular points. See: website of

*Mathe-in-Glas*, https://www.mathe-in-glas.de/ (accessed November 6, 2021). - 12.
The making (or, more precisely, folding) of polyhedra out of paper is a tradition that started in the sixteenth century. See: Michael Friedman,

*A History of Folding in Mathematics: Mathematizing the Margins*(Cham: Birkhäuser, 2018), 29–83. - 13.
See, for example: Anja Sattelmacher,

*Anschauen, Anfassen, Auffassen: Eine Wissensgeschichte Mathematischer Modelle*(Wiesbaden: Springer Spektrum, 2021); Anja Sattelmacher, “Geordnete Verhältnisse: Mathematische Anschauungsmodelle im frühen 20. Jahrhundert,”*Berichte zur Wissenschaftsgeschichte*36, no. 4, 2013, 294–312; Anja Sattelmacher, “Zwischen Ästhetisierung und Historisierung: Die Sammlung geometrischer Modelle des Göttinger mathematischen Instituts,”*Mathematische Semesterberichte*61, no. 2, 2014, 131–43; Gerd Fischer, ed.,*Mathematische Modelle aus den Sammlungen von Universitäten und Museen*, 2 vols. (Braunschweig: Vieweg, 1986). - 14.
The workshop took place on March 23 and 24, 2017, at the IZWT, Bergische Universität Wuppertal, and was organized by Maria Remenyi.

- 15.
For the mathematical form of a heart, see: Siobhan Roberts, “The Perfect Valentine? A Math Formula,”

*The New York Times*, https://www.nytimes.com/2019/02/14/science/math-algorithm-valentine.html (accessed November 6, 2021).

## Acknowledgements

Michael Friedman and Karin Krauthausen acknowledge the support of the Cluster of Excellence “Matters of Activity. Image Space Material” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy—EXC 2025—390648296.

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Friedman, M., Krauthausen, K., Matt, A.D. (2022). Interview with Andreas Daniel Matt: Real-Time Mathematics. In: Friedman, M., Krauthausen, K. (eds) Model and Mathematics: From the 19th to the 21st Century. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-97833-4_15

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