Aesthetic Preferences in Mathematics: A Case Study

Aesthetic Preferences in Mathematics: A Case Study ABSTRACT Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to their visualisations? Using an example from graph theory (the highly symmetric Petersen graph), this paper argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians’ sensitivity to aesthetics of the abstract. 1. INTRODUCTION What is mathematical beauty? How could beauty be found in such an abstract subject as mathematics? Not attempting to solve this problem, I will attempt to answer a pair of specific questions which have some bearing on it. These questions arise from mathematicians’ judgements about both the abstract objects they investigate and the visual artefacts they use to represent those objects. These seem to be aesthetic judgements, often using the word ‘beautiful’, about the objects and their representations. Could such judgements, taken literally, be correct? Specifically: (i) Can an abstract mathematical object be literally beautiful? (ii) Can one of its visual representations be more beautiful than others? The last question needs to be refined. A visual representation can be appreciated without regard for what it is intended to represent, and as such could be beautiful in a way that a non-mathematician could appreciate. But that is not what is meant. A better way to put the question is this: (ii) Can one visual representation of a mathematical object represent its object in a more beautiful way than another visual representation represents the same object? And what does this mean for a mathematician? Mathematicians’ talk suggests a positive answer to both questions. But the possibility of loose or metaphorical uses of aesthetic expressions entails that we should be cautious about taking such talk at face value. In what follows I will first very briefly set out some of the positions that philosophers working in aesthetics have taken related to these questions. I will then introduce a particular mathematical example, an abstract object and representations of it, and mathematicians’ apparently aesthetic judgements about them. Finally, I will discuss these examples with a view to answering my questions. 2. THE ON-GOING DISCUSSION IN AESTHETICS There is an on-going discussion about whether there is such a thing as mathematical beauty, since there seem to be two options to consider, namely that beauty, apart from sharing a propensity to give pleasure of a certain disinterested kind (i.e., non-instrumental pleasure), is: (1) only perceptual, i.e., dependent only on properties, which an object can be perceived to have; all the other talk about ‘beauty’ is metaphorical or loose or (2) not only perceptual, but also intellectual (and possibly of many kinds). Clearly it is only in case 2 that there could be mathematical beauty, strictly and literally speaking. Jerrold Levinson [2011] among others claims that there is an irreducible variety of beauty, supporting claim 2. Rafael De Clercq [2013] in his excellent survey briefly criticises the postulation of different kinds of beauty and calls attention to the problem of expert-related aesthetic judgements as a way to define beauty. Nick Zangwill [2003, 2001, 1999] supports claim 1 and, in considering mathematical proofs, denies that they are literally elegant. John Barker [2009] responds to Zangwill’s arguments and puts forward the idea that elegance of mathematical proofs is the same elegance as found in other objects. I will consider Zangwill’s points as they apply to certain mathematical entities which are not proofs. Most works discussing mathematical beauty, including Barker’s, concentrate on proofs. Zangwill builds his argument on the basis that a proof is an epistemically functional construction and all aesthetic-like appreciations of elegant proofs are strictly speaking epistemic rather than aesthetic. I will point out that a closer look at mathematical practice shows that mathematical beauty is not limited to proofs only: there are other entities appreciated as beautiful. In this paper I will consider mathematical graphs rather than proofs, where these are the objects of graph theory, not the diagrams found in calculus textbooks and elsewhere. I start the paper with some basic assumptions about beauty. I then explain the distinction between an abstract mathematical object and its particular visualisations. I distinguish beauty of the object and beauty of its visual representations. This is not a metaphysical but a methodological distinction, which will later help me to trace the contributors to mathematical beauty. Looking at particular cases, I identify the factors of beauty to which mathematicians respond and whether these factors are perceptual or not. I take some time to develop my case study and towards the end I return to the philosophical discussion and respond to some objections. 3. MY ASSUMPTIONS Let me begin with some minimal understanding of mathematical beauty and then try to develop this understanding by studying particular examples. As mentioned, Zangwill’s view is disputed by Barker and others, and I will take it as a premise in this paper that abstract mathematical entities can be beautiful, strictly and literally speaking. To judge something mathematical to be beautiful is to admit that this thing has the power to give an aesthetic pleasure and is valued to some positive degree. By saying ‘This is a beautiful proof!’ the mathematician implies that the proof has a positive emotional effect on her in comparison to other proofs she knows. This does not imply that beauty is a particular property of a mathematical entity independent of a mathematician’s cognition, like the curvature of a surface or the number of roots of an equation. Beauty seems to be more like a power, causing pleasure to mathematicians while they are intellectually engaged with the mathematical entity. In aesthetic terms, beauty is a response-dependent property. In this paper I constrain the discussion to aesthetic judgements made by specialists about their subject matter. While there is rarely perfect agreement about what is beautiful and what is not, there is a large measure of agreement among specialists when making comparisons.1 Another assumption I make is that (mathematical) beauty is a matter of degree. We can judge one entity to be more beautiful than another, and some can even appear ugly to us. In practice aesthetic judgements often arise in the context of comparing things. A mathematics lecturer will not miss a chance to mention to the students if somebody has found a more beautiful proof of a theorem than the one in their textbook; and she will be always delighted by a student’s more beautiful solution to a problem. Analogously, as I will show later, one representation of a mathematical object may be judged more beautiful than another. 4. MATHEMATICAL OBJECTS AND THEIR COGNITIVE REPRESENTATIONS Mathematical matter is abstract, but in practice mathematicians use representations of mathematical entities such as drawings of graphs, diagrams, written formulas, tables, and the like. These representations, along with definitions, help to fix the abstract content and help understanding and communication; they also facilitate various mathematical actions (such as matrix multiplication). For a simple example, a linear function is a mathematical object, but the formula printed here: ‘$$y= x + 5$$’ or a visualised graph of this function are cognitive representations of it. The distinction between an abstract mathematical object and its cognitive (visual) representation will be useful in analysing aesthetic judgements in mathematics. 5. PERCEPTUAL VS INTELLECTUAL BEAUTY I will distinguish perceptual and intellectual (or mathematical) beauty, with respect to mathematical visualisations and mathematical objects. Kant said that ‘beautiful representations of objects are to be distinguished from beautiful objects’, and I will come back to his view later in the discussion.2 I am not saying that they are two different kinds of beauty; I simply want to point out that mathematicians when judging a mathematical entity as beautiful, may be responding to abstract properties of mathematical objects or to visible properties of the representation. Perceptual beauty can be found in visually pleasing representations of mathematical entities, such as visualisations of geometric constructions, graphs, and arrays of formulas. Looking at a visual representation we can appreciate its perceptual beauty. In that case it is not the mathematical content of the representation that causes us pleasure. It is possible that somebody without any mathematical background and who does not know or understand the mathematical content of the representation can appreciate the overall visual effect of the mixture of visible properties, such as symmetry, smoothness of curves, and the combination of colours. To illustrate the case of intellectual beauty, I will use a puzzle called ‘tangram’, originating in China.3 It consists of seven flat pieces, which may not overlap and combine to form different shapes. The conventional default shape is a square as in Figure 1a. No particular shape is what makes it special (Figure 1b), but the fact that so many different shapes can be made is truly astonishing. Over 6500 different tangram problems can be found in nineteenth-century books alone, and the current number continues to increase [Slocum, 2001, p. 37]. New mathematical facts have been revealed about tangram. For example, there are only thirteen convex configurations (such that a line segment drawn between any two points on the configuration’s edge always passes through the configuration’s interior, as shown in Figure 1b [Wang and Hsiung, 1976]). So it is not the shapes themselves but their numerosity and diversity that is attractive. That could be classified as intellectually pleasing. Fig. 1. View largeDownload slide (a) Tangram: The standard view. (b) Tangram: The 13 possible convex shapes. Fig. 1. View largeDownload slide (a) Tangram: The standard view. (b) Tangram: The 13 possible convex shapes. A clash between visualisation and the geometrical combinatorics behind it can create paradoxes. Some visualisations surprise us: they seem to be broken, but in fact they contain the same seven geometric shapes. In Figure 2 depicting the Two Monks paradox, the first monk is missing a foot; as similarly the second and third cups in Figure 3 are missing a piece (the Magic Dice Cup paradox [Loyd, 1968, p. 25]). As soon as we understand the paradox geometrically we are able to admire the tangram’s mathematical beauty. Fig. 2. View largeDownload slide The Two Monks paradox Fig. 2. View largeDownload slide The Two Monks paradox Fig. 3. View largeDownload slide The Magic Dice Cup paradox. Fig. 3. View largeDownload slide The Magic Dice Cup paradox. There is another observation. It looks like this judgement comes to us at once without distinguishing whether we appreciate the simplicity, clarity, or arrangement of the diagram, or the clever management of perceptual elements of the diagram to serve as the justification. Although visual representations are involved and understanding of the mathematics does rely on them, it is clearly non-perceptual beauty that initiates aesthetic judgements. This last observation brings us to the questions: how do perceptual features of visual representations of something mathematical interact with the intellectual beauty of what is represented in the visualisation? Does the perceptual combined with the intellectual contribute to mathematical beauty? The strategy I will use to approach these questions will be (i) to consider a particular property, which has traditionally been associated with beauty both in the perceptual and intellectual senses, namely, symmetry and then (ii) to examine how visually perceived symmetries and symmetries known to exist but not perceived can contribute to beauty. For this purpose I will focus on a particular mathematical example. 6. SYMMETRY AS WE SEE IT AND SYMMETRY IN MATHEMATICS In this section I will point out that not all abstract symmetries can be easily perceived in visual representations. It will be useful to understand how the perceived and how merely calculated symmetries contribute to beauty. In fact only some symmetries of an object, usually reflection, rotation, and translation, but not all of them, can be perceived in its visual representation. Examples for limits of visible symmetry will follow. Rotations and reflections of a square and a repeated pattern in tiles are examples of symmetries we easily grasp. The distinction between symmetries visualised in representations and algebraically determined symmetries is sometimes designated as ‘geometrical and combinatorial symmetries’; I will use these expressions that way. In mathematics ‘symmetry’ has a precise definition.4 A symmetry of a mathematical object is a transformation that leaves the object essentially unchanged: a structure-preserving bijection of the object onto itself. From a mathematical perspective, there may be many more symmetries than we can be perceptually sensitive to, partly because our natural grasp of symmetry is often solely geometrical. However, symmetry has a broader scope than geometry alone. Consider the value-preserving permutations of the indices 1 and 2, 3 and 4, in $$x_1x_2+x_3x_4$$. These are algebraic symmetries. Symmetry in Graph Theory In discrete mathematics, a graph is an ordered pair $$G=(V,E)$$ where $$V$$ is the set of vertices of $$G$$ and $$E$$, the set of edges of $$G$$, is a subset of the 2-subsets of $$V$$; that is, the edges are pairs of the elements of $$V$$. A bipartite graph allows the partition of its set of vertices into two non-empty sets such that every edge runs from set to set; that is, no edge connects vertices in the same set. The complete bipartite graph $$K_{m,n}$$ has its partition into two sets with $$m$$ and $$n$$ vertices, and every vertex in each set connected to every vertex in the other, making $$mn$$ edges. For instance $$K_{1,3}=(\{u,v,w,x\},\{\{u,x\},\{v,x\},\{w,x\}\})$$. In practice it is often useful to give a graph a pictorial representation: that is to represent vertices as dots and edges as lines connecting them. Such pictorial representations (drawings) can be realised in an aesthetically more or less attractive way, and this is where the question of aesthetics about representations appears. In graph theory a symmetry is a vertex permutation that preserves graph structure, namely adjacency and non-adjacency of vertices. Such a permutation is also called an automorphism. Let me explicitly articulate the difference between (1) a geometrical symmetry and (2) a symmetry in graph theory. (1) For any geometrical object $$G$$ (taken as a set of points in a Euclidean space of the relevant dimension) a symmetry $$s$$ of $$G$$ is a distance-preserving bijection from $$G$$ onto $$G$$. $$s$$ is distance-preserving iff for any $$x,y$$ in $$G,d(s(x),s(y))=d(x,y)$$; (2) For any graph $$G=(V,E)$$ a symmetry $$s$$ of $$G$$ is an adjacency-preserving bijection of $$V$$ into $$V$$; $$s$$ is adjacency-preserving iff for any $$u,v$$ in $$V$$, $$\{u,v\}$$ is in $$E$$ iff $$\{s(u),s(v)\}$$ is in $$E$$ as well. The symmetries that we notice in a graph drawing are those geometrical symmetries which can easily be discerned by visualising rigid motions of the whole geometrical object. Motions of this kind which take vertices to vertices will clearly preserve adjacency. So the vertex-to-vertex geometrical symmetries which we notice in a drawing of a graph reveal to us some of the graph’s symmetries. 7. THE PETERSEN GRAPH Graphs are convenient objects for observing symmetries: they are combinatorial objects, and algebra provides us with sufficient information about combinatorial symmetries. At the same time, graphs have many visual representations, which are useful for studying perceptual aspects of symmetry including aesthetical aspects. Let me now introduce a graph which is considered by many graph-theorists to be ‘one of the most beautiful graphs in graph theory’, the Petersen graph. Although the graph first appeared in [Kempe, 1886], it was named after Petersen [1898]. There are many drawings of this graph, symmetric and asymmetric. In Petersen’s paper the drawing of this graph did not reflect its symmetries (Figure 4a). In Kempe’s paper the drawing appears with symmetry of order 3 (Figure 4b), meaning the 3 rotations and 3 reflections. The now usual and preferred drawing shows 10 symmetries: 5 rotations and 5 reflections, only one of which is one of Kempe’s 6, namely a reflection fixing two points. In fact, it can be shown that a drawing of the Petersen graph can have at most 10 symmetries.5 A striking fact is that the Petersen graph has 120 symmetries! Various drawings of the Petersen graph enable us to perceive some of its symmetries, but no drawing can show all the graph symmetries at once. We can visualise some rotations and reflections, but not all their compositions. So there is more mathematically detectable symmetry, or ‘invisible’ or ‘hidden’ beauty, as mathematicians like to put it. Having a variety of drawings of the Petersen graph, symmetric and asymmetric, is useful for different tasks, but interestingly one of them is considered to be ‘more beautiful’ than the others. Keeping in mind these two considerations: ‘hidden beauty’ and ‘the most beautiful drawing’, one about an abstraction and another about an artefact, I will try to explain this situation and explore the aesthetic impact of symmetry, starting with the impact of symmetries in an abstract mathematical sense, including those which are not visualised, as factors in beauty. Towards the end of the paper I will consider how the visually perceived symmetries can contribute to mathematical beauty. Fig. 4. View largeDownload slide Three drawings of the Petersen graph: (a) from Petersen’s paper; (b) from Kempe’s paper; (c) the now usual drawing. Fig. 4. View largeDownload slide Three drawings of the Petersen graph: (a) from Petersen’s paper; (b) from Kempe’s paper; (c) the now usual drawing. 8. IS THE PETERSEN GRAPH STRICTLY AND LITERALLY BEAUTIFUL? Having discussed the aesthetics of visual representations of mathematical entities I now move to the aesthetics of the abstract mathematical entities themselves. In this case the entity is the Petersen graph, independent of a particular representation, i.e., a collection of vertices connected by edges, but not the drawings. From this perspective, how could anything so abstract as that be beautiful? My general answer to this question is that a combination of rare and eminent mathematical properties of the abstract object shapes the intellectual beauty to which mathematicians respond emotionally. If one asks graph theorists who work with the Petersen graph why it is beautiful, they naturally give an extended list of properties. Compared to many important graphs, e.g., the Hoffman-Singleton graph (see Figure 5), the Petersen graph has relatively few vertices and edges. Yet mathematicians have discovered that it has a multitude of outstanding qualities, which one would not expect on initial acquaintance. It is first of all this combination of properties that motivates the strongly positive aesthetic response of mathematicians. They are responding to what they have discovered of the deep nature of the Petersen graph, as opposed to its surface appearance in visual representations. In order to give the reader a taste of what mathematicians are responding to, a brief description of several relevant properties will be given in the next few paragraphs (not all can be defined here).6 Fig. 5. View largeDownload slide Three known Moore graphs: (a) pentagon, (b) the Petersen graph, (c) the Hoffman-Singleton graph. Fig. 5. View largeDownload slide Three known Moore graphs: (a) pentagon, (b) the Petersen graph, (c) the Hoffman-Singleton graph. 9. RICH IN SYMMETRY The Petersen graph is highly symmetric. There are 120 symmetries — permutations of vertices preserving adjacency/non-adjacency. Graph theory also interprets them in terms of various regularities. The Petersen graph is strongly regular: (a) it is regular, meaning that each vertex has the same number of adjacent vertices (neighbours, in this case three), (b) every pair of adjacent vertices has the same number of common neighbours (zero), and (c) every pair of non-adjacent vertices have the same number of common neighbours (one). It is vertex-transitive, meaning there is a symmetry taking any vertex to any other (every vertex has the same adjacency/non-adjacency configuration). It is also edge-transitive, which means, analogously to vertex-transitivity, that there is a symmetry taking any edge to any other. In fact, it is 3-arc-transitive: every directed three-edge path (i.e., a path of three edges with a given ordering of vertices) in the Petersen graph can be transformed into every other such path by a symmetry of the graph. The Petersen graph is also a cubic graph. A cubic graph, a graph in which all vertices have degree three, is like the graph representing the edges and vertices of a cube. In combination with arc-transitivity, cubic graphs form a list called the Foster census of only thirteen cubic arc-transitive graphs with up to 30 vertices. Only ten of them are also distance-transitive, i.e., such that every pair of vertices may be mapped by a symmetry into any other pair of vertices that are the same distance apart.7 Remarkably the Petersen graph is one them! This high degree of symmetry in such a small graph is impressive, and regardless of our visual perception of it, symmetry makes a mathematical object more beautiful. Why? Firstly, because symmetry helps us to grasp mathematical structure and its higher generality. Indeed, on the basis of these different types of symmetry, one can form families of graphs and compare them by relating properties. Also some of these types form hierarchies: e.g., strong regularity implies regularity, distance transitivity implies arc-transitivity, which implies vertex- and edge-transitivity, etc. Thus the classification of these symmetries provides a general method for studying graphs. Secondly, symmetry may lead us to other significant properties and connections. Let me now indicate this with some examples. 9.1. One of Four Possible Moore Graphs The Petersen graph is one of only four possible Moore graphs. This is an exceptional feature, which is directly related to symmetry, namely strong regularity of the graph, and depends on two characteristics: diameter and girth. A graph diameter is the greatest of the shortest distances between two vertices, and since in the Petersen graph any two vertices can be connected by two edges the graph diameter is 2. The girth of a graph is the length of a shortest cycle contained in the graph (a shortest closed path without repeating vertices except the starting one). In an $$n$$-regular graph every vertex is adjacent to $$n$$ edges. For diameter 2 and girth 5, the only possible strongly regular graphs are: (1) The 5-vertex 2-regular pentagon; (2) The 10-vertex 3-regular Petersen graph; (3) The 50-vertex 7-regular Hoffman-Singleton graph; (4) A 3250-vertex 57-regular graph (open problem). The first three are depicted in Figure 5, and the fourth is not yet proven to exist. To be a Moore graph is therefore a rare property;8 it is intriguing and surprising. This is what may help to make these graphs seem beautiful to mathematicians. Moreover, the Petersen graph has a combination of virtues not shared by the other two Moore graphs known to exist (pentagon and the Hoffman-Singleton graph). The Petersen graph is not as trivial as a pentagon and has a more interesting structure. At the same time it is not as complicated as the Hoffman-Singleton graph: its configuration is very clear and can be grasped at a glance. It is very likely that our aesthetic responses are more favourable when we find a balance between simplicity and complexity. So even among the Moore graphs it may be the most attractive. Interestingly, one can notice in Figure 5 that the Hoffman-Singleton graph contains copies of the Petersen graph (in fact exactly 525) [Klin and Zivav, 2014, p. 121]. 9.2. Not a Cayley Graph (Not a Graph of a Group) The Petersen graph is vertex-transitive. All Cayley graphs, which are graphs of generated groups, have this property by construction. Curiously not all vertex-transitive graphs are Cayley. It was shown in [Biggs, 1993] that the Petersen graph is the smallest connected vertex-transitive graph that is not Cayley. $$K_5$$, in Figure 7b, is a Cayley graph and Petersen is not. This is quite a surprising fact. There are many vertex-transitive graphs, and many of them are Cayley graphs; we know how to construct those. Constructing vertex-transitive non-Cayley graphs, however, is a challenging ongoing project. Interestingly the Petersen graph falls into this special category. 9.3. Non-Planar Graphs A graph may have planar (with no edges crossing) and non-planar drawings in a Euclidean plane. If a graph has any planar drawing, it is a planar graph, as in Figure 6. Therefore a planar drawing proves the graph’s planarity; otherwise we have to prove that such a drawing is impossible. Fig. 6. View largeDownload slide Examples of planar graphs. Fig. 6. View largeDownload slide Examples of planar graphs. Fig. 7. View largeDownload slide The Petersen and $$K_5$$ graphs. Fig. 7. View largeDownload slide The Petersen and $$K_5$$ graphs. A graph is non-planar if and only if it contains graphs $$K_5$$ or $$K_{3,3}$$ as a minor [Wagner, 1937]. A minor of a given graph is another graph formed by deleting vertices or edges, and/or contracting edges. When an edge is contracted, its two end vertices are merged to form a single vertex. The Petersen graph has both of these as minors; so it is non-planar. Figure 7 helps us to see that the Petersen graph contains $$K_5$$ as a minor (by contracting the 5 edges between the pentagon and the pentagram). Figure 8 shows that it contains $$K_{3,3}$$ as a minor. Fig. 8. View largeDownload slide $$K_{3,3}$$ as a minor of the Petersen graph. Fig. 8. View largeDownload slide $$K_{3,3}$$ as a minor of the Petersen graph. 9.4. A Unit-Distance Graph The Petersen graph is a unit-distance graph: it can be drawn in the plane with each edge having unit length. Figure 9 helps us to see that the unit-distance visualisation of the Petersen graph can be obtained from the symmetry of the usual drawing. This can be done by rotation of the inner pentagram stretching the edges until they become the same length as the sides of the pentagon and the pentagram. Fig. 9. View largeDownload slide A unit-distance drawing of the Petersen graph. Fig. 9. View largeDownload slide A unit-distance drawing of the Petersen graph. 9.5. A Snark This is probably the most intriguing property of the Petersen graph; it is even associated with a fictional hero. Lewis Carroll’s snarks were hard to find and at the same time, highly captivating. That explains best Martin Gardner’s choice for a fictional character rather than for a precise abbreviation of ‘nontrivial uncolourable trivalent graph’ [Gardner, 1976, Introduction]. The name was borrowed from Lewis Carroll’s poem The Hunting of the Snark about a mysterious creature with an unusual combination of features. According to Gardner a snark is a graph in which every vertex has three neighbours (meaning it is a cubic graph), and the edges cannot be coloured by only three colours without two edges of the same colour meeting at a point. Again the Petersen graph stands out. Not only was it the first snark found, it is probably the smallest snark: all the other possible snarks are likely to be reducible to it (by deleting some edges and collapsing the contained vertices). This is a conjecture due to W.T. Tutte: every snark has the Petersen graph as a minor.9 Interestingly, this conjecture is a strengthened form of the famous four-colour theorem!10 10. DISCUSSION 10.1. About the Beauty of the Abstract The above are just some of the remarkable properties of the Petersen graph, the simplest to explain.11 Why should we accept that this combination of properties makes the Petersen graph beautiful? Cain Todd [2008] raises a related question, challenging philosophers to differentiate epistemic and aesthetic dimensions. In his more recent paper, Todd [2018] suggests that aesthetic and epistemic pleasures can overlap on the basis of the ‘feeling of fittingness’, which is similar to the feeling of understanding. I agree that in practice it is often difficult to separate epistemic and aesthetic aspects, but still there may be a difference. Perhaps we can distinguish fulfilling an epistemic function (e.g., proving or solving) from finding instances of some other features which mathematicians value, such as being illuminating, surprising, intriguing, and mysterious.12 One comes across positive emotional responses to the properties of the Petersen graph which can refer to both practical epistemic and aesthetic benefits. Graph theorists like testing conjectures on the Petersen graph and express their satisfaction in positively emotional terms (they ‘fall in love with this graph’, they are ‘faithful to it’, ‘want to marry it’ and the like). In the first place it seems they are talking about practical benefits one can obtain using the Petersen graph, but the emotions they have may be related to aesthetic aspects. It is often so in ethical cases: something immoral we also perceive as ugly and something moral as beautiful. Also there are moral or useful and not very beautiful things and vice versa, and mathematicians seem to be aware of this fact. For example, all proofs serve the primary epistemic function of establishing their conclusion, but not all of them are judged beautiful; also, some proofs which have the additional epistemic merit of explaining their conclusions are not judged beautiful, and some which are judged beautiful do not explain their conclusions [Giaquinto, 2016, §4]. One reason for accepting that these properties contribute to the beauty of the Petersen graph is that experts agree in citing them as relevant: mathematicians I have spoken with referred to these and other properties in defence of their claim that the Petersen graph is a beautiful graph. This convergence of opinion may be taken as fulfilling Kant’s universality criterion that anyone, given necessary abilities and experience, can appreciate this beauty. While it is possible that specialists describe the graph as ‘beautiful’ meaning only that it is fascinating or remarkable, we have to ask why in that case do they so often use an aesthetic expression here, when they are perfectly capable of finding non-aesthetic words to express their interest (‘fascinating’, ‘remarkable’, ‘noteworthy’, ‘intriguing’ and the like). The most plausible explanation of their using ‘beautiful’ is that they experience a kind and degree of pleasure that is similar to the pleasure experienced when attending to things which they feel are strictly beautiful, such as fine musical compositions or natural scenes. In this case, loose usage is not a serious possibility. Someone can ask, ‘But does the beauty of the Petersen graph not depend on its having a pretty representation (unlike the Hoffman-Singleton graph which is too complicated)?’ I think not. Even the most beautiful drawing of the Petersen graph is not perceptually strikingly beautiful. Its visual attraction is not sufficient to explain the strong emotional response that mathematicians have to the graph itself. Moreover, when they talk about what is beautiful about this graph, they do not mention the drawing: The Petersen graph is a beautiful graph. At least, that is what graph theorists will tell you, time and again $$\ldots$$ [I]n spite of its small size — only 10 vertices and 15 edges — its structure is beautifully symmetric, and this has far reaching consequences. [Erickson, 2014] If you are talking about theoretical beauty, Petersen graph would top the list. Simple, but almost always counterexample to the simple theorems you try to cook up. [Ashwin, 2014] This can serve as an example for believing that at least some mathematical aesthetic judgements may be correct, as I believe the two above are. When mathematicians judge the Petersen graph to be beautiful they do not seem to be concerned with using it. Now, given that most of the remarkable and rare properties of the graph are connected with its high symmetry, why would the fact that symmetries help us to grasp the mathematical structure and other properties imply that symmetries help to make the object beautiful? It is oversimple to say that symmetries help make an object beautiful in mathematics, just as it would be in painting. Perhaps what makes the Petersen graph beautiful is its combination of cognitive simplicity and richness of symmetry. Also, being a counterexample to many conjectures is an epistemic benefit. But this is not to be taken as a factor of beauty, but evidence of the Petersen graph’s aesthetic value. Zangwill has two objections to the view that mathematical entities can be beautiful. One objection relates only to entities which have an intrinsic non-aesthetic function, such as proofs. This objection is not a threat to the view that mathematical objects such as the Petersen graph can be beautiful, because mathematical graphs, like stones, can be used but have no intrinsic function. Zangwill’s second objection is that properties of non-sensory abstract objects simply fall outside the category of aesthetic. In support of this view Zangwill makes the following remark: ‘As the etymological origins of the word “aesthetic” suggest, aesthetic properties are those that we appreciate in perception’ [2001, p. 81]. This implies that, strictly speaking, mathematicians use aesthetic terms metaphorically when they have feeling about mathematical objects similar to feelings they have about art objects. ‘Beautiful’ then, applied to mathematical entities would express a non-aesthetic value, most probably an epistemic value. As Zangwill is no doubt aware, the etymological origin of a word does not determine what the word expresses today. The fact that ‘ethical’ originates from an ancient Greek word meaning ‘habit’ or ‘custom’ does not entail that only what is habitual or customary can be ethical. Similarly, the fact that ‘aesthetic’ originates from an ancient Greek word meaning ‘perceive’ does not entail that only what is perceivable can have aesthetic value. So the etymological consideration is not strong support for Zangwill’s second objection. Rather it helps to notice that our concepts and use of aesthetic terms develop with time and experience. And here I agree with Robert Thomas [2017, p. 121] that ‘beauty is not all there is to aesthetics in mathematics’ and that, in mathematical contexts, it is quite natural to take ‘interesting’ as an aesthetic term, where by ‘interesting’ we indicate a desire to pay attention to an object for its own sake independently of external aims or appetites. In defending the view that mathematical proofs can be elegant against Zangwill’s objection, Barker [2009] points out that one could simply stipulate that aesthetic properties and responses are sensory (Zangwill himself does not do this), but the gain would be small. But it is hard for me to see any real benefit in making such a stipulation. Quite the opposite: it will simply blind us to the real and important similarities that exist between mathematical and sensory beauty. [Barker, 2009, p. 14] Indeed, along with mathematics, most of conceptual art where artefacts are almost irrelevant, falls out of aesthetic considerations exactly because the beauty is not necessarily perceptual. Similarly in mathematics, perceptual attractiveness is, with rare exceptions, irrelevant, and perceptual mediators require mathematical interpretation to be appreciated, as in the case of tangrams and in the case I am going to consider next. Besides, in mathematics, it takes time to develop a good level of expertise, not to mention aesthetic sensitivity. 10.2. About the Beauty of Visual Mathematical Representations Remember that any planar representation of the Petersen graph cannot show all its 120 symmetries. This fact clearly implies a distinction between the mathematical object and its representations. Let us now consider mathematicians’ judgements about visualisations of mathematical objects, and try to understand what role perceptual properties play in such judgements. Compare the drawings of the Petersen graph in Figure 4. In the standard drawing (4c) 10 geometrical symmetries (5 rotations and 5 reflections) are easily noticed, whereas in the Kempe drawing (4b) only 6 geometrical symmetries (3 rotations and 3 reflections) are easily noticed, and the drawing (4a) is even less revealing. The standard, showing a maximum number of symmetries, is the favourite of graph theorists, and most often found in the literature. Could it be just perceptual attractiveness of visible symmetries that makes it the favourite? To tackle the relation of perceptual properties to aesthetic quality, I will consider a case where two drawings of the same graph have the same number of symmetries. Obviously by simply pulling the vertices of the pentagram outside of the pentagon one can obtain an isomorphic drawing as depicted in Figure 10b. (One could also have one with the pentagon within the central area of the pentagram.) This drawing is hardly ever used, it does not appear naturally in practice, and according to graph theorists, they come across it occasionally via computer visualisations. Fig. 10. View largeDownload slide Two isomorphic symmetry-five drawings of the Petersen graph. Fig. 10. View largeDownload slide Two isomorphic symmetry-five drawings of the Petersen graph. Both drawings have the same 5 reflective and 5 rotational symmetries. In fact 10 is the maximum number of symmetries visualisable in a plane drawing of the graph.13 However, there is a big difference in mathematicians’ aesthetic judgements of the two. Graph theorists judge the usual one to be significantly more beautiful than the other (Figure 10b) and sometimes even call them ‘the beautiful’ and ‘the ugly’. For a lay person they may appear equally beautiful or 10b may look prettier. In fact the majority prefers it ‘because it is a star’. What are the experts actually responding to when saying that the left drawing is ‘most beautiful’ and the other is ‘ugly’, given that they have the same symmetry and represent the same mathematical structure? Figure 10a has more ‘good’ properties, such as fewer edge intersections and more wide angles.14 More precisely, there are 5 intersection points in both drawings but in the right one 3 edges meet at each intersection point, instead of 2. So at each intersection point in 10b there are 3 edge-intersections: if $$a,b,c$$, are the intersecting edges, $$a$$ meets $$b$$, $$a$$ meets $$c$$, $$b$$ meets $$c$$. Also broader inner angles mean broader space between the edges incident at a node keeping the same size of the overall drawing. Fewer intersections with broader inner angles make clear that it is composed of a pentagon and a pentagram (and 5 edges bridging them). Perceiving this geometric structure that links the graph to geometry may add to the mathematical beauty of the drawing.15 This is an aesthetic merit and it supports my emphasis on the distinction between a representation of an object and the object itself. The two geometric subgraphs, are so well-articulated in relation one to another, that with a bit of mental transformation, one can easily turn ‘the beautiful’ drawing into $$K_5$$ and realise that $$K_5$$ is a minor of Petersen (see Figure 7). This visible similarity between the drawings of Petersen and $$K_5$$ immediately reveals that the Petersen graph is non-planar and makes it surprising that, while it is like $$K_5$$ in being vertex-transitive, it is unlike $$K_5$$ in being non-Cayley. Moreover, observing Figure 10 of ‘the beautiful’ and ‘the ugly’ drawings of the Petersen graph, one can notice that ‘the beautiful’ shows a lot of other interesting structural information which is obscured in ‘the ugly’. The most intriguing properties of the graph from Section 9 are visible in the usual drawing. For example, using (a magnified) image of the common visual representation of the Hoffmann-Singleton (Figure 5), one can see that the Petersen graph is a subgraph of the Hofmann-Singleton graph. Furthermore, a simple twist of the pentagram relative to the pentagon keeping the centre fixed and allowing the bridging edges to stretch shows that the graph is unit distance (see Figure 9). In the study of snarks, it is ‘the beautiful’ drawing which ‘reveals’ that other snarks have the Petersen graph as a minor. Notice that Figure 10b does not provide such insights. Therefore, the abstract beauty of the graph shines brightest through the ‘beautiful’ drawing and is obscured in the ‘ugly’. Similarly to mathematicians trying to visualise abstract structures in the most effective way, artists follow the ‘rules of perception’ (proportions, density of details, composition, symmetry) to help a perceiver recognise beauty in the artefact. This strengthens the case for saying that the mathematicians’ aesthetic judgement in favour of the usual drawing arises from intellectual pleasure as opposed to merely sensory pleasure. Some of the relevant mathematical properties are revealed by the visual properties of the drawing. So the intellectual pleasure in this case has a sensory mediator, but mathematicians seem to respond to the intellectual content. The question is whether the judgement in favour of the usual drawing is an aesthetic judgement, and what roles intellectual and sensory components have in this judgement. Since the function of the drawing is to help us understand the graph, this could still be a case in which the drawing performs its function more beautifully than other drawings. One can say that the value of the drawing in this case is merely instrumental, non-aesthetic, and then Zangwill’s objection applies. Zangwill uses the idea of beauty independent of fulfilling a purpose from Kant, although with modification [Zangwill, 1999]. However, in one passage Kant suggests that a feature which facilitates understanding may also contribute aesthetically: Objects that ease our perception of them give us pleasure and are beautiful $$\ldots$$ Symmetry eases our understanding and is the proportion of sensibility. Looking at a disproportional house, I find it difficult to conceive it as a whole $$\ldots$$ Uniformity of the parts helps my representation, increases my inner life, and I therefore must find it beautiful.16 Holding a subjectivist position about mathematical beauty, Kant would not say that understanding of the representation adds to mathematical beauty of the abstract.17 He wrote: While mathematical properties themselves are not beautiful, it is the demonstration of such properties that can be the object of aesthetic appreciation $$\ldots$$ The purposiveness of mathematical properties does not indicate beauty but a form of perfection.18 Wenzel in his comments on Kant’s view that there is no beauty in mathematical objects and that ‘mathematics by itself is nothing but rules’,19 supposes that Kant underestimated the richness of mathematical practice in its historical dynamics. I believe that the case I have presented in this paper supports the view that abstract mathematical objects can be beautiful. The interpretation in [Breitenbach, 2015] of Kant’s aesthetic theory sheds light on the Kantian rejection of beauty of mathematical abstract objects, as it is often understood and suggests that ‘the experience of beauty in mathematics is grounded not in an intellectual insight into particular properties of mathematical objects but in our felt awareness of the imaginative processes that lead to mathematical knowledge’ [Breitenbach, 2015, p. 957]. This implies that Kant aims to distinguish knowing the properties (perhaps including interpretation of visual representations) of the abstract object (properties that may converge to some perfection) and free (spontaneous) imaginative activities involving discovery of these properties. Only these activities, which are subject to creativity, make possible aesthetic appreciation of such perfection. These two types of pleasure: epistemic and aesthetic, can be distinguished, but in intellectual contexts they blur; and Zangwill’s objection is not such a threat. Kant’s view is a strongly subjectivist position, but it stresses an under-appreciated aspect of mathematical practice which is still important — the practice of visualising and representing mathematical abstracta in a most comprehensible way. Maybe more than that are also under-appreciated: responding emotionally and creatively, experimenting, going beyond standard ways in reasoning. Clearly many mathematical areas develop their specific styles of visual representations: consider geometry, graph theory and algebra. Apparently, the art of sculpting mathematics includes designing and refining the look of its elements. These definitely have an aesthetic character. In practice it is often difficult to produce a mathematically effective visual representation, one which easily conveys the content. Mathematicians may react in a very favorable way to such exemplars, as with the favourite drawing of the Petersen graph. Therefore, according to Kant, visible symmetry of an object may contribute both to understanding and to beauty. Indeed the examples above demonstrate that in mathematics an aesthetic response requires at least some and sometimes very profound understanding. If that is right, the same may be true of those properties of the favourite drawing of the Petersen graph which most help one understand the graph itself. So, the favoured drawing of the Petersen graph has intellectual beauty, i.e., beauty that requires mathematical interpretation of the drawing to be appreciated. But the perfect and exceptional properties of the graph itself still have to be there. To sum up this section, it is unlikely to be a mere coincidence that the best drawing is also considered most beautiful. On the graph theorists’ view the usual drawing wins. The reason for that is perhaps that experts respond to the combination of the perceptual and intellectual, but presumably intellectual factors have more weight in mathematical beauty than perceptual factors. The intellectual factors are those concerning the manner of representation and the mathematical structure of the object itself. One principle that may be operating here is how easy it is for us to extract structural information directly from the drawing, or in other words the signal-to-noise ratio. In Figure 10b there is more irrelevant information: three times as much for each intersection point, as explained above. Moreover, the structurally significant subgraphs are not clearly separated in the right-hand drawing, whereas the insignificant properties, e.g., three-edge crosses, become louder. These properties reduce the signal-to-noise ratio. Looking at the ‘beautiful’ drawing, a mathematician more easily and more quickly discerns the structure and therefore its beauty. In contrast, when the mathematician looks at the ‘ugly’ drawing, the extra noise diminishes her aesthetic pleasure. It is not just the number of ‘good’ perceptual properties that makes a drawing more beautiful. The mathematician responds emotionally perhaps to the structural features of the graph itself revealed to us by the drawing, such as the possession of significant substructures having shapes of geometric figures.20 In their empirical research Reber et al. [2004] claim that fluency in processing information always enforces aesthetic response. They also admit along with perceptual fluency there is conceptual fluency that can add or even outweigh the perceptual. Thus, mathematicians’ aesthetic preference is for the drawing of the Petersen graph displaying the graph structure more clearly, and lay people rank the ‘star-like’ drawing (Figure 10b) higher. Similarly in the tangram example in Figure 2, a lay person would prefer 2b because it resembles a complete human image. But the two drawings together direct our attention to the paradox. Then we are able to appreciate the beauty of the tangrams not simply for palette or proportions. On the basis of this case study we can put forward the following tentative hypotheses: Different drawings of the same object may reveal different mathematical properties (symmetries). Visible symmetries may contribute to the mathematical beauty of a representation of a mathematical object (though the effect is not proportional to the number of symmetries). Particular visual properties help one grasp mathematical properties (structural) and appreciate them aesthetically. If two drawings of an object have a similar degree of perceptual beauty, the one which reveals a greater degree of mathematical beauty of the object has greater mathematical beauty. Mathematicians respond to a combination of various factors, perceptual and intellectual, but intellectual factors prevail. 11. CONCLUSIONS Taking into account that in practice mathematicians work with visual representations, we must distinguish between beauty of mathematical objects and beauty of visual representations of mathematical objects. Visual mathematical representations may look aesthetically appealing both for a lay person and a mathematician. However, while a lay person appreciates solely the visual appearance of a representation, for a mathematician a visual representation of a mathematical object may be assessed for perceptual beauty, for intellectual beauty, or a combination of both. Visible symmetry is one mathematical property that contributes to perceptual beauty. Mathematicians find the most symmetric drawing of the Petersen graph the most beautiful. However, between two equally symmetric representations of the Petersen graph they choose the clearer one as more beautiful. It turns out that they find visible symmetry along with other perceptual properties of graph drawings (e.g., fewer edge intersections, broader angles between adjacent vertices) also aesthetically appealing. This suggests that when comparing diagrams of the Petersen graph in aesthetic terms, mathematicians respond to a combination of both perceptually appealing properties (e.g., visible symmetries) and the mathematical properties of the object represented. A representation’s possession of properties that are useful is not by itself sufficient for mathematical beauty. Mathematicians may also make aesthetic judgements about a mathematical abstract object as such, focusing on its mathematical properties. For example, in explaining why they describe the Petersen graph as beautiful, they tend to emphasise its high symmetry relative to its small size, and its outstanding mathematical properties, mostly related to symmetry. This implies that both visible symmetry and abstract symmetry contribute to mathematical beauty. REFERENCES Ashwin Jacob [ 2014 ]: Answer to ‘What is the most beautiful graph you have ever seen?’. http://www.quora.com/Whats-the-most-beautiful-graph-you-have-ever-seen. Accessed September, 2015. Barker John [ 2009 ]: ‘Mathematical beauty’, Sztuka i Filozofia (Art and Philosophy), Special Issue No. 35, ‘Symposium on Aestheticism’, 60 – 74 . http://www.sztukaifilozofia.uw.edu.pl/en/issue/35-2009-en/ Accessed March 2017 . Biggs Norman [ 1993 ]: Algebraic Graph Theory . Cambridge Mathematical Library. 2nd ed. Cambridge University Press . Google Scholar CrossRef Search ADS Breitenbach Angela [ 2015 ]: ‘Beauty in proofs: Kant on aesthetics in mathematics’, European Journal of Philosophy 23 , 955 – 977 . Google Scholar CrossRef Search ADS Budd Malcolm [ 2008 ]: Aesthetic Essays . Oxford University Press . 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Goucher Adam [ 2013 ]: ‘Ten things you (possibly) didn’t know about the Petersen graph’. https://cp4space.wordpress.com/2013/09/06/ten-things-you-possibly-didnt-know-about-the-petersen-graph/. Accessed September 2015 . Holton Derek Allan , and Sheehan James [ 1993 ]: The Petersen Graph . Cambridge University Press . Google Scholar CrossRef Search ADS Hong Seok-Hee , and Eades Peter [ 2005 ]: ‘Drawing planar graphs symmetrically, ii: Biconnected planar graphs’, Algorithmica 42 , 159 – 197 . Google Scholar CrossRef Search ADS Kempe Alfred [ 1886 ]: ‘A memoir on the theory of mathematical form’, Philosophical Transactions of the Royal Society of London 177 , 1 – 70 . Google Scholar CrossRef Search ADS Klin Mikhail , and Zivav Matan [ 2014 ]: ‘Computer algebra investigation of known primitive triangle-free strongly regular graphs’. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.418.1432. Accessed August 2016 . Levinson Jerrold [ 2011 ]: Beauty Is Not One: The Irreducible Variety of Visual Beauty . Oxford University Press . Google Scholar CrossRef Search ADS Lovász László [ 2014 ]: Geometric Representations of Graphs . Preprint at http://www.cs.elte.hu/ lovasz/bookxx/geomrep-old2.pdf. Accessed March 2017 . Loyd Sam [ 1968 ]: The Eighth Book of Tan — 700 Tangrams by Sam Loyd . With an introduction and solutions by Peter Van Note . New York : Dover Publications . Petersen Julius [ 1898 ]: ‘Sur le théorème de Tait’, L’Intermédiaire des Mathématiciens 5 , 225 – 227 . Pisanski Tomaz [ 2000 ]: ‘Bridges between geometry and graph theory’, in Gorini Catherine ed., Geometry at Work , pp. 174 – 194 . MAA Notes; 53. Washington, D.C. : Mathematical Assoc. of America . Purchase Helen , Pilcher Christopher and Plimmer Beryl [ 2012 ]: ‘Graph drawing aesthetics — created by users not algorithms’, IEEE Transactions on Visualization and Computer Graphics 18 , 81 – 92 . Google Scholar CrossRef Search ADS PubMed Reber Rolf , Schwarz Norbert and Winkielman Piotr [ 2004 ]: ‘Processing fluency and aesthetic pleasure: Is beauty in the perceiver’s processing experience?’, Personality and Social Psychology Review 8 , 364 – 382 . Google Scholar CrossRef Search ADS PubMed Slocum Jerry [ 2001 ]: The Tao of Tangram . New York : Barnes and Noble . Starikova Irina [ 2012 ]: ‘From practice to new concepts: Geometric properties of groups’, Philosophia Scientiae 16 , 129 – 151 . Google Scholar CrossRef Search ADS Tait Peter [ 1880 ]: ‘Remarks on the colourings of maps’, Proc. Roy. Soc. Edinburgh 10 , 501 – 503 . Google Scholar CrossRef Search ADS Thomas R.S.D. [ 2017 ]: ‘Beauty is not all there is to aesthetics in mathematics’, Philosophia Mathematica (3) 25 , 116 – 127 . Thomas Robin [ 1999 ]: ‘Recent excluded minor theorems for graphs’. http://people.math.gatech.edu/ thomas/PAP/bcc.pdf. Todd Cain S. [ 2008 ]: ‘Unmasking the truth beneath the beauty: Why the supposed aesthetic judgements made in science may not be aesthetic at all’, International Studies in the Philosophy of Science 22 , 61 – 79 . Google Scholar CrossRef Search ADS Todd Cain S. [ 2018 ]: ‘Fitting feelings and elegant proofs: On the psychology of aesthetic evaluation in mathematics’, Philosophia Mathematica (3) 26 , 211 – 233 . Tutte William Thomas [ 1966 ]: ‘On the algebraic theory of graph colorings’, Journal of Combinatorial Theory 1 , 15 – 50 . http://www.sciencedirect.com/science/article/pii/S0021980066800042. Accessed March 2017 . Google Scholar CrossRef Search ADS Wagner Klaus [ 1937 ]: ‘Über eine Eigenschaft der ebenen Komplexe’, Mathematische Annalen 114 , 570 – 590 . http://eudml.org/doc/159935. Google Scholar CrossRef Search ADS Wang Fu Traing , and Hsiung Chuan-Chih [ 1976 ]: ‘A theorem on the tangram’, The American Mathematical Monthly 49 , 596 – 599 . Google Scholar CrossRef Search ADS Wenzel Christian [ 2001 ]: ‘Beauty, genius, and mathematics: Why did Kant change his mind?’ History of Philosophy Quarterly 18 , 415 – 432 . Weyl Hermann [ 1952 ]: Symmetry . Princeton University Press . Google Scholar CrossRef Search ADS Zangwill Nick [ 1999 ]: ‘Feasible aesthetic formalism’, Noûs 33 , 610 – 629 . Google Scholar CrossRef Search ADS Zangwill Nick [ 2001 ]: The Metaphysics of Beauty . Cornell University Press . Zangwill Nick [ 2003 ]: ‘Beauty’, in Levinson Jerrold ed, Oxford Companion to Aesthetics , pp. 325 – 343 . Oxford University Press . Google Scholar CrossRef Search ADS Footnotes 1For proponents of the response-dependent view see [Goldman, 1995] and [Levinson, 2011]; for criticism see [Budd, 2008] and [De Clercq, 2013]. 2I borrow this quotation from [Wenzel, 2001, p. 426] and I am grateful to a referee for recommending it to me. 3I am grateful to a referee for suggesting this example. 4Weyl [1952] gives a nice exposition about symmetry and mathematics. 5This claim can be found in [Hong and Eades, 2005]. 6The choice of properties is suggested in [Goucher, 2013]. 7Distance here is the number of edges in a shortest path. 8Moore graphs are also optimal models for message-passing networks. 9See [Tutte, 1966]. Thomas [1999] announced a proof of this conjecture. 10This is because (i) any graph containing the Petersen graph as a minor must be non-planar by the non-planarity of the Petersen graph, and (ii) the four-colour theorem is equivalent to the statement that no snark is planar (the theorem that opens the study of snarks due to Tait [1880]) or in topological terms, no snark embeds on a sphere. 11There is an entire book devoted solely to the Petersen graph, where a colossal number of remarkable properties of the graph can be found, The Petersen Graph [Holton and Sheehan, 1993]. 12In some contexts aesthetic properties seem to play an epistemic role. For example, Cellucci [2015, p. 15] suggests: Mathematical beauty can have a role in the context of discovery, because it can guide us in selecting which hypothesis to consider and which to disregard. Therefore, the aesthetic factors can have an epistemic role qua aesthetic factors. 13[Hong and Eades, 2005] discusses mathematicians’ preferences in graph drawings. 14About efficient graph drawing see, e.g., [Purchase et al., 2012]. 15A referee noticed that 10b also suggests a geometric interpretation (a composition of 5 isosceles triangles). In fact, geometric graph theory and topological graph theory are based on geometric representations of graphs (for details see, e.g., [Lovász, 2014] and [Pisanski, 2000]). I give an example when graphs played a role of inter-mediators between group theory and hyperbolic geometry in [Starikova, 2012]. 16Quoted in [Wenzel, 2001, p. 426]. According to Wenzel, Kant eventually changed his mind towards rejecting idea of mathematical aesthetics, but at least until the 1790s Kant kept this view. 17Breitenbach [2015, Introduction] sharpens this contrast of Kant’s subjectivism with a Platonist’s objective beauty. 18As quoted by Breitenbach [2015, p. 957] from Critique of Judgment, §62. 19Quoted in [Wenzel, 2001]. 20Provided by figural goodness and figure-ground contrast in the picture. © The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Philosophia Mathematica Oxford University Press

Aesthetic Preferences in Mathematics: A Case Study

Philosophia Mathematica , Volume Advance Article (2) – Jul 20, 2017

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Abstract

ABSTRACT Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to their visualisations? Using an example from graph theory (the highly symmetric Petersen graph), this paper argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians’ sensitivity to aesthetics of the abstract. 1. INTRODUCTION What is mathematical beauty? How could beauty be found in such an abstract subject as mathematics? Not attempting to solve this problem, I will attempt to answer a pair of specific questions which have some bearing on it. These questions arise from mathematicians’ judgements about both the abstract objects they investigate and the visual artefacts they use to represent those objects. These seem to be aesthetic judgements, often using the word ‘beautiful’, about the objects and their representations. Could such judgements, taken literally, be correct? Specifically: (i) Can an abstract mathematical object be literally beautiful? (ii) Can one of its visual representations be more beautiful than others? The last question needs to be refined. A visual representation can be appreciated without regard for what it is intended to represent, and as such could be beautiful in a way that a non-mathematician could appreciate. But that is not what is meant. A better way to put the question is this: (ii) Can one visual representation of a mathematical object represent its object in a more beautiful way than another visual representation represents the same object? And what does this mean for a mathematician? Mathematicians’ talk suggests a positive answer to both questions. But the possibility of loose or metaphorical uses of aesthetic expressions entails that we should be cautious about taking such talk at face value. In what follows I will first very briefly set out some of the positions that philosophers working in aesthetics have taken related to these questions. I will then introduce a particular mathematical example, an abstract object and representations of it, and mathematicians’ apparently aesthetic judgements about them. Finally, I will discuss these examples with a view to answering my questions. 2. THE ON-GOING DISCUSSION IN AESTHETICS There is an on-going discussion about whether there is such a thing as mathematical beauty, since there seem to be two options to consider, namely that beauty, apart from sharing a propensity to give pleasure of a certain disinterested kind (i.e., non-instrumental pleasure), is: (1) only perceptual, i.e., dependent only on properties, which an object can be perceived to have; all the other talk about ‘beauty’ is metaphorical or loose or (2) not only perceptual, but also intellectual (and possibly of many kinds). Clearly it is only in case 2 that there could be mathematical beauty, strictly and literally speaking. Jerrold Levinson [2011] among others claims that there is an irreducible variety of beauty, supporting claim 2. Rafael De Clercq [2013] in his excellent survey briefly criticises the postulation of different kinds of beauty and calls attention to the problem of expert-related aesthetic judgements as a way to define beauty. Nick Zangwill [2003, 2001, 1999] supports claim 1 and, in considering mathematical proofs, denies that they are literally elegant. John Barker [2009] responds to Zangwill’s arguments and puts forward the idea that elegance of mathematical proofs is the same elegance as found in other objects. I will consider Zangwill’s points as they apply to certain mathematical entities which are not proofs. Most works discussing mathematical beauty, including Barker’s, concentrate on proofs. Zangwill builds his argument on the basis that a proof is an epistemically functional construction and all aesthetic-like appreciations of elegant proofs are strictly speaking epistemic rather than aesthetic. I will point out that a closer look at mathematical practice shows that mathematical beauty is not limited to proofs only: there are other entities appreciated as beautiful. In this paper I will consider mathematical graphs rather than proofs, where these are the objects of graph theory, not the diagrams found in calculus textbooks and elsewhere. I start the paper with some basic assumptions about beauty. I then explain the distinction between an abstract mathematical object and its particular visualisations. I distinguish beauty of the object and beauty of its visual representations. This is not a metaphysical but a methodological distinction, which will later help me to trace the contributors to mathematical beauty. Looking at particular cases, I identify the factors of beauty to which mathematicians respond and whether these factors are perceptual or not. I take some time to develop my case study and towards the end I return to the philosophical discussion and respond to some objections. 3. MY ASSUMPTIONS Let me begin with some minimal understanding of mathematical beauty and then try to develop this understanding by studying particular examples. As mentioned, Zangwill’s view is disputed by Barker and others, and I will take it as a premise in this paper that abstract mathematical entities can be beautiful, strictly and literally speaking. To judge something mathematical to be beautiful is to admit that this thing has the power to give an aesthetic pleasure and is valued to some positive degree. By saying ‘This is a beautiful proof!’ the mathematician implies that the proof has a positive emotional effect on her in comparison to other proofs she knows. This does not imply that beauty is a particular property of a mathematical entity independent of a mathematician’s cognition, like the curvature of a surface or the number of roots of an equation. Beauty seems to be more like a power, causing pleasure to mathematicians while they are intellectually engaged with the mathematical entity. In aesthetic terms, beauty is a response-dependent property. In this paper I constrain the discussion to aesthetic judgements made by specialists about their subject matter. While there is rarely perfect agreement about what is beautiful and what is not, there is a large measure of agreement among specialists when making comparisons.1 Another assumption I make is that (mathematical) beauty is a matter of degree. We can judge one entity to be more beautiful than another, and some can even appear ugly to us. In practice aesthetic judgements often arise in the context of comparing things. A mathematics lecturer will not miss a chance to mention to the students if somebody has found a more beautiful proof of a theorem than the one in their textbook; and she will be always delighted by a student’s more beautiful solution to a problem. Analogously, as I will show later, one representation of a mathematical object may be judged more beautiful than another. 4. MATHEMATICAL OBJECTS AND THEIR COGNITIVE REPRESENTATIONS Mathematical matter is abstract, but in practice mathematicians use representations of mathematical entities such as drawings of graphs, diagrams, written formulas, tables, and the like. These representations, along with definitions, help to fix the abstract content and help understanding and communication; they also facilitate various mathematical actions (such as matrix multiplication). For a simple example, a linear function is a mathematical object, but the formula printed here: ‘$$y= x + 5$$’ or a visualised graph of this function are cognitive representations of it. The distinction between an abstract mathematical object and its cognitive (visual) representation will be useful in analysing aesthetic judgements in mathematics. 5. PERCEPTUAL VS INTELLECTUAL BEAUTY I will distinguish perceptual and intellectual (or mathematical) beauty, with respect to mathematical visualisations and mathematical objects. Kant said that ‘beautiful representations of objects are to be distinguished from beautiful objects’, and I will come back to his view later in the discussion.2 I am not saying that they are two different kinds of beauty; I simply want to point out that mathematicians when judging a mathematical entity as beautiful, may be responding to abstract properties of mathematical objects or to visible properties of the representation. Perceptual beauty can be found in visually pleasing representations of mathematical entities, such as visualisations of geometric constructions, graphs, and arrays of formulas. Looking at a visual representation we can appreciate its perceptual beauty. In that case it is not the mathematical content of the representation that causes us pleasure. It is possible that somebody without any mathematical background and who does not know or understand the mathematical content of the representation can appreciate the overall visual effect of the mixture of visible properties, such as symmetry, smoothness of curves, and the combination of colours. To illustrate the case of intellectual beauty, I will use a puzzle called ‘tangram’, originating in China.3 It consists of seven flat pieces, which may not overlap and combine to form different shapes. The conventional default shape is a square as in Figure 1a. No particular shape is what makes it special (Figure 1b), but the fact that so many different shapes can be made is truly astonishing. Over 6500 different tangram problems can be found in nineteenth-century books alone, and the current number continues to increase [Slocum, 2001, p. 37]. New mathematical facts have been revealed about tangram. For example, there are only thirteen convex configurations (such that a line segment drawn between any two points on the configuration’s edge always passes through the configuration’s interior, as shown in Figure 1b [Wang and Hsiung, 1976]). So it is not the shapes themselves but their numerosity and diversity that is attractive. That could be classified as intellectually pleasing. Fig. 1. View largeDownload slide (a) Tangram: The standard view. (b) Tangram: The 13 possible convex shapes. Fig. 1. View largeDownload slide (a) Tangram: The standard view. (b) Tangram: The 13 possible convex shapes. A clash between visualisation and the geometrical combinatorics behind it can create paradoxes. Some visualisations surprise us: they seem to be broken, but in fact they contain the same seven geometric shapes. In Figure 2 depicting the Two Monks paradox, the first monk is missing a foot; as similarly the second and third cups in Figure 3 are missing a piece (the Magic Dice Cup paradox [Loyd, 1968, p. 25]). As soon as we understand the paradox geometrically we are able to admire the tangram’s mathematical beauty. Fig. 2. View largeDownload slide The Two Monks paradox Fig. 2. View largeDownload slide The Two Monks paradox Fig. 3. View largeDownload slide The Magic Dice Cup paradox. Fig. 3. View largeDownload slide The Magic Dice Cup paradox. There is another observation. It looks like this judgement comes to us at once without distinguishing whether we appreciate the simplicity, clarity, or arrangement of the diagram, or the clever management of perceptual elements of the diagram to serve as the justification. Although visual representations are involved and understanding of the mathematics does rely on them, it is clearly non-perceptual beauty that initiates aesthetic judgements. This last observation brings us to the questions: how do perceptual features of visual representations of something mathematical interact with the intellectual beauty of what is represented in the visualisation? Does the perceptual combined with the intellectual contribute to mathematical beauty? The strategy I will use to approach these questions will be (i) to consider a particular property, which has traditionally been associated with beauty both in the perceptual and intellectual senses, namely, symmetry and then (ii) to examine how visually perceived symmetries and symmetries known to exist but not perceived can contribute to beauty. For this purpose I will focus on a particular mathematical example. 6. SYMMETRY AS WE SEE IT AND SYMMETRY IN MATHEMATICS In this section I will point out that not all abstract symmetries can be easily perceived in visual representations. It will be useful to understand how the perceived and how merely calculated symmetries contribute to beauty. In fact only some symmetries of an object, usually reflection, rotation, and translation, but not all of them, can be perceived in its visual representation. Examples for limits of visible symmetry will follow. Rotations and reflections of a square and a repeated pattern in tiles are examples of symmetries we easily grasp. The distinction between symmetries visualised in representations and algebraically determined symmetries is sometimes designated as ‘geometrical and combinatorial symmetries’; I will use these expressions that way. In mathematics ‘symmetry’ has a precise definition.4 A symmetry of a mathematical object is a transformation that leaves the object essentially unchanged: a structure-preserving bijection of the object onto itself. From a mathematical perspective, there may be many more symmetries than we can be perceptually sensitive to, partly because our natural grasp of symmetry is often solely geometrical. However, symmetry has a broader scope than geometry alone. Consider the value-preserving permutations of the indices 1 and 2, 3 and 4, in $$x_1x_2+x_3x_4$$. These are algebraic symmetries. Symmetry in Graph Theory In discrete mathematics, a graph is an ordered pair $$G=(V,E)$$ where $$V$$ is the set of vertices of $$G$$ and $$E$$, the set of edges of $$G$$, is a subset of the 2-subsets of $$V$$; that is, the edges are pairs of the elements of $$V$$. A bipartite graph allows the partition of its set of vertices into two non-empty sets such that every edge runs from set to set; that is, no edge connects vertices in the same set. The complete bipartite graph $$K_{m,n}$$ has its partition into two sets with $$m$$ and $$n$$ vertices, and every vertex in each set connected to every vertex in the other, making $$mn$$ edges. For instance $$K_{1,3}=(\{u,v,w,x\},\{\{u,x\},\{v,x\},\{w,x\}\})$$. In practice it is often useful to give a graph a pictorial representation: that is to represent vertices as dots and edges as lines connecting them. Such pictorial representations (drawings) can be realised in an aesthetically more or less attractive way, and this is where the question of aesthetics about representations appears. In graph theory a symmetry is a vertex permutation that preserves graph structure, namely adjacency and non-adjacency of vertices. Such a permutation is also called an automorphism. Let me explicitly articulate the difference between (1) a geometrical symmetry and (2) a symmetry in graph theory. (1) For any geometrical object $$G$$ (taken as a set of points in a Euclidean space of the relevant dimension) a symmetry $$s$$ of $$G$$ is a distance-preserving bijection from $$G$$ onto $$G$$. $$s$$ is distance-preserving iff for any $$x,y$$ in $$G,d(s(x),s(y))=d(x,y)$$; (2) For any graph $$G=(V,E)$$ a symmetry $$s$$ of $$G$$ is an adjacency-preserving bijection of $$V$$ into $$V$$; $$s$$ is adjacency-preserving iff for any $$u,v$$ in $$V$$, $$\{u,v\}$$ is in $$E$$ iff $$\{s(u),s(v)\}$$ is in $$E$$ as well. The symmetries that we notice in a graph drawing are those geometrical symmetries which can easily be discerned by visualising rigid motions of the whole geometrical object. Motions of this kind which take vertices to vertices will clearly preserve adjacency. So the vertex-to-vertex geometrical symmetries which we notice in a drawing of a graph reveal to us some of the graph’s symmetries. 7. THE PETERSEN GRAPH Graphs are convenient objects for observing symmetries: they are combinatorial objects, and algebra provides us with sufficient information about combinatorial symmetries. At the same time, graphs have many visual representations, which are useful for studying perceptual aspects of symmetry including aesthetical aspects. Let me now introduce a graph which is considered by many graph-theorists to be ‘one of the most beautiful graphs in graph theory’, the Petersen graph. Although the graph first appeared in [Kempe, 1886], it was named after Petersen [1898]. There are many drawings of this graph, symmetric and asymmetric. In Petersen’s paper the drawing of this graph did not reflect its symmetries (Figure 4a). In Kempe’s paper the drawing appears with symmetry of order 3 (Figure 4b), meaning the 3 rotations and 3 reflections. The now usual and preferred drawing shows 10 symmetries: 5 rotations and 5 reflections, only one of which is one of Kempe’s 6, namely a reflection fixing two points. In fact, it can be shown that a drawing of the Petersen graph can have at most 10 symmetries.5 A striking fact is that the Petersen graph has 120 symmetries! Various drawings of the Petersen graph enable us to perceive some of its symmetries, but no drawing can show all the graph symmetries at once. We can visualise some rotations and reflections, but not all their compositions. So there is more mathematically detectable symmetry, or ‘invisible’ or ‘hidden’ beauty, as mathematicians like to put it. Having a variety of drawings of the Petersen graph, symmetric and asymmetric, is useful for different tasks, but interestingly one of them is considered to be ‘more beautiful’ than the others. Keeping in mind these two considerations: ‘hidden beauty’ and ‘the most beautiful drawing’, one about an abstraction and another about an artefact, I will try to explain this situation and explore the aesthetic impact of symmetry, starting with the impact of symmetries in an abstract mathematical sense, including those which are not visualised, as factors in beauty. Towards the end of the paper I will consider how the visually perceived symmetries can contribute to mathematical beauty. Fig. 4. View largeDownload slide Three drawings of the Petersen graph: (a) from Petersen’s paper; (b) from Kempe’s paper; (c) the now usual drawing. Fig. 4. View largeDownload slide Three drawings of the Petersen graph: (a) from Petersen’s paper; (b) from Kempe’s paper; (c) the now usual drawing. 8. IS THE PETERSEN GRAPH STRICTLY AND LITERALLY BEAUTIFUL? Having discussed the aesthetics of visual representations of mathematical entities I now move to the aesthetics of the abstract mathematical entities themselves. In this case the entity is the Petersen graph, independent of a particular representation, i.e., a collection of vertices connected by edges, but not the drawings. From this perspective, how could anything so abstract as that be beautiful? My general answer to this question is that a combination of rare and eminent mathematical properties of the abstract object shapes the intellectual beauty to which mathematicians respond emotionally. If one asks graph theorists who work with the Petersen graph why it is beautiful, they naturally give an extended list of properties. Compared to many important graphs, e.g., the Hoffman-Singleton graph (see Figure 5), the Petersen graph has relatively few vertices and edges. Yet mathematicians have discovered that it has a multitude of outstanding qualities, which one would not expect on initial acquaintance. It is first of all this combination of properties that motivates the strongly positive aesthetic response of mathematicians. They are responding to what they have discovered of the deep nature of the Petersen graph, as opposed to its surface appearance in visual representations. In order to give the reader a taste of what mathematicians are responding to, a brief description of several relevant properties will be given in the next few paragraphs (not all can be defined here).6 Fig. 5. View largeDownload slide Three known Moore graphs: (a) pentagon, (b) the Petersen graph, (c) the Hoffman-Singleton graph. Fig. 5. View largeDownload slide Three known Moore graphs: (a) pentagon, (b) the Petersen graph, (c) the Hoffman-Singleton graph. 9. RICH IN SYMMETRY The Petersen graph is highly symmetric. There are 120 symmetries — permutations of vertices preserving adjacency/non-adjacency. Graph theory also interprets them in terms of various regularities. The Petersen graph is strongly regular: (a) it is regular, meaning that each vertex has the same number of adjacent vertices (neighbours, in this case three), (b) every pair of adjacent vertices has the same number of common neighbours (zero), and (c) every pair of non-adjacent vertices have the same number of common neighbours (one). It is vertex-transitive, meaning there is a symmetry taking any vertex to any other (every vertex has the same adjacency/non-adjacency configuration). It is also edge-transitive, which means, analogously to vertex-transitivity, that there is a symmetry taking any edge to any other. In fact, it is 3-arc-transitive: every directed three-edge path (i.e., a path of three edges with a given ordering of vertices) in the Petersen graph can be transformed into every other such path by a symmetry of the graph. The Petersen graph is also a cubic graph. A cubic graph, a graph in which all vertices have degree three, is like the graph representing the edges and vertices of a cube. In combination with arc-transitivity, cubic graphs form a list called the Foster census of only thirteen cubic arc-transitive graphs with up to 30 vertices. Only ten of them are also distance-transitive, i.e., such that every pair of vertices may be mapped by a symmetry into any other pair of vertices that are the same distance apart.7 Remarkably the Petersen graph is one them! This high degree of symmetry in such a small graph is impressive, and regardless of our visual perception of it, symmetry makes a mathematical object more beautiful. Why? Firstly, because symmetry helps us to grasp mathematical structure and its higher generality. Indeed, on the basis of these different types of symmetry, one can form families of graphs and compare them by relating properties. Also some of these types form hierarchies: e.g., strong regularity implies regularity, distance transitivity implies arc-transitivity, which implies vertex- and edge-transitivity, etc. Thus the classification of these symmetries provides a general method for studying graphs. Secondly, symmetry may lead us to other significant properties and connections. Let me now indicate this with some examples. 9.1. One of Four Possible Moore Graphs The Petersen graph is one of only four possible Moore graphs. This is an exceptional feature, which is directly related to symmetry, namely strong regularity of the graph, and depends on two characteristics: diameter and girth. A graph diameter is the greatest of the shortest distances between two vertices, and since in the Petersen graph any two vertices can be connected by two edges the graph diameter is 2. The girth of a graph is the length of a shortest cycle contained in the graph (a shortest closed path without repeating vertices except the starting one). In an $$n$$-regular graph every vertex is adjacent to $$n$$ edges. For diameter 2 and girth 5, the only possible strongly regular graphs are: (1) The 5-vertex 2-regular pentagon; (2) The 10-vertex 3-regular Petersen graph; (3) The 50-vertex 7-regular Hoffman-Singleton graph; (4) A 3250-vertex 57-regular graph (open problem). The first three are depicted in Figure 5, and the fourth is not yet proven to exist. To be a Moore graph is therefore a rare property;8 it is intriguing and surprising. This is what may help to make these graphs seem beautiful to mathematicians. Moreover, the Petersen graph has a combination of virtues not shared by the other two Moore graphs known to exist (pentagon and the Hoffman-Singleton graph). The Petersen graph is not as trivial as a pentagon and has a more interesting structure. At the same time it is not as complicated as the Hoffman-Singleton graph: its configuration is very clear and can be grasped at a glance. It is very likely that our aesthetic responses are more favourable when we find a balance between simplicity and complexity. So even among the Moore graphs it may be the most attractive. Interestingly, one can notice in Figure 5 that the Hoffman-Singleton graph contains copies of the Petersen graph (in fact exactly 525) [Klin and Zivav, 2014, p. 121]. 9.2. Not a Cayley Graph (Not a Graph of a Group) The Petersen graph is vertex-transitive. All Cayley graphs, which are graphs of generated groups, have this property by construction. Curiously not all vertex-transitive graphs are Cayley. It was shown in [Biggs, 1993] that the Petersen graph is the smallest connected vertex-transitive graph that is not Cayley. $$K_5$$, in Figure 7b, is a Cayley graph and Petersen is not. This is quite a surprising fact. There are many vertex-transitive graphs, and many of them are Cayley graphs; we know how to construct those. Constructing vertex-transitive non-Cayley graphs, however, is a challenging ongoing project. Interestingly the Petersen graph falls into this special category. 9.3. Non-Planar Graphs A graph may have planar (with no edges crossing) and non-planar drawings in a Euclidean plane. If a graph has any planar drawing, it is a planar graph, as in Figure 6. Therefore a planar drawing proves the graph’s planarity; otherwise we have to prove that such a drawing is impossible. Fig. 6. View largeDownload slide Examples of planar graphs. Fig. 6. View largeDownload slide Examples of planar graphs. Fig. 7. View largeDownload slide The Petersen and $$K_5$$ graphs. Fig. 7. View largeDownload slide The Petersen and $$K_5$$ graphs. A graph is non-planar if and only if it contains graphs $$K_5$$ or $$K_{3,3}$$ as a minor [Wagner, 1937]. A minor of a given graph is another graph formed by deleting vertices or edges, and/or contracting edges. When an edge is contracted, its two end vertices are merged to form a single vertex. The Petersen graph has both of these as minors; so it is non-planar. Figure 7 helps us to see that the Petersen graph contains $$K_5$$ as a minor (by contracting the 5 edges between the pentagon and the pentagram). Figure 8 shows that it contains $$K_{3,3}$$ as a minor. Fig. 8. View largeDownload slide $$K_{3,3}$$ as a minor of the Petersen graph. Fig. 8. View largeDownload slide $$K_{3,3}$$ as a minor of the Petersen graph. 9.4. A Unit-Distance Graph The Petersen graph is a unit-distance graph: it can be drawn in the plane with each edge having unit length. Figure 9 helps us to see that the unit-distance visualisation of the Petersen graph can be obtained from the symmetry of the usual drawing. This can be done by rotation of the inner pentagram stretching the edges until they become the same length as the sides of the pentagon and the pentagram. Fig. 9. View largeDownload slide A unit-distance drawing of the Petersen graph. Fig. 9. View largeDownload slide A unit-distance drawing of the Petersen graph. 9.5. A Snark This is probably the most intriguing property of the Petersen graph; it is even associated with a fictional hero. Lewis Carroll’s snarks were hard to find and at the same time, highly captivating. That explains best Martin Gardner’s choice for a fictional character rather than for a precise abbreviation of ‘nontrivial uncolourable trivalent graph’ [Gardner, 1976, Introduction]. The name was borrowed from Lewis Carroll’s poem The Hunting of the Snark about a mysterious creature with an unusual combination of features. According to Gardner a snark is a graph in which every vertex has three neighbours (meaning it is a cubic graph), and the edges cannot be coloured by only three colours without two edges of the same colour meeting at a point. Again the Petersen graph stands out. Not only was it the first snark found, it is probably the smallest snark: all the other possible snarks are likely to be reducible to it (by deleting some edges and collapsing the contained vertices). This is a conjecture due to W.T. Tutte: every snark has the Petersen graph as a minor.9 Interestingly, this conjecture is a strengthened form of the famous four-colour theorem!10 10. DISCUSSION 10.1. About the Beauty of the Abstract The above are just some of the remarkable properties of the Petersen graph, the simplest to explain.11 Why should we accept that this combination of properties makes the Petersen graph beautiful? Cain Todd [2008] raises a related question, challenging philosophers to differentiate epistemic and aesthetic dimensions. In his more recent paper, Todd [2018] suggests that aesthetic and epistemic pleasures can overlap on the basis of the ‘feeling of fittingness’, which is similar to the feeling of understanding. I agree that in practice it is often difficult to separate epistemic and aesthetic aspects, but still there may be a difference. Perhaps we can distinguish fulfilling an epistemic function (e.g., proving or solving) from finding instances of some other features which mathematicians value, such as being illuminating, surprising, intriguing, and mysterious.12 One comes across positive emotional responses to the properties of the Petersen graph which can refer to both practical epistemic and aesthetic benefits. Graph theorists like testing conjectures on the Petersen graph and express their satisfaction in positively emotional terms (they ‘fall in love with this graph’, they are ‘faithful to it’, ‘want to marry it’ and the like). In the first place it seems they are talking about practical benefits one can obtain using the Petersen graph, but the emotions they have may be related to aesthetic aspects. It is often so in ethical cases: something immoral we also perceive as ugly and something moral as beautiful. Also there are moral or useful and not very beautiful things and vice versa, and mathematicians seem to be aware of this fact. For example, all proofs serve the primary epistemic function of establishing their conclusion, but not all of them are judged beautiful; also, some proofs which have the additional epistemic merit of explaining their conclusions are not judged beautiful, and some which are judged beautiful do not explain their conclusions [Giaquinto, 2016, §4]. One reason for accepting that these properties contribute to the beauty of the Petersen graph is that experts agree in citing them as relevant: mathematicians I have spoken with referred to these and other properties in defence of their claim that the Petersen graph is a beautiful graph. This convergence of opinion may be taken as fulfilling Kant’s universality criterion that anyone, given necessary abilities and experience, can appreciate this beauty. While it is possible that specialists describe the graph as ‘beautiful’ meaning only that it is fascinating or remarkable, we have to ask why in that case do they so often use an aesthetic expression here, when they are perfectly capable of finding non-aesthetic words to express their interest (‘fascinating’, ‘remarkable’, ‘noteworthy’, ‘intriguing’ and the like). The most plausible explanation of their using ‘beautiful’ is that they experience a kind and degree of pleasure that is similar to the pleasure experienced when attending to things which they feel are strictly beautiful, such as fine musical compositions or natural scenes. In this case, loose usage is not a serious possibility. Someone can ask, ‘But does the beauty of the Petersen graph not depend on its having a pretty representation (unlike the Hoffman-Singleton graph which is too complicated)?’ I think not. Even the most beautiful drawing of the Petersen graph is not perceptually strikingly beautiful. Its visual attraction is not sufficient to explain the strong emotional response that mathematicians have to the graph itself. Moreover, when they talk about what is beautiful about this graph, they do not mention the drawing: The Petersen graph is a beautiful graph. At least, that is what graph theorists will tell you, time and again $$\ldots$$ [I]n spite of its small size — only 10 vertices and 15 edges — its structure is beautifully symmetric, and this has far reaching consequences. [Erickson, 2014] If you are talking about theoretical beauty, Petersen graph would top the list. Simple, but almost always counterexample to the simple theorems you try to cook up. [Ashwin, 2014] This can serve as an example for believing that at least some mathematical aesthetic judgements may be correct, as I believe the two above are. When mathematicians judge the Petersen graph to be beautiful they do not seem to be concerned with using it. Now, given that most of the remarkable and rare properties of the graph are connected with its high symmetry, why would the fact that symmetries help us to grasp the mathematical structure and other properties imply that symmetries help to make the object beautiful? It is oversimple to say that symmetries help make an object beautiful in mathematics, just as it would be in painting. Perhaps what makes the Petersen graph beautiful is its combination of cognitive simplicity and richness of symmetry. Also, being a counterexample to many conjectures is an epistemic benefit. But this is not to be taken as a factor of beauty, but evidence of the Petersen graph’s aesthetic value. Zangwill has two objections to the view that mathematical entities can be beautiful. One objection relates only to entities which have an intrinsic non-aesthetic function, such as proofs. This objection is not a threat to the view that mathematical objects such as the Petersen graph can be beautiful, because mathematical graphs, like stones, can be used but have no intrinsic function. Zangwill’s second objection is that properties of non-sensory abstract objects simply fall outside the category of aesthetic. In support of this view Zangwill makes the following remark: ‘As the etymological origins of the word “aesthetic” suggest, aesthetic properties are those that we appreciate in perception’ [2001, p. 81]. This implies that, strictly speaking, mathematicians use aesthetic terms metaphorically when they have feeling about mathematical objects similar to feelings they have about art objects. ‘Beautiful’ then, applied to mathematical entities would express a non-aesthetic value, most probably an epistemic value. As Zangwill is no doubt aware, the etymological origin of a word does not determine what the word expresses today. The fact that ‘ethical’ originates from an ancient Greek word meaning ‘habit’ or ‘custom’ does not entail that only what is habitual or customary can be ethical. Similarly, the fact that ‘aesthetic’ originates from an ancient Greek word meaning ‘perceive’ does not entail that only what is perceivable can have aesthetic value. So the etymological consideration is not strong support for Zangwill’s second objection. Rather it helps to notice that our concepts and use of aesthetic terms develop with time and experience. And here I agree with Robert Thomas [2017, p. 121] that ‘beauty is not all there is to aesthetics in mathematics’ and that, in mathematical contexts, it is quite natural to take ‘interesting’ as an aesthetic term, where by ‘interesting’ we indicate a desire to pay attention to an object for its own sake independently of external aims or appetites. In defending the view that mathematical proofs can be elegant against Zangwill’s objection, Barker [2009] points out that one could simply stipulate that aesthetic properties and responses are sensory (Zangwill himself does not do this), but the gain would be small. But it is hard for me to see any real benefit in making such a stipulation. Quite the opposite: it will simply blind us to the real and important similarities that exist between mathematical and sensory beauty. [Barker, 2009, p. 14] Indeed, along with mathematics, most of conceptual art where artefacts are almost irrelevant, falls out of aesthetic considerations exactly because the beauty is not necessarily perceptual. Similarly in mathematics, perceptual attractiveness is, with rare exceptions, irrelevant, and perceptual mediators require mathematical interpretation to be appreciated, as in the case of tangrams and in the case I am going to consider next. Besides, in mathematics, it takes time to develop a good level of expertise, not to mention aesthetic sensitivity. 10.2. About the Beauty of Visual Mathematical Representations Remember that any planar representation of the Petersen graph cannot show all its 120 symmetries. This fact clearly implies a distinction between the mathematical object and its representations. Let us now consider mathematicians’ judgements about visualisations of mathematical objects, and try to understand what role perceptual properties play in such judgements. Compare the drawings of the Petersen graph in Figure 4. In the standard drawing (4c) 10 geometrical symmetries (5 rotations and 5 reflections) are easily noticed, whereas in the Kempe drawing (4b) only 6 geometrical symmetries (3 rotations and 3 reflections) are easily noticed, and the drawing (4a) is even less revealing. The standard, showing a maximum number of symmetries, is the favourite of graph theorists, and most often found in the literature. Could it be just perceptual attractiveness of visible symmetries that makes it the favourite? To tackle the relation of perceptual properties to aesthetic quality, I will consider a case where two drawings of the same graph have the same number of symmetries. Obviously by simply pulling the vertices of the pentagram outside of the pentagon one can obtain an isomorphic drawing as depicted in Figure 10b. (One could also have one with the pentagon within the central area of the pentagram.) This drawing is hardly ever used, it does not appear naturally in practice, and according to graph theorists, they come across it occasionally via computer visualisations. Fig. 10. View largeDownload slide Two isomorphic symmetry-five drawings of the Petersen graph. Fig. 10. View largeDownload slide Two isomorphic symmetry-five drawings of the Petersen graph. Both drawings have the same 5 reflective and 5 rotational symmetries. In fact 10 is the maximum number of symmetries visualisable in a plane drawing of the graph.13 However, there is a big difference in mathematicians’ aesthetic judgements of the two. Graph theorists judge the usual one to be significantly more beautiful than the other (Figure 10b) and sometimes even call them ‘the beautiful’ and ‘the ugly’. For a lay person they may appear equally beautiful or 10b may look prettier. In fact the majority prefers it ‘because it is a star’. What are the experts actually responding to when saying that the left drawing is ‘most beautiful’ and the other is ‘ugly’, given that they have the same symmetry and represent the same mathematical structure? Figure 10a has more ‘good’ properties, such as fewer edge intersections and more wide angles.14 More precisely, there are 5 intersection points in both drawings but in the right one 3 edges meet at each intersection point, instead of 2. So at each intersection point in 10b there are 3 edge-intersections: if $$a,b,c$$, are the intersecting edges, $$a$$ meets $$b$$, $$a$$ meets $$c$$, $$b$$ meets $$c$$. Also broader inner angles mean broader space between the edges incident at a node keeping the same size of the overall drawing. Fewer intersections with broader inner angles make clear that it is composed of a pentagon and a pentagram (and 5 edges bridging them). Perceiving this geometric structure that links the graph to geometry may add to the mathematical beauty of the drawing.15 This is an aesthetic merit and it supports my emphasis on the distinction between a representation of an object and the object itself. The two geometric subgraphs, are so well-articulated in relation one to another, that with a bit of mental transformation, one can easily turn ‘the beautiful’ drawing into $$K_5$$ and realise that $$K_5$$ is a minor of Petersen (see Figure 7). This visible similarity between the drawings of Petersen and $$K_5$$ immediately reveals that the Petersen graph is non-planar and makes it surprising that, while it is like $$K_5$$ in being vertex-transitive, it is unlike $$K_5$$ in being non-Cayley. Moreover, observing Figure 10 of ‘the beautiful’ and ‘the ugly’ drawings of the Petersen graph, one can notice that ‘the beautiful’ shows a lot of other interesting structural information which is obscured in ‘the ugly’. The most intriguing properties of the graph from Section 9 are visible in the usual drawing. For example, using (a magnified) image of the common visual representation of the Hoffmann-Singleton (Figure 5), one can see that the Petersen graph is a subgraph of the Hofmann-Singleton graph. Furthermore, a simple twist of the pentagram relative to the pentagon keeping the centre fixed and allowing the bridging edges to stretch shows that the graph is unit distance (see Figure 9). In the study of snarks, it is ‘the beautiful’ drawing which ‘reveals’ that other snarks have the Petersen graph as a minor. Notice that Figure 10b does not provide such insights. Therefore, the abstract beauty of the graph shines brightest through the ‘beautiful’ drawing and is obscured in the ‘ugly’. Similarly to mathematicians trying to visualise abstract structures in the most effective way, artists follow the ‘rules of perception’ (proportions, density of details, composition, symmetry) to help a perceiver recognise beauty in the artefact. This strengthens the case for saying that the mathematicians’ aesthetic judgement in favour of the usual drawing arises from intellectual pleasure as opposed to merely sensory pleasure. Some of the relevant mathematical properties are revealed by the visual properties of the drawing. So the intellectual pleasure in this case has a sensory mediator, but mathematicians seem to respond to the intellectual content. The question is whether the judgement in favour of the usual drawing is an aesthetic judgement, and what roles intellectual and sensory components have in this judgement. Since the function of the drawing is to help us understand the graph, this could still be a case in which the drawing performs its function more beautifully than other drawings. One can say that the value of the drawing in this case is merely instrumental, non-aesthetic, and then Zangwill’s objection applies. Zangwill uses the idea of beauty independent of fulfilling a purpose from Kant, although with modification [Zangwill, 1999]. However, in one passage Kant suggests that a feature which facilitates understanding may also contribute aesthetically: Objects that ease our perception of them give us pleasure and are beautiful $$\ldots$$ Symmetry eases our understanding and is the proportion of sensibility. Looking at a disproportional house, I find it difficult to conceive it as a whole $$\ldots$$ Uniformity of the parts helps my representation, increases my inner life, and I therefore must find it beautiful.16 Holding a subjectivist position about mathematical beauty, Kant would not say that understanding of the representation adds to mathematical beauty of the abstract.17 He wrote: While mathematical properties themselves are not beautiful, it is the demonstration of such properties that can be the object of aesthetic appreciation $$\ldots$$ The purposiveness of mathematical properties does not indicate beauty but a form of perfection.18 Wenzel in his comments on Kant’s view that there is no beauty in mathematical objects and that ‘mathematics by itself is nothing but rules’,19 supposes that Kant underestimated the richness of mathematical practice in its historical dynamics. I believe that the case I have presented in this paper supports the view that abstract mathematical objects can be beautiful. The interpretation in [Breitenbach, 2015] of Kant’s aesthetic theory sheds light on the Kantian rejection of beauty of mathematical abstract objects, as it is often understood and suggests that ‘the experience of beauty in mathematics is grounded not in an intellectual insight into particular properties of mathematical objects but in our felt awareness of the imaginative processes that lead to mathematical knowledge’ [Breitenbach, 2015, p. 957]. This implies that Kant aims to distinguish knowing the properties (perhaps including interpretation of visual representations) of the abstract object (properties that may converge to some perfection) and free (spontaneous) imaginative activities involving discovery of these properties. Only these activities, which are subject to creativity, make possible aesthetic appreciation of such perfection. These two types of pleasure: epistemic and aesthetic, can be distinguished, but in intellectual contexts they blur; and Zangwill’s objection is not such a threat. Kant’s view is a strongly subjectivist position, but it stresses an under-appreciated aspect of mathematical practice which is still important — the practice of visualising and representing mathematical abstracta in a most comprehensible way. Maybe more than that are also under-appreciated: responding emotionally and creatively, experimenting, going beyond standard ways in reasoning. Clearly many mathematical areas develop their specific styles of visual representations: consider geometry, graph theory and algebra. Apparently, the art of sculpting mathematics includes designing and refining the look of its elements. These definitely have an aesthetic character. In practice it is often difficult to produce a mathematically effective visual representation, one which easily conveys the content. Mathematicians may react in a very favorable way to such exemplars, as with the favourite drawing of the Petersen graph. Therefore, according to Kant, visible symmetry of an object may contribute both to understanding and to beauty. Indeed the examples above demonstrate that in mathematics an aesthetic response requires at least some and sometimes very profound understanding. If that is right, the same may be true of those properties of the favourite drawing of the Petersen graph which most help one understand the graph itself. So, the favoured drawing of the Petersen graph has intellectual beauty, i.e., beauty that requires mathematical interpretation of the drawing to be appreciated. But the perfect and exceptional properties of the graph itself still have to be there. To sum up this section, it is unlikely to be a mere coincidence that the best drawing is also considered most beautiful. On the graph theorists’ view the usual drawing wins. The reason for that is perhaps that experts respond to the combination of the perceptual and intellectual, but presumably intellectual factors have more weight in mathematical beauty than perceptual factors. The intellectual factors are those concerning the manner of representation and the mathematical structure of the object itself. One principle that may be operating here is how easy it is for us to extract structural information directly from the drawing, or in other words the signal-to-noise ratio. In Figure 10b there is more irrelevant information: three times as much for each intersection point, as explained above. Moreover, the structurally significant subgraphs are not clearly separated in the right-hand drawing, whereas the insignificant properties, e.g., three-edge crosses, become louder. These properties reduce the signal-to-noise ratio. Looking at the ‘beautiful’ drawing, a mathematician more easily and more quickly discerns the structure and therefore its beauty. In contrast, when the mathematician looks at the ‘ugly’ drawing, the extra noise diminishes her aesthetic pleasure. It is not just the number of ‘good’ perceptual properties that makes a drawing more beautiful. The mathematician responds emotionally perhaps to the structural features of the graph itself revealed to us by the drawing, such as the possession of significant substructures having shapes of geometric figures.20 In their empirical research Reber et al. [2004] claim that fluency in processing information always enforces aesthetic response. They also admit along with perceptual fluency there is conceptual fluency that can add or even outweigh the perceptual. Thus, mathematicians’ aesthetic preference is for the drawing of the Petersen graph displaying the graph structure more clearly, and lay people rank the ‘star-like’ drawing (Figure 10b) higher. Similarly in the tangram example in Figure 2, a lay person would prefer 2b because it resembles a complete human image. But the two drawings together direct our attention to the paradox. Then we are able to appreciate the beauty of the tangrams not simply for palette or proportions. On the basis of this case study we can put forward the following tentative hypotheses: Different drawings of the same object may reveal different mathematical properties (symmetries). Visible symmetries may contribute to the mathematical beauty of a representation of a mathematical object (though the effect is not proportional to the number of symmetries). Particular visual properties help one grasp mathematical properties (structural) and appreciate them aesthetically. If two drawings of an object have a similar degree of perceptual beauty, the one which reveals a greater degree of mathematical beauty of the object has greater mathematical beauty. Mathematicians respond to a combination of various factors, perceptual and intellectual, but intellectual factors prevail. 11. CONCLUSIONS Taking into account that in practice mathematicians work with visual representations, we must distinguish between beauty of mathematical objects and beauty of visual representations of mathematical objects. Visual mathematical representations may look aesthetically appealing both for a lay person and a mathematician. However, while a lay person appreciates solely the visual appearance of a representation, for a mathematician a visual representation of a mathematical object may be assessed for perceptual beauty, for intellectual beauty, or a combination of both. Visible symmetry is one mathematical property that contributes to perceptual beauty. Mathematicians find the most symmetric drawing of the Petersen graph the most beautiful. However, between two equally symmetric representations of the Petersen graph they choose the clearer one as more beautiful. It turns out that they find visible symmetry along with other perceptual properties of graph drawings (e.g., fewer edge intersections, broader angles between adjacent vertices) also aesthetically appealing. This suggests that when comparing diagrams of the Petersen graph in aesthetic terms, mathematicians respond to a combination of both perceptually appealing properties (e.g., visible symmetries) and the mathematical properties of the object represented. A representation’s possession of properties that are useful is not by itself sufficient for mathematical beauty. Mathematicians may also make aesthetic judgements about a mathematical abstract object as such, focusing on its mathematical properties. For example, in explaining why they describe the Petersen graph as beautiful, they tend to emphasise its high symmetry relative to its small size, and its outstanding mathematical properties, mostly related to symmetry. 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Google Scholar CrossRef Search ADS Footnotes 1For proponents of the response-dependent view see [Goldman, 1995] and [Levinson, 2011]; for criticism see [Budd, 2008] and [De Clercq, 2013]. 2I borrow this quotation from [Wenzel, 2001, p. 426] and I am grateful to a referee for recommending it to me. 3I am grateful to a referee for suggesting this example. 4Weyl [1952] gives a nice exposition about symmetry and mathematics. 5This claim can be found in [Hong and Eades, 2005]. 6The choice of properties is suggested in [Goucher, 2013]. 7Distance here is the number of edges in a shortest path. 8Moore graphs are also optimal models for message-passing networks. 9See [Tutte, 1966]. Thomas [1999] announced a proof of this conjecture. 10This is because (i) any graph containing the Petersen graph as a minor must be non-planar by the non-planarity of the Petersen graph, and (ii) the four-colour theorem is equivalent to the statement that no snark is planar (the theorem that opens the study of snarks due to Tait [1880]) or in topological terms, no snark embeds on a sphere. 11There is an entire book devoted solely to the Petersen graph, where a colossal number of remarkable properties of the graph can be found, The Petersen Graph [Holton and Sheehan, 1993]. 12In some contexts aesthetic properties seem to play an epistemic role. For example, Cellucci [2015, p. 15] suggests: Mathematical beauty can have a role in the context of discovery, because it can guide us in selecting which hypothesis to consider and which to disregard. Therefore, the aesthetic factors can have an epistemic role qua aesthetic factors. 13[Hong and Eades, 2005] discusses mathematicians’ preferences in graph drawings. 14About efficient graph drawing see, e.g., [Purchase et al., 2012]. 15A referee noticed that 10b also suggests a geometric interpretation (a composition of 5 isosceles triangles). In fact, geometric graph theory and topological graph theory are based on geometric representations of graphs (for details see, e.g., [Lovász, 2014] and [Pisanski, 2000]). I give an example when graphs played a role of inter-mediators between group theory and hyperbolic geometry in [Starikova, 2012]. 16Quoted in [Wenzel, 2001, p. 426]. According to Wenzel, Kant eventually changed his mind towards rejecting idea of mathematical aesthetics, but at least until the 1790s Kant kept this view. 17Breitenbach [2015, Introduction] sharpens this contrast of Kant’s subjectivism with a Platonist’s objective beauty. 18As quoted by Breitenbach [2015, p. 957] from Critique of Judgment, §62. 19Quoted in [Wenzel, 2001]. 20Provided by figural goodness and figure-ground contrast in the picture. © The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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Published: Jul 20, 2017

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