## Abstract

Does the materiality of a three-dimensional model have an effect on how this model operates in an exploratory way, how it prompts discovery of new mathematical results? Material mathematical models were produced and used during the second half of the nineteenth century, visualizing mathematical objects, such as curves and surfaces—and these were produced from a variety of materials: paper, cardboard, plaster, strings, wood. However, the question, whether their materiality influenced the status of these models—considered as exploratory, technical, or representational—was hardly touched upon. This article aims to approach this question by investigating two case studies: Beltrami’s paper models vs. Dyck’s plaster ones of the hyperbolic plane; and Chisini’s string models of braids vs. Artin’s and Moishezon’s algebraization of these braids. These two case studies indicate that materiality might have a decisive role in how the model was taken into account mathematically: either as an exploratory or rather as a technical or pedagogical object.

## 1. Introduction

Between the 1860s and 1920s an influential movement of three-dimensional material mathematical models flourished in especially Germany, visualizing mathematical objects, such as complex curves and surfaces. One may think that these material models were considered mainly as a second-order concretization of abstract objects, which were manufactured after the theorems and properties of such objects were already discovered. Although this understanding of the material model was also common, a different understanding was also present, considering such material models as exploratory, as what enables either the discovery of new, unforeseen properties or prompts the proof of new theorems. This unique understanding of the material mathematical model not only was initiated by the production process and the haptic character of the model, but also was dependent on the material itself, in the sense that if the model was either drawn (as a two-dimensional sketch) or made of another material, then its exploratory character either disappeared or was marginalized. This points to a surprising understanding that the material from which the models were produced was essential to the discovery and development of mathematical theories. This is surprising, since mathematical objects are usually considered abstract and independent of materiality.

In this paper I will concentrate on the role materiality played with respect to these models. The choice of material influenced first and foremost the ability of the produced objects to be physically manipulated—in what may be considered a twist of what De Toffoli and Giardino call “manipulative imagination” regarding the role played by two-dimensional diagrams. I will return later to this conception, but it is clear that the question from which matter the diagrams were drawn (e.g., ink or charcoal) had hardly any effect regarding their manipulability. This stands contrary to the material three-dimensional model. Paper, to give one example, could be more easily manipulated than plaster. I therefore follow Mary S. Morgan’s remark, that “representations become models only when they have the resources for manipulation” (Morgan 2012, p. 27).^{1} This manipulative materiality is what enables an exploration with and of the model. However, to follow Axel Gelfert and to emphasize right from the beginning, exploration should be thought of not only as “an activity that aims at the discovery of new facts” (Gelfert 2016, p. 74), but also as “not [necessarily] directed at a specific object, question, or stimulus” (2016, p. 75). In the context of mathematical research, this exploration without a specific aim was also expressed by the mathematician Alexander Grothendieck (1928–2014). Grothendieck, in his monumental autobiographical manuscript *Récoltes et Semailles*, describes two methods of mathematical research he favors. The first is characterized by solving a specific problem, reaching a goal, as an arrow directed towards a problem. But the mathematical work itself, of factually dealing with mathematical problems, is described by Grothendieck otherwise: “There is no longer an arrow, hastening towards a target, but a wave that extends very far and advances to some unknown place, […] a wave followed by another wave, followed by yet another wave […]. In every moment there is a progression, one cannot say towards what […] and there is no goal. The very idea of a ‘goal’ here seems strangely absurd” (Grothendieck 1985–1987, p. 594).^{2} In this sense, exploration does not necessarily lead to a (goal oriented) discovery but rather opens new spaces of inquiry and representation for the mathematical object. Whether or not this opening would prompt new results, may depend also on the materiality of the model.

I aim in this paper to examine two case studies which support this claim: the first from the end of the 1860s, the second from the 1930s. The first case study concentrates on the work of the Italian mathematician Eugenio Beltrami, who made paper models of the hyperbolic plane. Beltrami considered his models exploratory, but as we will see, once the German mathematician Walther Dyck produced the same model in plaster, its exploratory character disappeared. The second case study focuses on another Italian mathematician, Oscar Chisini, who worked with models of strings in order to research plane curves embedded in the complex plane. For Chisini, the aim was not to represent the curves in an exact manner, but rather to concentrate on what he considered essential for the research of the relations between singular points of the curve. According to Chisini, a method that should not be trusted was drawing these strings or using symbolical formulas.

With these two case studies I aim to show that the role that materiality played was decisive in the way material mathematical models could serve in an exploratory capacity. Even if this material-exploratory understanding of the material model was being marginalized already in the 1920s, it nevertheless points towards new possibilities of how materiality could have an important role in the mathematical research.

## 2. The End of the Nineteenth Century: Mathematical Models from Paper and Plaster and the Stagnation of Beltrami’s Model

As mentioned in the introduction, at the turn of the twentieth century, numerous material mathematical models were manufactured in Europe. Where are the origins of this tradition of material models of complex mathematical objects, which reached its peak at the 1870–1890s in Germany, to be found? A possible answer is France. Starting from the second quarter of the nineteenth century, several French mathematicians began to manufacture three-dimensional models. Gaspard Monge and Théodore Olivier were its main promoters, both supporting the teaching of descriptive geometry and having the view that for pedagogical purposes one must see, touch and experience mathematical objects.^{3} The tradition of models became influential in Germany, where these material-mathematical models—from plaster, wood, cardboard, strings etc.—became prolific, especially starting from the 1860s.^{4} One might claim that the role of these models was mainly pedagogical, serving to represent or rather to visualize complicated mathematical objects (mainly curves or surfaces of a high degree), in order to help the student understand such objects in a more comprehensive fashion (Sattelmacher 2021, pp. 27–8). To give a specific example: taking the theorem, proved in 1848, that on every smooth complex cubic surface, there are 27 lines, one can pose the following question: is it possible to find an equation of a cubic surface, such that all of these lines would be real, and hence, one could build a material model of (the real part of) the cubic surface, where all of these lines could be seen? Indeed, an equation of such a surface was discovered at the end of 1860s and in 1869 a plaster model of a smooth cubic surface was constructed, showing the 27 real lines lying on it (cf. Tobies 2017; Rowe 2013). However, the question that immediately arises is whether the material model was only considered as a mere haptic representation, a materialization in a concrete form of an abstract (mathematical) object.

In Germany, Felix Klein and Alexander Brill were among the driving forces of construction and acquisition of models. They advanced a reform in the education of future engineers and mathematicians, where the usage of models was necessary in order to exemplify the new concepts and objects that were used during class (Sattelmacher 2021, p. 7). Their conception of the material model, especially during the 1870s and the 1880s, was unique, and certainly could not be reduced to the understanding of the model as a mere visualization of the abstract mathematical object. This can be seen in a remark Klein made in 1872: “For geometry a model – be it realized and observed or only vividly imagined—is not a means to an end but the thing itself [die Sache selbst]” (Klein 1872, p. 42).^{5} The model tradition was flourishing during the 1880s: four editions of catalogues of models (*Catalog mathematischer Modelle*) were published by Brill throughout this decade; between the second and the third edition (between 1882 and 1885) hundreds of models were added to the catalogue, as well elaborate theoretical explanations.^{6} Ludwig Brill, the brother of Alexander Brill, writes in the introduction to the third edition that the models “support the promotion and revival [Förderung und Belebung] of mathematical studies” (Brill 1885, IV). In 1886 Alexander Brill noted the success of the model movement, claiming in the lecture “Über die Modellsammlung des mathematischen Seminars der Universität Tübingen,” that models serve not only as a means for visualization of tedious computations but also as what initiates the emergence of new knowledge: “a model,” according to Brill, may “cause retrospective investigations on the peculiarities of the presented shape” (Brill 1887, p. 77).^{7}

The models, therefore, did not only support the spatial imagination of the students, they also were considered—at least for a short time—to give an indication of the direction in which new research questions could be posed and to prompt the discovery of new theories. In this sense they were considered exploratory. However, the growing tendency, or the restriction of what Brill called “the word,” i.e., of the formal, axiomatic and symbolic language, led to an underestimation of the material model as an exploratory tool for mathematics: “this limitation to the *word*, which the synthetic geometers fondly favored for a time, could not fail to lead to an underestimation of the *image* [Bild] and the technical skills required to produce it” (Brill 1887, p. 71).^{8}

Klein and Brill thought therefore of the mathematical material model as the “thing itself,” which meant that the model was not a derivative of the describing formula or a materialization of it, as an object belongs to a second order, but what rather transmits *complementary* knowledge concerning the mathematical object. The material model during the 1870s and the 1880s was hence considered as what may lead to the discovery of new truths, which may even be formulated before the complete theory describing it was.^{9} One of the questions that stand at the center of this paper is whether one of the factors that made these models exploratory is their manual construction and how they were manually constructed. It is clear that the mathematician had to construct the model with her own hands (see also Sattelmacher 2021, p. 13), but merely being the result of manual construction does not (yet) confer any epistemologically distinct exploratory quality upon the models. However, as I aim to show, the material, from which one made the models, was in certain models a decisive factor, whether the material model could either prompt and transmit new mathematical knowledge or, on the contrary, if it would rather be considered as a pedagogical tool, only enabling the mathematical object to be more, “anschaulich,” “intuitive,” or more visualizable. Anja Sattelmacher (2021, pp. 176–7) notes, for example, that while a model constructed from several layers of cardboard or from strings could be changed via movement and hence could present several surfaces or curves, models from plaster—while presenting only one type of surface—could be easily re-produced and were more easily transportable and better preserved. That is, the choice of the material already influenced the functions the model may have.

It is therefore worth considering in more details a specific example, stemming from the end of the 1860s, of how the materiality of a mathematical model prompted new mathematical discoveries, hence functioning in an exploratory fashion.^{10} As I will show, the same example was then re-conceptualized twice: turning first into a mere visualizing representation once the material changed, and second, forgotten within a very different conception of the model: one in which semantics took priority of place.

### 2.1. Material Models of Non-Euclidean Geometry I: Beltrami and the Exploration via Paper Models

With the appearance of the non-Euclidean geometries and later, with Riemann’s differential geometry, new types of surfaces were examined with respect to these geometries; of special importance was the geometry of the hyperbolic plane, known as the Bolyai-Lobachevsky plane. The surfaces associated with the hyperbolic plane were surfaces of constant negative curvature. The Italian mathematician Eugenio Beltrami (1835–1910) sought material models to these geometries, in terms of three-dimensional surfaces. However, in contrast to the German tradition, it is essential to note that Beltrami’s models were more of an exception: As Nicola Palladino and Franco Palladino remark (2009, p. 71), mathematical models in Italy were mainly imported from Germany, England and France and were hardly ever produced in Italy itself;^{11} it is for this reason that Palladinos call the models of Beltrami an “isolated realization.” although it is clear, as they describe, that the Italian mathematicians were aware of this tradition in Germany (2009, p. 46).

Nevertheless, Beltrami’s models were highly influential. But were they exploratory, and if so, how did their exploratory character depend on their materiality? Beltrami’s aim, as David E. Rowe notes, was “to build an actual physical model of his surface (the pseudosphere), and then to use this to study geometric theorems valid in the Bolyai-Lobachevsky plane” (2017, p. 8). While the history of the models of Beltrami has already been thoroughly explored (see Capelo and Ferrari 1982; Boi, Giacardi and Tazzioli 1998; Arcozzi 2012; Epple 2016; Rowe 2017, among others), I would like to focus on the exploratory and the epistemic role that materiality played in Beltrami’s models and later in other material models of the hyperbolic plane.

In 1868 and 1869 Beltrami published two seminal papers: “Saggio di interpretazione della geometria non-euclidea” (Essay on the Interpretation of Non-Euclidean Geometry) and “Teoria fondamentale degli spazii di curvatura costante” (Fundamental Theory of Spaces with Constant Curvature). These papers presented surfaces of revolution that showed how non-Euclidean geometry could be—to a certain extent—realized on such surfaces of constant curvature, surfaces that Beltrami called pseudospherical. Beltrami aimed to construct a surface (or a part of it), embedded in three-dimensional Euclidean space, where the theorems of the Bolyai-Lobachevski hyperbolic plane could be easily verified. Beltrami presents three surfaces of revolution, whose curvature are constant and negative, proving that parts of the pseudosphere are isometric to these surfaces. He emphasizes at the beginning of the “Saggio” that his goal is to “to find a real substrate [*substrato reale*] for this theory [of the hyperbolic plane] before admitting the need for a new order of entities and concepts to support it” (Beltrami 1868, p. 375).^{12} The “real substrate” is a surface in the three-dimensional space; the work and the publication of “Saggio” were accompanied by the construction of material models of the different surfaces of revolution. But how did Beltrami come to the idea of materially constructing these models? In March 1869 Beltrami wrote Jules Hoüel about his first attempt to construct such a model: “I had a strange idea [idée bizarre] […]. I wanted to attempt to *construct materially* the pseudosphere, on which the theorems of non-Euclidean geometry are realized. […] by appropriately cut pieces of paper, one can reproduce […] from them the [pseudospheric] surface” (Boi, Giacardi and Tazzioli 1998, p. 80).^{13}

The paper models that Beltrami built served to verify the results that he achieved in “Saggio,” as well as to lead to new results, which then could be proved later.^{14} Hence, the material model brings about the becoming of the theoretical results. As Beltrami writes to Hoüel in another letter from March 1869, the theoretical aspects and formulation are not (yet) known:

This morning […] I cut out quite successfully a cardboard model, which will serve for a new attempt at constructing a pseudospheric surface. You speak of empirical propositions [

propositions empiriques] which might be found by this means, and you are perfectly right, for here we are dealing with surfaces whose general equations we do not possess. (Boi, Giacardi and Tazzioli 1998, p. 86)^{15}

Beltrami also sent to the mathematician Luigi Cremona two of his models (see figure 1 for one of these models). Along with these models he included instructions on how to fold them (in the sense of bending), in order to form the surface: “[t]he surface of revolution, according to which it [the model] is folded […] [–its] meridian is a transcendent curve whose finite equation is unobtainable” (Boi, Giacardi and Tazzioli 1998, p. 202).^{16} In a word, manual and empirical bending can succeed where algebraic equations fail, and where the surface can only be described via transcendental equations.^{17} I claim that it is the material itself—paper—that prompted the discovery of these results. To explicate: it is not (just) due to the paper as a representational medium that these discoveries were prompted, but rather the paper’s material properties; this is since constructing the model from plaster could not have enabled the same manual operations. Beltrami emphasizes in several letters (either to Cremona or to Hoüel) that the discovery of new theorems regarding geodesic lines and other curves on the pseudosphere was enabled by means of trace curves (“with a pen”; Boi, Giacardi and Tazzioli 1998, p. 91)^{18} on his models. Were Beltrami to produce his models from another material, this would have been hardly possible. To explicate this issue, it serves to reference a letter from Beltrami to Hoüel where he notes that, after describing the material construction of his model, one of the theorems concerning the hyperbolic plane was discovered by him after “tracing” a line on the model (Boi, Giacardi and Tazzioli 1998, p. 82).^{19} A similar instruction—of tracing curves on the paper model—appears in a letter to Cremona from April 1869 (Boi, Giacardi and Tazzioli 1998, p. 201). Once this model was produced from other materials—plaster, for example, as was done starting at the end of 1870s (see the following subsection)—drawing on the model itself was either extremely technical or impossible.

In the following years, Beltrami continued to emphasize the importance of the material constructions of models (see also Beltrami 1872, p. 394, 397). Yet from 1872 onwards he ceased his research on hyperbolic geometry, and hence no longer produced any models. As I will show, for Beltrami it was the material medium itself – now as a representational medium (regardless of the material from it was made)—that was now considered as an obstacle rather than as having an exploratory character. For other mathematicians, who continued working with material models of these surfaces, it was the change of material medium, which prevented these models from being exploratory.

### 2.2. Material Models of Non-Euclidean Geometry II: The Insufficiency of Material Models and the Change to Plaster

To reiterate, Beltrami initially called the production of his models an “idée bizarre”, and one might see in this “strangeness” an indication to what Beltrami thought about later in terms of limitations and obstacles, which material models posed with respect to the understanding of non-Euclidean geometry. These limitations were already implied in two papers, the first from 1867 and the second from 1868. In 1867 Beltrami noted clearly that one has to leave aside any concept or image that implies a concrete determination of its shape when investigating a surface and its metric: “It is useful to recall from the outset that when it concerns a surface defined by the mere expression of its linear element [i.e., of the metric], one should leave aside any concept or image which implies a concrete determination of its form in relation to external objects” (Beltrami 1867, p. 318).^{20} No doubt this statement also applied to the material (concrete) models, but one has to recall that the paper was written in December 1867 (Beltrami 1867, p. 353), that is, before Beltrami constructed his models. In addition, although Beltrami called for “leaving aside” any “image which implies a concrete determination,” the 1867 paper contains several images and figures (Beltrami 1867, pp. 337, 345, 346, 350), and obviously Beltrami’s concrete paper models, constructed in 1869, prompt a tension with the above statement, exactly because Beltrami uses it to investigate and determine few properties of the metric.

This tension comes into expression in yet another paper. In an 1868 paper investigating manifolds, whose curvature is constant and negative, Beltrami notes that while two-dimensional manifolds are (materially) constructible, the manifolds, whose dimension is three or more, have only an analytic representation.^{21} This is also expressed in a letter Beltrami wrote to Hoüel in April 1869, which, to emphasize, was sent after the models were already constructed in March 1869: “I frankly admit that, when the number of variables in these expressions is greater than 2 their construction generally exceeds the limits of geometric experience […]” (Boi, Giacardi and Tazzioli 1998, p. 92).^{22} The “geometric experience” may be limited to the two-dimensional surfaces, which can be constructed materially, whereas in dimensions higher than two, one should turn either to an analytical representation or to formal, analytical tools, as Beltrami indicates later in this letter to Hoüel. Indeed, while Beltrami thought of his material models as an exploratory “concrete interpretation” for two-dimensional non-Euclidean geometry (Beltrami 1868, p. 7),^{23} for three-dimensional non-Euclidean geometry the situation was completely different. In the latter instance, Beltrami indicates that, while for the two-dimensional case a proper and true interpretation is possible (“un’interpretazione vera e propria”), for a three-dimensional hyperbolic geometry, “only an analytic representation” (“una rappresentazione analitica”) is possible (Beltrami 1868–1869, p. 427).

Even if material, paper models were considered for a short period as exploratory, eventually Beltrami thought of them as a limited and limiting technique. And while sending his two-dimensional paper models to his colleagues, he, at the same time, also promoted a less concrete and more theoretical approach. However, even if we still consider the material models used to make the hyperbolic two-dimensional plane visualizable, one might note another reason for the disappearance of the exploratory aspect of these models: the change of the material medium.

Taking a look at the first edition of the *Catalog mathematischer Modelle*, edited in 1881 by Ludwig Brill, presenting mostly the models made by Klein und Alexander Brill and their students, one discovers a plaster model of a “surface of revolution of a constant negative curvature […] with parallel geodesic lines and geodesic circles […] [done by the] mathematics student W. [Walther] Dyck” (Brill 1881, p. 12). The model belongs to the second series of plaster models, and Ludwig Brill includes a photo of it on the inner cover of the catalog (see figure 2, the model at the middle, having no curves carved on it).

Examining the small booklet that accompanied the model, written in 1877 by Walther Dyck, it is clear that he knew Beltrami’s work (Dyck 1877, p. 1). However, Beltrami’s exploratory approach is not to be found in Dyck’s booklet. The explanations are mainly theoretical, and when it comes to a description of how one can materially draw the geodesic lines on the model based on mapping them from curves on the plane, Dyck notes the following: “On the model, geodetic lines and geodesic circles were applied in such a way that straight lines and circles were drawn in the plane of the image in a suitable manner, then a [….] net was constructed, which facilitated a point by point transmission” (Dyck 1877, p. 8).^{24} Any indication as to whether new theorems might be deduced from this procedure go unmentioned; moreover, a description of the production process itself—how one actually produces the plaster models or how one carves the geodesic lines—is completely absent from Dyck’s booklet.

It is obvious that the plaster model presented is one of Beltrami’s surfaces of constant negative curvature. It is also obvious that in order to enable a mass production of the model and to enable its sale, shipping, and handing it over to the public, one had to manufacture it from stable, sustainable materials (such as plaster and not paper).^{25} This is also indicated in the introduction to this series of the plaster models, written in 1877 by Alexander Brill, which was published in the 1885 catalog of mathematical models.^{26} However, the change of the material medium prompted also a change in its exploratory character, as was already seen above. Although Alexander Brill expressed in 1877 his wish that the plaster models would lead to “new and interesting results” (Brill 1885, p. 4), this was hardly the case. Firstly, the manufacturing of the plaster models already required a special expertise in plaster casting—a casting mold had to be prepared first.^{27} Even if after the mold was ready and the models were prepared for mass-production, exact mathematical calculations had to be made before preparing the mold itself. Only in this way would the plaster model be an exact replica of the mathematical object. Hence, the plaster mold was a technical object, as there was neither room for experimentation nor for manipulation with it.^{28}

Secondly, the student who obtained the model, could observe it, but not necessarily draw on it, since the geodesic lines were already carved into the model. Hence, the mass production of the plaster model prevented in some sense any exploration with it. As can be seen from figure 3, the models were sold when these curves were already carved on them. As Sattelmacher remarks (2021, pp. 204–6), the process of carving lines and curves into a plaster model required practice and experience: an error in carving could only be corrected by abrading the surface, which could cause inaccuracies. Since the models themselves, which were sold to various universities, were error-free, numerous calculations had to be made—not as an exploratory operation but rather as a technical one. In addition, since the material model was not made from paper, one could not bend or fold it. These two material operations: bending the (paper) model and drawing special curves on it, two operations which were for Beltrami essential in order to think in an exploratory way with the model—became irrelevant from an exploratory point of view with the production of the model from plaster. In this sense the model became technical and pedagogical, hence losing its epistemic character; that is—it was used only for showing the geodesic curves, but not drawing them, and certainly not for drawing conclusions or future theorems regarding them.

The fact that the models from plaster were considered more in a pedagogical fashion than in an exploratory way is to be seen also in the third edition of *Catalog mathematischer Modelle*, issued in 1885. As mentioned above, this edition included detailed theoretical explanations concerning the models, which did not appear in the first and second edition. Within the section on surfaces with constant curvature, the following description is to be found: “The geometry on the surfaces […] of constant-negative curvature is called non-Euclidean geometry and coincides with the one established by Lobachevsky, which lacks the eleventh axiom of Euclid. […] The following surface models are intended for the study of this geometry” (Brill 1885, p. 42).^{29} Again, the same model, made by Dyck, is presented (see figure 3) and it is clear that on the purchased model the geodesic lines are already carved.

### 2.3. Material Models of Non-Euclidean Geometry III: The Complete Disappearance of the Material Model

The paper models of Beltrami did not completely disappear from the awareness of mathematicians at the beginning of the twentieth century. I will partially follow here David E. Rowe (2017, pp. 7–9) and Moritz Epple’s accounts on the concept of the model (2016, pp. 14–18), though I will focus more on the disappearance of the material model of the hyperbolic plane. As Rowe emphasizes (2017, p. 8), in 1906 Roberto Bonola published his influential book *La geometria non euclidea: Esposizione storico critica del suo sviluppo*, which was translated to German in 1908 and to English in 1912 (as *Die nichteuklidische Geometrie. Historisch-kritische Darstellung ihrer Entwicklung* resp. *Non-Euclidean Geometry: A Critical and Historical Study of Its Development*). In his book, Bonola indicates that the model for non-Euclidean geometry is a material model,^{30} presenting a photo of one Beltrami’s paper models; see figure 4. But immediately following this discussion, after which Bonola supplies the equations for the surface of revolution with negative constant curvature and presents Beltrami’s paper model, he emphasizes another way of thinking of non-Euclidean geometry: “There is an analogy between the geometry on a surface of constant [negative] curvature and that of a portion of a plane, both taken within suitable boundaries. We can make this analogy clear by *translating* the fundamental definitions and properties of the one into those of the other”^{31} (1906, p. 125; 1908, p. 141; 1912, p. 134). Bonola however did not use the word “model” to term the above-mentioned “translation” of these definitions and properties. How then did the term “model” shift into denoting a semantic interpretation of axioms? Epple, Rowe and Erhard Scholz (Rowe 2017, p. 8; Epple 2016, p. 24) suggest that it was due to Hermann Weyl’s books and lectures and especially Weyl’s 1918 book *Raum Zeit Materie*, which initiated a shift in how the term “model” was understood in mathematics. This deserves a closer look, in order to see whether Weyl did refer to Beltrami’s material model in any way.^{32}

In 1913, five years after the publication of the German translation of Bonola’s book, Hermann Weyl published his influential book *Die Idee der Riemannschen Fläche*. When discussing the non-Euclidean plane, Weyl ignores completely the material models of Beltrami. He however does emphasize a similar “translation” between basic concepts (points, lines), as Bonola does; after this translation, “the complete *Bolyai-Lobatschefsky geometry* holds for these ‘points’ and ‘straight lines’” (Weyl 1913, p. 152). As a reference for this “model of the plane non-Euclidean geometry” (Weyl 1913, p. 152), Weyl provides the title of Bonola’s book. In his book *Raum Zeit Materie* Weyl refined his point of view (1919, p. 83): he presents Beltrami’s “Euclidean model” of non-Euclidean plane geometry (as what holds on a surface of revolution), but he once more principally stresses models of Euclidean and non-Euclidean geometry as a kind of translation between two languages: one constructs a “Euclidean model for the non-Euclidean geometry” and “sets a lexicon, by which one translates the concepts of the Euclidean geometry into a foreign language [fremde Sprache], that of non-Euclidean geometry” (Weyl 1919, pp. 71–2).^{33} A similar ignorance appears in Weyl’s 1927 book *Philosophie der Mathematik und Naturwissenschaft*.^{34} However, the references to a possible material model of the hyperbolic plane had not completely disappeared: Hilbert and Cohn-Vossen note in their 1932 book *Anschauliche Geometrie* that from three surfaces with negative constant curvature (see figure 5) one may “construct models,” the geometry of which is called hyperbolic geometry;^{35} but although mentioning other material models in their book, they also do not mention that these models can be materially constructed. Although they did take into consideration material models, they also refer to “model” in the axiomatic-semantic sense (Hilbert and Cohn-Vossen 1932, pp. 103, 216), as an instantiation of axioms.

The decline and disappearance of Beltrami’s material model, and eventually also of Dyck’s, which were at best considered pedagogical tools,^{36} was therefore also due to the focus given to the term “model” within a set-theoretical and semantic framework, as an interpretation of axioms and basic concepts.^{37} This emphasis—what Alexander Brill called the growing tendency to the “word”—caused a decrease in the significance of other uses of models and aspects within the scientific practice of exploration. In addition, one might say that it was also materiality itself that was an obstacle standing in the way continuing exploration. It is essential to remember that Hilbert proved (1901) that there exists no complete regular (i.e., continuous and smooth) surface having a constant negative curvature immersed in three-dimensional Euclidean space. Beltrami hence proved that the surfaces of revolution, whose models he constructed, having a constant negative curvature, are isometric only to parts of the pseudosphere. This points out that the models attempting to represent in an exact manner the mathematical object might at first function in an exploratory way, but most likely eventually transform into purely technical uses. To recall: Beltrami, as well as Dyck, aimed to eventually find an exact, error-free model. The question that arises is whether material models of mathematical objects might not have become technical or disappeared if they were not aiming at an exact representation of the object, but rather at giving an account of only a part of the properties of the object in question. I will deal with this issue in the next section, which focuses on another model stemming from Italy.

## 3. The 1930s and the 1980s: Braids from Strings and the Algebraization of Chisini’s Model

Beltrami’s model was not a complete exception in Italy. Although the production of material mathematical models was not common in Italy as in Germany, during the first decades of the twentieth century few Italian mathematicians, such as Guido Castelnuovo and his student Federigo Enriques, did appreciate this mathematical tradition, considering it even in an exploratory way, as Beltrami had also done. There is something remarkable that testifies to this fact in the following citation from Castelnuovo in 1928:

We had constructed […] a large number of surface models […] [placed] in two showcases. One contained the regular surfaces for which everything proceeded as in the best of all possible worlds […]. But when we tried to verify these properties on the surface of the other window, the irregular ones, trouble began and there were exceptions of every kind. In the end, the assiduous study of our models had led us to divinate some properties that had to exist, with appropriate modifications, for the surfaces of both showcases; we then put these properties into practice with the construction of new models. If they resisted the test, we were looking for the logical justification for the last phase. With this procedure, which resembles the one carried in the

experimental sciences, we have succeeded in establishing some distinctive traits for families of surfaces.^{38}(Castelnuovo 1928, 1:194)

### 3.1. 1930s: Chisini and the String Models of Algebraic Curves

During the 1930s Chisini was interested in how one can obtain a real representation of a plane algebraic (complex) curve.^{39} This is to be seen clearly already in the title of his 1933 article “Una suggestiva rappresentazione reale per le curve algebriche piane”. Indeed, ever since Bernhard Riemann introduced the now well-known *Riemann surfaces* in his 1851 doctoral dissertation, as the covering of the complex line (or of the projective complex line) for multi-valued analytic functions, attempts have been made to visualize these coverings.^{40} Chisini refers directly to Riemann’s approach—considering the curve as a covering of the complex line – as what remained “always a less significant model [poco significativo]” (1933, p. 1141) for a representation of complex plane curves.

This leads Chisini to a search for a more significant model. Already in the introduction to the 1933 article he declares that his model leads to “remarkable results” (1933, p. 1142; Chisini’s construction is described on pp. 1145–47). The question remains, however, as to what actually is being modeled? Chisini begins with a curve of degree *n*: *f*(*x*, *y*) = 0 which might be singular. Examining its equation as a function of *x*: *y* = *y*(*x*) and the projection *p*: (x, y) ↦ *x*, he looks at the *N* branch points on the *x*-axis, which are points whose *n* preimages are not distinct.^{41} Noting that the *x*-axis is a complex line, hence homeomorphic to the real plane, he draws *N* loops on the plane denoted as *γ*_{1}, …, *γ*_{N}, all exiting from a given point *O*, when each loop encircles another branch point. The point *O*, not being a branch point, has *n* preimages *p*^{−1}(*O*) = {*y*_{1}, *y*_{2}, …, *y*_{n}}. Taking one of the loops *γ*_{i} starting (and ending) at *O*, Chisini looks at what happens to these *n* preimages as one goes along this loop. More specifically, the loop is given as a function *γ*_{i}: [0, 1] → ℝ^{2}, when *γ*_{i}(0) = *γ*_{i}(1) = *O* and hence *p*^{−1}(*O*) = {*y*_{1}(0), *y*_{2},(0), …, *y*_{n}(0)}, when *y*_{j}(0) = *y*_{j}, 1 ≤ *j* ≤ *n*. Chisini then looks on the movement of all the *y*_{j}(*t*) as *t* runs from 0 to 1. Considering the complex points *y*_{j}(*t*) as points in ℝ^{2}, what one obtains is a movement of *n* points in ℝ^{3}, having coordinates as (*t*, *Re*(*y*), *Im*(*y*)).^{42} When one looks at the movement of these points as delineating curves in three-dimensional space, one obtains a braid, composed of *n* strings; see figure 6 for an example of a braid composed of two curves, induced from a loop encircling the branch point of a “double point [punto doppio]” (1933, 1150) of a curve *f* (i.e. encircling the image (under the projection to the *x*-axis) of the singular point (0, 0), locally of the form (*y* − *x*)(*y* + *x*) = 0).

The next step for Chisini is to look at all of the loops *γ*_{1}, …, *γ*_{N} together—i.e., concatenating them into one loop *γ* encircling all of the branch points—and then performing the same process as above to obtain a braid which is termed as the “characteristic bundle [fascio caratteristico] of the curve *f*(*x*, *y*) = 0” (1933, p. 1146). This bundle, according to Chisini, “constitutes the real representation of an algebraic curve” (1933, p. 1147). In figure 7 one can find one of Chisini’s depictions of how this braid looks for the curve *y*^{4} − 4*y* + *x*^{4} = 0, having 12 branch point, for the concatenation of only three loops (of the 12 in total).

Chisini calls this construction a “model” in several of his papers (Chisini 1933, 1934, 1937). Why was this model so essential for Chisini? There are a few reasons for this, which indicate Chisini’s understanding of what a model is, and why it can be thought in an exploratory manner. Firstly, Chisini emphasizes throughout the 1933 article that this model becomes more accessible and convenient to work with once one works with “material models [modelli materiali],” with “material threads with notable thickness” (1933, pp. 1146–7). According to Piera Manara, the daughter of Carlo Felice Manara, one of Chisini’s students, these models—that were in Chisini’s office—consisted of “simple wooden boxes, with metal nails and braids of threads of various colors, intertwined in various ways,” and were “an instrument of concrete visualization.”^{43} The fact that Chisini considered his models as material is also to be seen in the thickness of the curves depicted in figures 6 and 7, as if they were real strings. As was noted above, Chisini must have been exposed to the material models constructed by Castelnuovo and Enriques. Additionally, already in 1915, Enriques presented in the first volume of his book *Teoria Geometria delle Equazioni e delle funzioni algebriche*, a similar representation of a complex plane curve. Chisini, who was at the same time an assistant of Enriques, edited the book. Enriques indeed named this representation “a model,” indicating: “The mode of [this] passage [of a loop below a branch point] can be made evident with a model” (1915, p. 360),^{44} describing how the preimages change their position while going below a simple branch point, adding a figure (see figure 8). Hence, it is clear that Chisini was influenced by Enriques’ approach.

However, Enriques did not hint at any point in his book that this model was supposed to be material. This stands in contrast to Chisini’s approach. Moreover, Chisini’s material models do not aim at an exact representation (i.e., how the curve is exactly embedded in the complex plane ℂ^{2}), but rather aim to describe the topological relations between the different *y* values (of the preimages) while encircling the branch points: “[…] in the actual representations (made with drawings or models) it is not necessary that the characteristic lines [the curves composing the braid] correspond exactly to the equations […]” (Chisini 1933, p. 1146). The suggestion of Chisini – of making the models from strings – was not just a metaphor or an empty proposal. Naming the “making visible” (Manara 1987, p. 21) as the mark of Chisini’s genius, Manara recalls that Chisini did not only construct materially string models of braids but considered them as the most trustworthy instrument to prove claims, far more trustworthy than “drawings and formulas”, in a sharp contrast to the algebraic turn taking place in algebraic geometry during these years:

[…] the invention of the ‘braid’ bears the initials of his creativity; but this was constantly held back by his critical spirit, which led him to persist in the construction of material models, not to be content with drawings and formulas; those who met him in those years remember that one of the most frequently repeated phrases was ‘… I do not trust’. And this distrust led him to build tangible and material models, on which he could verify the validity of his inventions.

^{45}(Manara 1987, 28)

*O*may be any point which is not a branch point (Chisini 1933, p. 1151); the system of loops

*γ*

_{1}, …,

*γ*

_{N}encircling the branch point may be chosen in a different manner (Chisini 1937, pp. 60–61). These choices do not affect the final result: the different characteristic braids obtained while changing the above choices (the point

*O*or the system of loops) would be the same.

The third reason concerns how Chisini and his students regarded this material model as exploratory, as a research tool, which does not only “make visible” complex plane curves or that serves the “visual intuition” (Chisini 1937, p. 50). Chisini indicates that the “value of the model” is that with it “one can deduce easily and clearly few remarkable propositions” (1933, p. 1153),^{46} or that “[one can] draw interesting conclusions regarding the curve *f*(*x*, *y*) = 0 from the characteristic bundle,” i.e., from the braid (Chisini 1937, p. 50). Moreover, how this braid could be used and which research questions might be prompted by it, “cannot be established a priori” (Chisini 1937, p. 50). This already indicates the exploratory aspect of Chisini’s model: “exploration as a precursor to theorizing” (Gelfert 2016, p. 78). This aspect—which results in propositions of braid theory *proved* by the material string models—can be seen, for example, either with the 1950 paper of Modesto Dedò “algebra of the characteristic braid [Algebra delle trecce caratteristiche]” or with a 1955 paper of Cesarina Marchionna Tibiletti, both of whom were Chisini’s students. Continuing Chisini’s research on braids associated to complex plane curves, both authors note that identities between two different braids can be proven materially: these identities can be “verified immediately [with] the material model” (Tibiletti 1955, p. 31) by moving strings, left, right or one on top (or below) of the other (without tearing them); see also (Dedò 1950, pp. 238, 247, 258).

Chisini’s models enabled exploration since they offered the possibility of being manipulated—this stems directly from their materiality as string models. Hence they not only could be used to check the validity of known theoretical results, but also as a way to deduce new results. However, as I will show presently, during the 1980s, when Chisini’s results were rediscovered and reworked by Boris Moishezon, Chisini’s material approach was absent completely.

### 3.2. 1980s: Artin, Moishezon and the Algebraization of Chisini’s Braids

Although Chisini published numerous papers during the 1940s and the 1950s on his “characteristic bundle” as a way to investigate plane curves (and especially branch curves; e.g., Chisini 1952), only in the 1933 paper did he mention his material models. Although Chisini did construct and also implicitly promote usage and exploration with his string models, the fact that any mention of these models disappeared from his writings after 1933 might explain why, when Moishezon introduced and developed the concept of “braid monodromy” in the 1980s, no reference to Chisini’s models is to be found.

How Chisini’s theory of braids was developed by his students is outside the scope of this paper, but it should be noted that one of the factors contributing to the total disappearance of the material string models as exploratory was the partial ignorance of Chisini and his students of Emil Artin’s 1925 paper “Theorie der Zöpfe,” which was later revised and published in English in 1946 as “Theory of Braids” (1947). Artin’s treatment attempts to achieve an algebraic formulation of the braid group (Artin 1925, p. 50), though for some of the propositions he does turn to topological arguments, see Friedman (2019b).

One of the references of Chisini to Artin’s work is in 1952, where Chisini distances himself from Artin’s algebraic formulation of the braid group, saying that the Chisini’s braid is different from Artin’s, in the sense that the braid of Chisini is associated to the branch points of a curve (Chisini 1952, p. 19). However, as Artin himself points out, some of the algebraic arguments presented by him can be also seen by looking at his sketches (Artin 1925, pp. 54, 62; see also figure 9; see also Epple 1999, pp. 318–19).

Taking this into account, it is nevertheless a fact that Artin’s theory became far better known than Chisini’s. Artin’s theory concentrated on braids as such; Chisini focused on braids associated to plane complex curves. Up until the 1980s, there was not significant research on the topic as developed by Chisini, and needless to say that the material models of Chisini slipped into oblivion. Artin’s theory was further developed, gaining popularity since it solved essential algebraic problems (such as the word problem) as well also due to its connections with knot theory. And when Boris Moishezon develops Chisini’s ideas in a series of papers (1981, 1983, 1985), the formulation of the braids associated to the complex plane curve is completely algebraic and relies on Artin’s formulation. Moishezon begins his paper “Algebraic Surfaces and the Arithmetic of Braids I” with a description—though being more algebraic and more precise—which is almost identical to Chisini’s characteristic braid. Moishezon defines the braid monodromy factorization (1983, pp. 201–202), which, when—using Chisini’s terms—choosing the loops *γ*_{1}, …, *γ*_{N} in a certain order, is the factorization of all the braids associated to the *γ*_{i}’s. This factorization is eventually an *algebraic* factorization of the braid associated to the loop *γ* encircling all of the branch points of the plane curve, an idea that already appears in Chisini’s papers. Moishezon, one should add, later knew Chisini’s work: in 1994 Moishezon disproved one the conjectures of Chisini^{47} which relied explicitly on Chisini’s work on braids associated to plane curves. In the 1994 paper Moishezon fails to mention any material models that Chisini used and for the obvious reasons, as mentioned above—simply because Chisini himself stopped writing about them. But what is essential to emphasize that another type of manipulable materiality appears with Moishezon, namely, materiality of the graphical symbols themselves, where the script itself becomes an “operative writing” (Krämer 2003, p. 522). That is, a shift in the space of mathematical inscription occurred: while Chisini introduced his braid both materially and graphically (using material models and drawings), Moishezon wrote it down as a symbolical factorization of algebraic elements from the braid group: according to Moishezon, the braid monodromy of a rational nodal plane curve “is *symbolically* given by the [above] formula.” (1981, p. 125; see figure 10).

## 4. Conclusion

If indeed—following Gelfert—a model is exploratory in the sense that “modeling need not always come *after* a fundamental theory has been established or an empirical phenomenon has been stabilized,” (2016, p. 94) i.e. it is exploratory also when an underlying theory is not (yet) available, then we may ask how much of this exploration depends on the materiality of the model. Paying attention to the two case studies presented in this article, I claim that different materials enable different hybrids between material three-dimensional models, two-dimensional figures and text. Indeed, all of the models were accompanied by a textual explanation and most of them were accompanied either by two-dimensional sketches or by photos of the models, although a few texts did not explicitly use two-dimensional figures. This combination had a different character when considering different types of model, hence it is what I call a hybrid, which also changed during the time.^{48}

Thus, for example, Dyck’s plaster models marginalized the exploratory aspects of the paper models of Beltrami. The subsequent shift to the semantic model emphasized this lack of exploratory materiality from another direction: the meaning and interpretation of the symbols could be manipulated, and not the material of the (non-existent) three-dimensional model. To take another example, the material string models of Chisini enabled a manipulative material model, while Artin applied with his diagrams at best a “manipulative imagination” (De Toffoli and Giardino 2014, p. 829), which may prompt a visualization of local motions of the strings only in the mind of the reader. With Moishezon one can notice a passage to a symbolic, graphic manipulation of the symbols themselves, indicating that an attempt to a complete formalization took place. Hence, a change of the medium (or its complete absence) necessarily causes a change in the exploratory character of the model and therefore reveals that exploration is what De Toffoli and Giardino call “representationally heterogeneous” (2015, p. 326). It is heterogeneous not only because symbolical one-dimensional text, two-dimensional figures and three-dimensional models should be considered together as a hybrid, but also since the relations within this triad are unstable and cannot be defined a priori. As I tried to show, materiality was a decisive factor determining these relations, and whether either a model would be exploratory or would become a technical representation and eventually vanish.

## Notes

Although Morgan discusses mainly economical models, she does also note the existence of material models during the nineteenth and twentieth century (see Morgan 2012, pp. 10–13).

“Il n’y a plus de flèche, se hâtant vers une cible, mais une vague qui s’étend très loin et qui s’avance on ne sait où, […] une vague suivie par une autre vague, suivie par une autre encore […]. En chaque moment il y a une progression, on ne saurait dire vers quoi, […] et il n’y a pas de but. L’idée même d’un ‘but’ ici paraît étrangement saugrenue.”

The string models Olivier manufactured are especially famous. Remarkable is also the following quote from Olivier in 1845: “It is necessary to learn to represent the idea of surfaces and the curves […] to see them through the eyes of the mind, to cut, to touch, to wrap […]” (Olivier 1845, p. 64).

This was mainly due to the strong connections between French and German mathematicians during the nineteenth century. On how the model building tradition was transferred from France to Germany, see Sattelmacher 2021, pp. 116–21.

This quote is in an endnote in Klein’s 1872 famous *Erlangen Programm*, which attempted the use an abstract, group theoretic approach to investigate different manifolds in space. As Mehrtens remarks, in this endnote Klein distances himself from the abstract approach of his program (Mehrtens 2004, p. 209).

The second edition of 1882 had only 30 pages; the third edition of 1885 had 48; the fourth edition of 1888 had 62 pages.

“Öfter veranlaßte […] das Modell nachträgliche Untersuchungen über Besonderheiten des dargestellten Gebildes.” (All of the translations by the author (M.F.) unless mentioned otherwise). Brill also points out (ibid.) that the making of a model of a surface with negative curvature (i.e., a model of an Enneper surface) has brought about the discovery of new forms of such surfaces, referring to what is what known today as the Kuen surface. According to Theodor Kuen (1884), the mathematician J. Mack modeled a surface with constant negative curvature. Kuen, then an assistant at the Mathematical Institute of the Technical University of Munich, assumed that due to the external shape of Mack’s model, this surface had a family of plane curvature lines and thus belonged to a family of surfaces examined by Enneper. After examining Mack’s models, Kuen could prove his assumption in 1881 with the help of the equations of the surface concerned. The surface investigated, today called Kuen surface, was thus a previously overlooked case of an Enneper area (Fischer 1986, pp. 40–41).

“[…] diese von den Synthetikern eine Zeit lang mit Vorliebe gepflegte Beschränkung auf das *Wort* konnte nicht ermangeln, zu einer Unterschätzung des *Bildes* und der technischen Fertigkeiten, die zu dessen Herstellung erforderlich sind, zu führen.”

See also (Mehrtens 2004). This understanding was not unique to Germany, but also could be found in England during the 1880s. Thus, for example one can find in the Encyclopaedia Britannica of 1883, under “Mathematical Drawing and Modeling,” the following explanation: “As a means of education, the model is lively and suggestive, forming in this way a completing factor in the course of instruction. […] The study of the model raises new and unexpected questions, and can even do valuable service in leading to new truths” (Encyclopaedia Britannica 1883, p. 628).

Exploratory material models were not a unique phenomenon to mathematics; they also appeared at the end of the nineteenth century in chemistry, with the folded paper models of Sachse and van ‘t Hoff, prompting discoveries is stereochemistry (see Friedman 2017); on the role of material models in chemistry during the second half of the nineteenth century (see, for example, Francouer 1997). For how these material models stood in connection to Berzelian chemical “paper tools” of the first half of the nineteenth century and to their iconicity, see Klein 2003, pp. 24–40.

Giacardi (2015) explains that this rejection is due to the mathematical tradition regarding geometry in Italy during the nineteenth and the twentieth century that consisted of several rather abstract mathematical approaches concerning geometry.

“Abbiamo tentato di trovare un substrato reale a quella dottrina, prima di ammettere per essa la necessità di un nuovo ordine di enti e di concetti.”

“J’ai eu […] une idée bizarre […] J’ai voulu tenter de *construire matériellement* la surface pseudosphérique, sur laquelle se réalisent les théorèmes de la géométrie non-euclidienne. […] on peut, par des morceaux de papier convenablement découpés, reproduire […] la surface véritable.”

Cf. Boi, Giacardi and Tazzioli 1998, p. 82, concerning a theorem that Beltrami proved only three years after its discovery using the model: Beltrami discovers that the “[t]he envelope of these [drawn] lines is the meridian of the pseudospheric surface.”

“Ce matin […] j’ai découpé un modèle en carton qui est assez bien réussi, et qui me servira pour un nouvel essai de construction d'une surface pseudosphérique. Vous parlez de propositions empiriques qu'on pourrait trouver par ce moyen, et vous avez parfaitement raison, car ici il s'agit de surfaces dont on ne possède pas les équations générales.”

“La superficie di rotazione seconda cui esso è ripiegato […], e il suo meridiano è una curva trascendente la cui equazione non può aversi in termini finiti.”

For a survey of the constructions of Beltrami’s models and the correspondence between him and Cremona (see Capelo and Ferrari 1982).

“Encore, j’ai tracé à la plume, sur la surface […]”

“Voici un résultat assez élégant que j’ai rencontré en m’occupant de cela: Que l’on trace une droite AB […]”

“Non sarà inutile il rammentare fin dal principio che quando si riguarda una superficie come definita dalla sola espressione del suo elemento lineare, bisogna prescindere da ogni concetto od imagine che implichi una concreta determinazione della sua forma in relazione ad oggetti esterni.”

See Beltrami 1868–1869, p. 427: “It should be observed […] that, while the concepts belonging to simple planimetry receive in this manner a true and proper interpretation, since they turn out to be *constructive* upon a *real* surface, those which embrace three dimensions will only admit an analytical representation, because the space in which such a representation would come to be concretized [*concretarsi*] is different from that to which we generally give that name.”

“Je vous avoue franchement que, lorsque le nombre des variables dans ces expressions est plus grand que 2, leur construction dépasse, en général, les bornes de l’expérience géométrique […].”

Regarding models as interpretations, see also Epple 2016.

“Auf dem Modell wurden nun geodätische Linien und geodätische Kreise in der Weise angetragen, dass man in der Bildebene Gerade und Kreise in passender Weise einzeichnete, dann für Fläche und Bild das […] Netz construirte und so eine punktweise Uebertragung der Abbildung ermöglichte.”

Following (Sattelmacher 2013), one can suggest that since the models became merchandised and an object to consume, they lost their exploratory character.

One of the aims of the plaster models was to be “handed over to the public” (“der Oeffentlichkeit übergeben”) (cf. Brill 1885, p. 4).

See Sattelmacher 2021, pp. 200–204 regarding the complexity of the process of manufacturing a model of plaster. It is clear that some of the stages involved in making a model from plaster did not involve any mathematical knowledge, but rather pure technical know-how.

Note that I do not mean that this as a sharp separation between the two domains: a technical object may also become an exploratory one or have epistemic implications, i.e., prompt new research results. However, in the above case study one may see the turning of an exploratory model into a technical object, which in turn stops functioning as exploratory.

“Die Geometrie auf den Flächen […] von constanter negativer Krümmung wird die Nicht-Euklidische Geometrie genannt und deckt sich mit der durch Lobatschewsky begründeten, welche des elften Axioms von Euklid entbehrt. […] Die nachfolgenden Flächenmodelle sind unter anderem dem Studium dieser Geometrie zu dienen bestimmt.”

(Bonola 1912, p. 132): For negative curvature, “we have the surfaces applicable to the Pseudosphere, which can be taken as a model for the surfaces of constant negative curvature.”

“It turned out, too, that the origins can be identified in the progressive context of Göttingen mathematics: in the strong focus on axiomatisation of mathematics, spurred by the set theory paradoxes, and in the interaction between mathematics and physics as intensified since the new focus there on relativity theory. Hermann Weyl’s book *Raum–Zeit–Materie* of 1918, elaborated as lecture notes of his course on relativity theory, evidences this breeding ground” (Schubring 2017, p. 275; see also Schubring 2017, pp. 270–274 for Weyl’s conception of “model”).

It might be possible that Weyl adopted this term from Hausdorff’s 1903 *Antrittsvorlesung*, where Hausdorff talks about the “Euclidean models” of non-Euclidean geometry (Hausdorff [1903] 1987, p. 85); see Epple 2021, chapter 4.3). Hence, contrary to Schubring’s claim, Weyl’s 1918 book was certainly not “[t]he first publication using ‘model’” (Schubring 2017, p. 270).

In this book, Weyl describes models of Euclidean and non-Euclidean geometries in the axiomatic-semantic sense as well as arithmetic models for Euclidean geometry, which were described by Hilbert (Weyl talks about the “construction of a model” (see Weyl [1927] 2009, pp. 37–40). John Von Neumann also used the term “model” in his paper “Eine Axiomatisierung der Mengenlehre” in 1925, when describing the “set theory models” (“Mengenlehrenmodelle”) (Neumann 1925, pp. 235–37). If both von Neumann and Weyl are taken into account here one can claim that mathematicians have begun to use the term “model” as a main concept in the discourse of axiomatics (see also Epple 2016, p. 24), regarding the re-conceptualization of Klein’s lecture “Nicht-euklidische Geometrie” in the year 1928. Paolo Mancosu notes that “[o]nce introduced in the axiomatical literature by Weyl, the word ‘model’ finds a favorable reception.” It occurs in Carnap’s papers from the late 1920s, and “in articles by Gödel, Zermelo and Tarski” from the 1930s (Mancuso 2006, p. 210n2). Alfred Tarski’s 1936 paper denotes an important moment when the term “model” was introduced in model theory (Tarski 1936, Vol. 7). In 1933, the traces of what was meant by the term “model” in relation to non-Euclidean geometry during Beltrami’s and Dyck’s period are few. In his 1933 lecture “Die vierte Dimension und der krumme Raum”, Georg Nöbeling comments: “Klein succeeded in specifying a system of things [System von Dingen] within the three-dimensional Euclidean space, which, if one terms these things ‘points’, ‘straight lines’ and ‘planes’, then all of the non-Euclidean axioms hold. In order to obtain this *model*, one considers the interior of a sphere of a three-dimensional Euclidean space […]” (Nöbeling 1933, pp. 77–8). In addition, Nöbeling points out that material or illustrative aspects of “point,” “line,” or “plane” are irrelevant and all that matters are the relations between the concepts (Nöbeling 1933, p. 79).

“Man nennt die Fläche, von denen wir solche Modelle konstruieren wollen, die *hyperbolische Ebene* und ihre Geometrie die *hyperbolische Geometrie*” (Hilbert and Cohn-Vossen 1932, p. 214). Hilbert and Cohn-Vossen present several photos of material models of surfaces in their book (see Hilbert and Cohn-Vossen 1932, pp. 17, 191, 193–4, 253, 267). Moreover, it is essential to recall that the book was based on an introductory course taught by Hilbert in 1920–21, hence it did not reflect the state of mathematical *research* on models during the 1930s.

See also Mehrtens’ account (2004, pp. 293, 301) on Klein’s changing approach to models “later […] interpret[ing] them as applied mathematics.”

This is not to imply that material-mathematical models were not used anymore after the 1930s in research contexts, though they were certainly marginalized. That these models were still used may be seen with the material models built by Alan Schoen during the late 1960s of triply periodic minimal surfaces, or with the Costa Surface, being a minimal surface of genus 1, discovered Celso Costa in 1982 and visualized by David Hoffman and William Meeks at the same decade; with their computer based visualization one could observe properties, which were only later proven. With the rise of these new methods of visualizations, one could say that the tradition of material models was revived, though the technological and medial means are different. However, a discussion on this revival is outside the scope of this paper.

“Avevamo costruito […] un gran numero di modelli di superficie […] e questi modelli avevamo distribuito […] in due vetrine. Una conteneva le superficie regolari per le quali tutto procedeva come nel migliore dei mondi possibili […]. Ma quando cercavamo di verificare queste proprietà sulle superficie dell'altra vetrina, le irregolari, cominciavano i guai e si presentavano eccezioni di ogni specie. Alla fine lo studio assiduo dei nostri modelli ci aveva condotto a divinare alcune proprietà che dovevano sussistere, con modificazioni opportune, per le superficie di ambedue le vetrine; mettevamo poi a cimento queste proprietà con la costruzione di nuovi modelli. Se resistevano alla prova, ne cercavamo, ultima fase, la giustificazione logica. Col detto procedimento, che assomiglia a quello tenuto nelle *scienze sperimentali*, siamo riusciti a stabilire alcuni caratteri distintivi tra le famiglie di superficie.”

I discuss Chisini’s work more thoroughly in Friedman 2020b.

A survey of the attempts to visualize Riemann surfaces or their branch points is outside the scope of this paper. However, see Friedman 2019a.

An example would be the curve *y*^{2} = *x* of degree 2; the point *x* = 0 is branch point, since it has only one preimage, being (0, 0). For this curve, any other point on the *x*-axis is not a branch point; for example, *x* = 3 has two preimages: (3, √3) and (3, −√3).

Given a complex number *w*, *Re*(*w*) denotes the real part of it and *Im*(*w*) the complex part.

Private communication with Piera Manara, email from 21 February 2018. I thank warmly Piera Manara for her help.

“Il modo del passaggio può essere reso evidente con un modello.”

“Invero la invenzione delle ‘trecce’ porta la sigla della sua creatività; ma questa era continuamente tenuta a freno dal suo spirito critico, che lo conduceva ad ostinarsi nella costruzione di modelli materiali, a non accontentarsi dei disegni e delle formule; chi l'ha incontrato in quegli anni ricorda che una delle frasi più frequentemente ripetute era ‘… non mi fido’. E questa diffidenza lo portava a voler costruire modelli tangibili e materiali, sui quali poter verificare la validità delle sue invenzioni.”

As examples Chisini gives the proof of Bezout’s theorem concerning the intersection of two complex plane curves or the computation of the class of a plane curve.

See also Friedman 2019b, pp. 54–57.

## References

*Algebra delle trecce caratteristiche*: Relazioni fondamentali e loro applicazioni

*modelli*matematici costruiti per l'insegnamento delle matematiche superiori pure ed applicate

## Author notes

The author acknowledges the support of the Cluster of Excellence »Matters of Activity. Image Space Material« funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2025 – 390648296.