Since the pandemic began “Love in the Time of Corona” has become a popular meme in mass culture that superficially resonates with a large cohort. While cultural acquisition attests to the transmission of the iconography of Gabriel Garcia Marquez’s 1985 Love in the Time of Cholera—only a small percentage of this cohort have actually read Marquez’s text. It is a novel based aesthetically in its own critical moment, at the threshold of experimental modernist and postmodernist imperative, and imbued with ideas about mathematical metaphor, mechanism, formal logic, pattern, sequence, and popularizations about chaotic and dynamic systems. In the same way, mathematics that is deeply rooted in cultural imagination can be loosely understood through the visual, symbolic, metaphoric, or even apocryphal uses in popular culture. Marquez’s work suggests a colourful experimental tension between logic and imagination, during plague, as a process of abstraction set against a backdrop of popularized 20th century fiction and society. In kee** with the theme of this Springer volume on “math in the time of corona,” and in concert with the underlying math of Marquez’s novel, this paper explores the idea of visualization in popular and digital uses during COVID-19, to examine how experimental approaches may or may not differ from pre-COVID conditions, as examples of the way experimental math exists through visualization for outreach and popularizing as it moves from one set of platforms to another. COVID is impacting outreach and learning pathways, through experimentation and innovative translation of visual math between platforms.

Modern experimental mathematics has flourished in a digital age, of computation supported by connectivity and distance, but media forms and approaches—nature of collaboration, digital resources, technological literacy, inquiry-based models, and experimentation—have undergone considerable change. Thus this paper attempts to answer the following questions by inspecting if the same hold true for outreach. What part does COVID-19 have to play in that transition? The stimulation of ideas in real time? In the digital environment? In experimental and virtual analysis?

1 Multi-sensory Experiments Before COVID-19

Consider experimental math as a research practice that is e-collaborative and is both computational and digitally assisted. Experimental math in popularization can be done through a broad spectrum of visual approaches, for example, mathematical pictures, the arts, music and immersive spaces. In these spaces, multi-sensory elements incorporate a toolbox of experimental methods. Take the example of a pre-COVID environment, a geometric outreach activity from the priority research centre for Experimental and Constructive Mathematics (CECM) at Simon Fraser University in the 1990s. This geometric and spatial concept progression in an outreach session is grounded in ideas about learning pathways that follow ‘concept progression’, or the gradual progression of concepts from early learning, through elementary and secondary learning into university learning (see studies by Daro et al. [8] and Shapiro [24]). It assumes both experimentation as a tool for conceptual knowledge, and experimental math as a digitally or concretely assisted mechanism underlying these progressions.

The outreach at CECM was not constrained by grade specific-curriculum learning expectations. What is interesting in this outreach is that the Grade 11–12 students were given the task of building 4–12-sided shapes using paper and balsa, with calculation of the internal angles. This progressed to origami folding and associated measurements, then on to viewing solid geometric objects that aided student visualization of internal angles, area, and volume. Visual proofs were given by rotating geometric images on touch screens capable of building 2D and 3D, then 4D images that furthered the understanding of concepts of space.

The experimental context of these outreach activities allowed for a transition in representations of various objects—for instance, from hollow to solid to virtual or immersive. The intention was to build conceptual thinking from the primary math of lower elementary shapes, through areas and volumes of secondary math in the 3D plane to academic research math in 4D.

Clearly some of outreach explorations translate more or less well into the digital resources that continue to be available and of use, even as platforms for visualization change.

Models that are underlying these forms of experimental outreach, including fun, innovation, math-in-action models, or multi-sensory learning forms, are often targeting an audience that spans multiple grades and includes an interested general public. An excellent example would be Helaman Ferguson who produces sculptural math in nature.

1.1 On-Line in the Digital Pre-COVID Era

A wealth of digital and/or interactive material for popularization and outreach assuredly existed before COVID. The Topic Study Group No. 07 explored “Popularization of Mathematics” at the International Congress of Mathematical Education (ICME), later published in the Proceedings of the 13th International Congress on Mathematical Education 2017, where the editors emphasize virtual forums, visual arts, and new technology alongside inquiry/research based projects. The 2012 volume Raising Public Awareness of Mathematics showcases an impressive range of popularization projects that happened in various countries. Globally, forms of math popularization employ a variety of experimental, embodied, or sensory approaches; these include topics and platforms such as

This diverse and playful subset of resources creates part of the foundation for extensions in COVID-era outreach.

2 Preliminary Examples During COVID-19

There is a move afoot by individual mathematicians and scholars to teach, intrigue, or enliven the constrained environment we find ourselves within today. This impulse is nothing new.

But, as if writing “The story of COVID-19” [23] as an experimental transition of highly visual elements across platforms in popular culture, consider the shift from visualizations of π to E8 in the media circa 2020. In American Mathematical Society (AMS) blogs, a reshuffling of earlier material, means students can access visualizations of a variety of mathematical ideas, not just from π and E8, but cooking Gaussian curvature, and “Journeys to the Distant Fields of Prime,” all drawing on popular interest in mathematical constants.

Under ‘Math in the Media’ the AMS offers visual math. Highlighting how preexisting outreach platforms are being negotiated during COVID, listed on the AMS site under Visual Math an interactive virtual exhibition tour by the sculptor Anton Bakker can be accessed at the National Museum of Mathematics in New York, written about by Denise M. Watson for The Virginian-Pilot, November 10, 2020.

Here the AMS, like the Canadian Mathematical Society (CMS), engages with an active and evolving conversation on “Mathematics and COVID-19,” such as in the Discussion Series: Mac Hyman of Tulane University talks Mathematical Modeling and COVID-19.

Internet math is a virtual space of mainstream and academic data and collaboration in the shadows of social media—on Facebook, Twitter, instagram, and more. There exists a large repository of visual math exploration of fractals, mandalas, and various popular engagements with π and e, as much as simply visualizing the pandemic, from “COVID Math” to “COVID Visualizers” that depict global morbidity and infection rates.

Experimentation resonates across cultural spaces. For instance, Eric Andrew-Gee writing “What Quebec’s COVID-19 experiment can teach us about the second wave” (Globe and Mail, November 20, 2020) underscores the perhaps unanticipated functionality of active, action-based research and experiment in mathematical modeling of disease: “That’s our country for you,” said economics professor Pierre-Carl Michaud, “We have a packet of experiments, and we can try to find the best recipe” [1]. His description of a recipe in this media piece suggests a pattern rule, or a toolbox through experiment, merging the modelling pandemics and disease contagions with other popular ideas such as chaos or apocalypse, like Mathematical Modelling of Zombies [25]. Conversely, in the journal Chaos, Solutions & Fractals Volume 140, November 2020, “Mathematical models of Ebola and COVID-19” “virus pathogen in the environment” by Zizhen Zhang and Sonal Jain describes randomness of disease vectors while citing ‘random walks’ [28].

What we see then showcases the broad uses and abuses of visualization and experimentation in popularization and research during COVID.

In 2020, Barry Mazur’s “Math in the Time of Plague,” published in The Mathematical Intelligencer, draws on Euclid’s construction of the perpendicular bisector of a line segment. He explores the idea of basic conceptual understanding, by exploring the idea of “axiomatic setup”—that there is no hint that the lines intersect. Contrasted to these “primitive bedrocks of thought,” what Mazur calls “common mathematical sensibility” recalls popular math knowledge. Mazur describes the process of “design[ing] fundamental online techniques to accommodate this moment” [18] and drawing on an experimental digital tool box that intersects with popular culture and in the connection has injected some of this sense of chaos into experimental space and practice.

Kevin Hartnett describes “Math After COVID-19” at the beginning of the Western pandemic in February, in Quanta Magazine; and, the byline reads “Modern mathematics relies on collaboration and travel. COVID-19 is making it increasingly difficult” [12]. With growing interest and the need to function in a virtual space, it is necessary to incorporate digital resources and form experimental research communities through new technologies, or new uses of pre-existent technologies—from ZOOM to Microsoft TEAMS to Cinderella. Digital spaces for popularization and interactive sites provide excellent interactive spaces for the dissemination of math games and support systems that are pop** up to help the mathematically inclined person suffering from ennui. All of these games and sites become part of a new COVID-era learning pathway.

3 One Learning Pathway

Mathematics sometimes called the quest for patterns? [22, p. 15]

Basic visualizations are a staple of popular math—pre-COVID, during the pandemic, and certainly after. In virtual outreach patterning, tiling, and fractal geometry are often used to educate students while attracting a math literate but non-specialized audience. One pathway for fractal patterning and visualization, that I have developed, which is useful for looking at a transmission between digital platforms, moves from the math behind the honeycomb formation and hexagons, to pinwheel fractal tiling and pinwheel fractal stages, to The L Shape Problem (where we ask what the area of a given shape is?) to the more complex tessellations of The Golden Bee Problem (see Michael Barnsley’s work) as a discussion of rotation and tiling, pattern/non-pattern, and fractal geometry. Clearly it enables a popular exploration of Euclidian geometry (e.g., hexagons) with fractal geometry.

To describe the common use of outreach in more depth, take the example of the hexagon honeycomb problem, as popular math widespread in mass media. It can be used to teach patterning and the writing of mathematical expressions, which in turn moves quite naturally to the pinwheel fractal; the student is asked to write an expression to show the fractal formation of the pinwheel structure, visualized in stages from the first level represented by a single square to the third level, a simple fractal, where clusters of squares form fractal nodes—with parallels in the formation of the snowflake. The next level of this basic visualization is the L-Shape Problem, an elementary problem where the L shape is used to learn, for example about area. Through emphasizing area, geometry, and patterning, it creates a step** stone, conceptually, to further visualize the Golden Bee Problem, as an example of Penrose tiling and fractal geometry, here, multiple tiles in the shape of a ‘b’ (in two sizes) make the pattern.

Photo by Naomi Borwein

For a hexagonal tessellation like in a honeycomb—imagine how ceramic tiles repeat on a kitchen floor forming a pattern. Consider one visualization that is a learning sequence or pathway for exploring such patterning. This outreach task instructs a student or interested general audience to use 22 tiles to begin a pattern. They must then explain how that pattern is continued; and by having them create a non-pattern with those 22 tiles, as a comparison, they are engaging with inquiry-based learning. In any number of interactive or immersive digital spaces, this outreach can be achieved. Much like the CECM exercise, they can reproduce the original pattern with hexagonal cubes or produce a quadrilateral honeycomb, as original or solid objects or virtual manipulatives (concrete learning aids), and muse about how it becomes more complicated.

There is ironically a lot of rhetoric in the mainstream about these ideas, and they even tie in, tangentially, to the butterfly effect and colony collapse. Equally, fractals often connect to popular chaos theories (e.g., J. W. Bloom) as much as being rooted in the appeal of visualizations.

4 An Extension of a Learning Path

Interactive implementation of random walks are an example of pre-pandemic outreach that continues to endure. Representing the idea of randomness, a color-coded visualization of π to 100,000 digits, entitled “walking on real numbers” was published as a random walk in Wired Magazine in an article by Samuel Arbesma:

Randomness as a mathematical feature is often misunderstood in mass media. But the pictures are not only mathematical, they are visual candy for an interested general audience.

The following is a screenshot of “Walking on Real Numbers A Multiple Media Mathematics Project.” It is an Interactive Random Walk, as a digital tool and website:

There are plenty of existing digital interfaces that help build public awareness of mathematics, such as IMAGINARY an Open Source Math Exhibition Platform [20], as well as numerous other resources that are actively evolving under the new constraints in which we find ourselves; such constraints encourage acts of inquiry, experiment, and discovery.

5 Modern Experimental Math

Far removed from the realm of popular visualization, the impact of modern experimental math as a theory and method is paradigmatically felt in the Philosophy of Mathematics. Here “Mathematical knowledge” once “regarded as certain since antiquity, because of formal proofs and deductive reasoning with respect to valid rules and axioms” is inherent in the very structure of knowledge systems that governed life. However, “this belief was shaken several times during the history of mathematics” and “most recently through computational experimental mathematics” itself emblematic of, or even eliciting, this shift in mathematics and mathematics education [21, p. 232].

Perhaps the most succinct definition of what exactly modern experimental mathematics is, is given in The Computer as Crucible.

Experimental mathematics is the use of a computer to run computations that are sometimes no more than trial-and-error tests to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions that may themselves arise by computational means, including search [5, p. 9].

During COVID-19 this transformative approach pushes the practitioner into further uncharted waters where the very dichotomies between beautiful proof and the ugly underbelly of mathematical research have to be renegotiated as much as virtualizing the element of research and outreach practice.

While in “About This Journal” in the first issue of Experimental Mathematics published in 1992, David Epstein, Silvio Levy, and Rafael de la Llave wrote

Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was “through systematic experimentation”.) Yet this tends to be concealed by the tradition of presenting only elegant, well-rounded and rigorous results… . we consider it anomalous that an important component of the process of mathematical creation is hidden from public discussion [9, p. 1].

Almost thirty years later, much has changed, and much has remained poignantly static.Footnote 1

While experimental math is no longer as contentious in the academic terrain of applied mathematics, in the journal Experimental Mathematics (2020) it is still noted that

many mathematicians have been reluctant to publish experimental results. Those who have tried it have sometimes found the best-known mathematical journals unwilling to accept such material, regardless of merit. Experimental Mathematics is an effort to change this situation. We envision it as something akin to a journal of experimental science: a forum where experiments can be described, conjectures posed, techniques debated, and standards set. We strongly believe that such a forum will further the healthy development of mathematics [2].

This “systematic” experimentation and “the process of mathematical creation” [2] can be rethought in light of new impediments caused by the pandemic, where mathematical discovery can still be framed within the constraints of scientific experimentation, through platforms, and digital and virtual analysis.

6 Conclusion

Will COVID become a paradigm shifter?

Clearly pre-COVID outreach is being extended, and with these extensions learning pathways are being altered—much like, for example, spatial and geometric visualizations. It is far too soon to know if there is or will be a dramatic difference in how experimental math exists in popular culture, or outreach after COVID, or which digital resources and hybrid methods will last. As visualization models move from one set of platforms to another, what does a pandemic driven transition in modern experimental math look like so far? The embodied, tactile, and multi-sensory nature of experimental outreach is replaced by many interactive hybrid models, various degrees of guided inquiry, and aspects of play. And the online experience offers a shield, from social pressure and peer-benchmarking that often exacerbates math fear. There is a redistribution of the student experience of outreach math in embodied, immersive, and digital formats. In popularization activities such as tiling and fractals visualization, or random walks, there is a suggestion both of increasing transmission and transition of visual math and its platforms today, where real-time and asynchronous forms of interaction, distance collaboration, lockdowns, virtual symposiums, interactive resources and outreach models, draw on a toolbox of experimental approaches as Bazar and others note. In the 1990s, the World Wide Web and computational power were foundational to modern experimental mathematics. Today, visible are the virtual extensions of multi-sensory learning pathways used in CECM outreach, as described at the beginning of this paper, and resources adapted through experimental translation in various new platforms. These interfaces and pathways will continue to co-evolve under current conditions.Footnote 2

figure d

Naomi Simone Borwein