What Is the Shape of a Cupola?

arXiv:2111.07920v1 [math.HO] 5 Nov 2021 WHAT IS THE SHAPE OF A CUPOLA? RAFAEL L´ OPEZ Abstract. This article examines the shape of a surface obtained by a hanging flexible, inelastic material with prescribed area and boundary curve. The shape of this surface, after being turned upside down, is a model for cupolas (or domes) under the simple hypothesis of compression. Investigating the rotational examples, we provide and illustrate a novel design for a roof which has the extraordinary property that its shape, although natural, is modeled by a surface of revolution whose axis of rotation is horizontal. 1. Introduction. Historically, the shape of a cupola (or dome) has been of enduring interest. The Greek’s use of columns and the Roman’s use of arches as a basic element in construction enabled architects to build ever larger walls and pillars, increasing the relevance of the cupola as the crowning element of the entire edifice. The use of flying buttresses to distribute loads and tensions in walls over a large area transformed the low windowless Romanesque churches into the tall, slender Gothic cathedrals that embellish the cities of Europe. The construction of cupolas involves an intricate interplay of artistic and structural issues requiring the architect to specify a variety of variables such as the choice of materials and the desired stylistic effect. The essential engineering problem to be solved is to build a large structurally stable, aesthetically appealing roof that rises over a large, empty space. In order to achieve this, architects certainly required the support of the sciences. In the 15th and 16th centuries, the Renaissance was a period of scientific and artistic development propitious for the building of domes. The examples of the Florentine Cathedral of Santa Maria del Fiore by Filippo Brunelleschi and the 2020 Mathematics Subject Classification. 53A10,49J35,00A67. 1 2 RAFAEL L ´ OPEZ Vatican’s Basilica Papale di San Pietro of Michelangelo (Figure 1) demonstrate the resulting triumphant achievements that fascinate us up to the present day. Figure 1. Domes of Santa Maria del Fiore in Florence (left) and the Basilica Papale di San Pietro in Vaticano (right). The first image is licensed under the Creative Commons Attribution-Share Alike 4.0 International license at commons.wikimedia.org/wiki/File:Florence duomo fc10.jpg. The second image is licensed under the public domain at commons.wikimedia.org/wiki/File:Petersdom von Engelsburg gesehen.jpg. Owing to the issues outlined above, it is not clear how one should go about formulating the problem of finding the optimal shape for a cupola. Here we take a mathematician’s perspective. A first thought that comes to mind is that the cupola is sustained along its boundary by its own weight. As a first approximation, we imagine a bounded, massive, homogenous piece of cloth whose boundary is represented by a fixed prescribed curve. Supported by this curve, the cloth evolves under the force of gravity to a static equilibrium. The reason the surface of the cloth is closely related to that of a cupola is that a surface suspended solely by its own weight experiences only tensional forces tangent to its interior. When this surface is inverted, it produces the optimal shape of a cupola. The inversion transforms the tensional force into a force of compression. In our context of cupolas, we can rephrase the words of Robert Hooke about the shape of an arch by saying that as hangs the flexible surface so inverted will stand the rigid WHAT IS THE SHAPE OF A CUPOLA? 3 cupola.1The inverted surface satisfies the same equation of equilibrium as the original surface so the only question to be answered is: What is the shape of a flexible hanging surface of uniform mass acted upon solely by gravity? As is often the case, some insight can be achieved by considering the one dimensional analog of the problem stated above, which is to determine the optimal shape of a hanging cable. The answer, as is well known, is a catenary curve given by the simple expression y(x) = a−1cosh(ax), for a positive constant a. The optimal shape of arches has also attracted the interest of mathematicians, where catenaries and parabolas have competed for this role; see, for example the beautiful discussion of R. Osserman on the shape of the Gateway Arch in Saint Louis, Missouri ([23]). The renowned Spanish architect Antonio Gaud´ı (1852-1926), who included many beautiful catenary shaped corridors, was an avid enthusiast of this shape (see Figure 2). The list of mathematicians who have investigated the shape of surfaces hanging under their own weight includes the names of Beltrami, Germain, Jellet, Lagrange and Poisson ([2, 11, 15, 16, 27, 31]). Surely, it was Cisa de Gresy who stated most lucidly [12, p. 260]: “Si on suppose, par example, une surface en ´equilibre, sollicit´ee uniquement par la gravit´e, et suspendue `a la circonf´erence d’un cercle fix´e horizontalement, it est clair que les ´el´ements de cette surface n’´eprouveront qu’une simple tension dans le sens des m´eridiens ou de la courbe g´en´eratrice.” [If we suppose, for example, that a surface is subjected only to the force of gravity and it is suspended from a circular perimeter, it is clear that the elements of this surface will only exert a simple tension in all directions of the meridians or the generating curve.] However, it is possible that there is not a minimum of the height of the center of gravity for surfaces with prescribed area and boundary curve. We present an example which is a slightly simplified version of the one given by Nitsche in [21]. 1Actually Hooke considered the problem of the hanging cable writing an anagram in Latin that deciphers to “ut pendet continuum flexile, sic stabit contiguum rigidum inversum” which translates as “as hangs the flexible line, so but inverted will stand the rigid arch” ([14, p. 31]). 4 RAFAEL L ´ OPEZ Figure 2. Left: Corridor in the Colegio Teresiano, Barcelona. Right: loft in La Pedrera, Barcelona. The first image is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license at commons.wikimedia.org/wiki/File:049 Col·legi de les Teresianes, arcs parab`olics.JPG. The second image is licensed under the GNU Free Documentation License at commons.wikimedia.org/wiki/File:LaPedreraParabola.jpg. Example 1. Let Γ be the circle in the plane z= 0 of radius 1 and centered at the origin. For 0 < R < 1, let ΩRbe the annulus {(x, y, 0) ∈R3:R2≤x2+y2≤1}. Consider the surface SRformed by ΩRtogether with the cone CRunderneath Ω with boundary CR={(x, y, 0) ∈R3:x2+y2=R2}and height h=p2 + 1/R2. We can parametrize SRin polar coordinates (r, θ) by u(r) = −hR−r R,0≤r < R 0, R ≤r≤1. See Figure 3. The boundary of SRis the circle Γ and with these choices of Rand h, the area of SRis constantly 2πindependently of R(the value 2πis only for convenience; any area greater than πcan be taken). It is well known that the center of gravity of the (hollow) cone of height his h/3 from the base. So the center of gravity of SRis at height −h 3·area(CR) area(SR)=−h 3πR√h2+R2 2π=−1 + R2 6r2 + 1 R2. WHAT IS THE SHAPE OF A CUPOLA? 5 In particular, the center of gravity can be made as low as one likes, by taking RsuffiCR z=0 h Γ ΩR Figure 3. The problem of minimizing the height of the center of gravity has no solution when the prescribed area is 2πand the boundary curve is the circle Γ. See Example 1. ciently small. This example can obviously also be made smooth by small modifications. This example contrasts with the one dimensional version of the problem because, as was proved by Jacob Bernouilli, the catenary has the property that its center of gravity is lower than that of any curve of equal length, and with the same fixed endpoints. As is usual in optimization problems, and in light of the above example, we approach the problem by requiring something less than an absolute minimum for the height of the center of gravity. Indeed, when a (flexible, inelastic) material hangs under its weight, the surface that is formed is a local extremum for the height of the center of gravity, in the space of smooth surfaces with given area and boundary. Therefore techniques from the Calculus of Variations are key for deriving the differential equation of the surface. In relation to this, Joseph-Louis Lagrange states in his M´ecanique Analytique: “[...] on verra par l’uniformit´e et la rapidit´e des solutions combien ces m´ethodes sont sup´erieures `a celles que l’on avait employ´ees jusqu’ici dans la Statique”. [[...] one will see by the consistency and speed of solvability, how these methods are greater than to those that have been employed until now in Statics.] See [16, p. 113]. 6 RAFAEL L ´ OPEZ But it was Sim´eon Denis Poisson who definitively found the equilibrium equation for the surface, improving the assumptions and calculations of Lagrange. What’s more, Joseph Bertrand, who edited the collected works of Lagrange, added a footnote: ‘Cette mani`ere d’´evaluer l’ensemble des forces que d´eveloppe l’´elasticit´e sur un point n’est pas suffisamment justifi´ee [...] Nous pouvons mˆeme ajouter que cela n’est pas exact. Poisson en a fait la remarque dans le M´emoires de l’Institut pour l’ann´ee 1812”. [This way of evaluating the collection of forces, which develop the elasticity at a point, is not sufficiently justified [...]. We may even add that it is not exact. Poisson made this observation in M´emoires de l’Institut in 1812]. See [16, p. 158]. Indeed, Poisson considered a much more general problem of a surface under different forces and tensions. As a particular case, he derived the correct equation of the surface stretched by its weight, which we will see in the next section. So, assuming only the effect of the weight, he asserts: “Consid´erons enfin la surface pesante, et prenons l’axe des zvertical et dirig´e dans le sens de la pesanteur”. [Let us finally consider the heavy surface. We take the vertical axis pointing along the direction of the gravitational field.] See [27, p. 185]. Then he successfully derived the equation for a nonparametric surface z=z(x, y) (see Figure 4), where k2= 1 + p2+q2,p=zx,q=zy,gis the gravitational acceleration and ǫis the density of the surface. Finally, he writes: Figure 4. The equation deduced by Poisson that satisfies a surface z= (x, y) acted upon solely by gravity. “Cette ´equation d’´equilibre de la surface pesante et ´egalement ´epaisse, doit comprendre l’´equation ordinaire de la chaˆınette, qui s’en d´eduit, en effect, en y supposant zind´ependante de l’une des deux variables xou y, WHAT IS THE SHAPE OF A CUPOLA? 7 de y, par exemple”. [This equilibrium equation of the heavy surface with uniform thickness must include the known equation of the suspended chain, which is deduced from it by assuming that zis independent of one of the two variables xor y, say of y]. See [27, p.186]. All the aforementioned works were apparently nearly forgotten until the 1980’s, when there was an explosion of interest in the evolution of surfaces by functions of their mean curvature. There is also the issue of the elasticity of the materials used in the construction. As the reader can well imagine, a dome’s actual material is not nearly so flexible as the cloth example discussed above. See a historical approach in [30]. Here, we would like to take note of the paper [7] by Ulrich Dierkes, which was surely motivated by the work of the German architect Frei Otto ([24]). Later, the problem was revisited by Bemelmans, B¨ohme, Dierkes, Hildebrandt, and Huisken in their works ([3, 4, 6, 7, 8]). The literature in architecture on the shape of cupolas is extensive and cannot be catalogued here. We refer only to [9, 13, 22, 26, 29]. Since we lack expertise in the fields of architecture and engineering, we have approached the problem from the perspective of differential geometry, although we have avoided its technical concepts, such as shape operator, principal curvatures, and second fundamental form, to maintain accessibility for a larger readership. The surfaces we discuss below will be graphs, surfaces of revolution, or cylindrical surfaces whose parameterizations are simple. In Section 2 we will employ the calculus of variations to derive the equation that a function z=u(x, y) must satisfy for its graph to define a surface whose shape is determined only by its own weight. These surfaces are called singular minimal surfaces. We will see that the boundary of the surface imposes geometric restrictions to the shape of the entire surface and this question will be briefly discussed. In Section 3 we focus on singular minimal surfaces that are surfaces of revolution, thinking of the shape of cupolas. Finally, in Section 4 we will present a new roof design modeled by a singular minimal surface. The novelty is that the roof is a surface of revolution but its rotation axis is horizontal, which is contrary to our common sense. 8 RAFAEL L ´ OPEZ 2. Singular minimal surfaces. Consider (x, y, z) the canonical coordinates of the three-dimensional Euclidean space R3where zindicates the vertical direction. Let Γ be a closed curve and A > 0 a fixed positive number. We wish to determine the differential equation that governs a surface Sspanning Γ with area Awhich is suspended from Γ by its weight. Suppose that Sis made of a flexible, incompressible material of uniform density σper unit area. In order to simplify the arguments, we restrict our attention to surfaces given by the graph of a smooth function z=u(x, y) defined on Ω, a bounded planar domain with smooth boundary ∂Ω. The weight per unit area of Sis σp1 + u2 x+u2 y, where the subscripts indicate the derivatives with respect to the corresponding variables. Under the effect of the weight, the surface Sattains a point of equilibrium when the height of its center of gravity is a local extremum. Assume that the gravitational potential at one point (x, y, z) is simply the distance zto the xy-plane. In particular, all our geometric objects (curves and surfaces) lie over the plane of equation z= 0. Let us also observe that the problem is invariant under translations in any horizontal direction. The height of the center of gravity is 1 AZΩ σ uq1 + u2 x+u2 ydxdy. The minimization is understood to be in the class of smooth functions uwith prescribed boundary u=ϕ > 0, where the graph of ϕ:∂Ω→Ris just the boundary curve Γ. We can assume that Aand σtake the value 1. We now consider simple arguments of calculus of variations and make an infinitesimal change in the surface z=u(x, y) given by u(x, y) + th(x, y), t∈Rand h: Ω →Ra smooth function vanishing on ∂Ω. Adopting a Lagrange multiplier for the constraint on the area of the surface, define the functional (1) J(u) = ZΩ up1 + |∇u|2dxdy +λZΩp1 + |∇u|2dxdy, where ∇u= (ux, uy) stands for the gradient of uand λ∈R. The domain of Jis the set of all smooth functions udefined on Ω with boundary condition u=ϕalong ∂Ω and fixed surface area equal to 1. The class Xof admissible variations is formed by the smooth functions h: Ω →Rwhich vanish on the boundary of Ω, h= 0 along ∂Ω. Thus an extremal uof Jimplies d dtt=0J(u+th) = 0 WHAT IS THE SHAPE OF A CUPOLA? 9 for all h∈ X. Set the Lagrangian L(x, y, u, p, q) = (u+λ)p1 + p2+q2, the integrand in (1), with p=uxand q=uy, and let F=∂L ∂p ,∂L ∂q = (u+λ) ux p1 + |∇u|2,uy p1 + |∇u|2!. Using div(h·F) = h∇h, Fi+h·divF, where h·,·i denotes the usual scalar product, we have d dtt=0J(u+th) = ZΩ∂L ∂u h+hx ∂L ∂p +hy ∂L ∂q dxdy =ZΩ∂L ∂u h+h∇h, Fidxdy =ZΩ h·∂L ∂u −divFdxdy +ZΩ div(h·F)dxdy. The Divergence Theorem allows us to rewrite the last integral as an integral over the boundary ∂Ω. So, using h= 0 on ∂Ω, we have ZΩ div(h·F)dxdy =Z∂Ω h·hF, ni= 0, where nis the unit outward-pointing normal of ∂Ω. As a consequence of the Fundamental Lemma of the calculus of variations, uis an extremal if and only if ∂L ∂u −divF=∂L ∂u −∂L ∂p x−∂L ∂q y = 0. By the definition of L, this identity can be expressed as p1 + |∇u|2− (u+λ)ux p1 + |∇u|2!x− (u+λ)uy p1 + |∇u|2!y = 0. Rewriting this identity, we conclude that an extremal uof the variational problem satisfies the Euler-Lagrange equation ux p1 + |∇u|2!x + uy p1 + |∇u|2!y =1 (u+λ)p1 + |∇u|2, 16 RAFAEL L ´ OPEZ by the direction ~v = (a1, a2, a3) with |~v|2= 1. All points p= (x, y, u(x, y)) ∈Sthat lie at the same horizontal plane, are circles centered at the z-axis of radius r=px2+y2. Thus uis a radial function u=u(r). Let us parametrize Sby introducing polar coordinates x=rcos θ,y=rsin θ, (7) X(r, θ) = (rcos θ, r sin θ, u(r)). We express (4) in terms of the derivatives of uwith respect to r. A change of variables transforms the left-hand side of (2) (equivalent to the mean curvature Hin (4)) into (8) u′(1 + u′2) + ru′′ r(1 + u′2)3/2. We now compute the right-hand side in (4). The unit normal vector field of Sis N=Xr×Xθ |Xr×Xθ|=1 √1 + u′2(−u′cos θ, −usin θ, 1). Since hX,~vi=a1rcos θ+a2rsin θ+a3u, Equation (4) is u′ r+u′′ 1 + u′2=−a1u′cos θ−a2u′sin θ+a3 a1rcos θ+a2rsin θ+a3u. After some manipulations, this equation can be written as A(r) cos θ+B(r) sin θ+ C(r) = 0, where A(r) = a1ru′(1 + u′2) + ru′′ +u′(1 + u′2) B(r) = a2ru′(1 + u′2) + ru′′ +u′(1 + u′2) C(r) = a3u(u′(1 + u′2) + ru′′)−r(1 + u′2). Since the functions {1,cos θ, sin θ}are linearly independent, the functions A,Band Cmust vanish in their domain. One case is that a1=a2= 0. Then the direction of gravity is parallel to the rotation axis and, in addition, C= 0 becomes (9) u′′ 1 + u′2=1 u−u′ r. Suppose now that a16= 0 (resp. a26= 0). Then we deduce from A= 0 (resp. B= 0), (10) u′′ 1 + u′2=−2u′ r. For the equation C(r) = 0, we distinguish two subcases. If a3= 0, then ~v is orthogonal to the axis of rotation. If a36= 0, then u(u′(1+u′2)+ru′′)−r(1+u′2) = 0 and combining with (10), we deduce uu′=−r. Thus u(r) = √r2+c, for some constant c. However, WHAT IS THE SHAPE OF A CUPOLA? 17 this function does not satisfy (10). This establishes the following theorem which is now written when the direction of the gravity is given, as usual, by the vertical axis ([17]). Theorem 2. If a surface of revolution is a singular minimal surface, then the axis of rotation is vertical or the axis of rotation is contained in the plane z= 0. The second case is striking because we have discovered a model of a rotational cupola whose rotation axis is horizontal! We separate the two cases and, in this section, we focus on the case where the rotation axis is parallel to the force of gravity. Here ~v in the proof of the theorem is actually the vertical direction of R3and usatisfies (9). From the standard theory of ordinary differential equations, the solution of the ordinary differential equation (9) is obtained once we give initial conditions (11) u(r0) = u0, u′(r0) = ¯u0, r0>0, u0>0. Let us observe that (9) is singular at r= 0, so r0must be positive. However, keeping in mind the shape of cupolas, our interest is that a solution meets the rotation axis. So we want to know if the solution ucan be prolongated until r= 0. This question is problematic. It is possible that under some initial conditions in (11), the solution does not meet the z-axis (see the example in Remark 1 below). In such a case, after rotating the graphic of u=u(r) about the z-axis, we would obtain a cupola with a “hole” at the top. However we are interested in those solutions whose initial conditions in (11) occur at r0= 0. An argument using the Banach Fixed Point Theorem proves the existence of a solution with u(0) = u0>0 ([17]). In such a case, the intersection of the surface with the rotational axis must be orthogonal by smoothness of the surface. This is equivalent to u′(0) = 0. Rotational singular minimal surfaces whose axis is vertical have been studied in the literature: see, for example, [6, 7]. Recall that in Section 2 we showed the singular solution u(r) = r, a cone with vertex at the origin. In this case, after inverting the surface, the shape of the cupola looks like a Native American teepee. In Figure 7, left, we show, using Mathematica, some numerical solutions of (9)-(11) when r0= 0, u′(0) = 0, and for different values of u0. All these curves will give shapes of domes once we invert them as Figure 7, right, shows. Remark 1. If r0>0 in (11), then the standard theory of ODE’s ensures the existence and uniqueness of solutions. In such a case, the maximal domain of the solution around r0may not reach the value 0, that is, the solution may not meet the rotation axis. This happens when we choose u′(r0) = 0 for a fix value r0>0. Indeed, if the domain of u 18 RAFAEL L ´ OPEZ Figure 7. Left: numerical solutions of (9)-(11), where u′(0) = 0 for different values of u(0): 0.5, 1 and 1.5. Right: the solutions viewed as cupolas after a symmetry about the horizontal line z= 2. Figure 8. Left: a solution of (9)-(11), where u(2) = 1 and u′(2) = 0. Right: the corresponding rotational surface. contains the value 0, we know that u′(0) = 0. From (9), u′′(r0) = 1/u(r0)>0, so r=r0 is a strict local minimum. We now see that r= 0 is another strict local minimum. By L’Hˆopital’s rule, letting r→0 in (9), we get u′′(0) = 1 u(0) −lim r→0 u′(r) r=1 u(0) −lim r→0 u′′(r) 1=1 u(0) −u′′(0). Hence u′′(0) = 1/(2u(0)) >0. Thus the function urestricted to the interval [0, r0] must have a local maximum at some point c∈(0, r0), which must also be a local maximum of u=u(x, y). This contradicts property (2) of Section 2. In Figure 8 we show an example of a rotational surface that does not meet the z-axis. WHAT IS THE SHAPE OF A CUPOLA? 19 4. A new design for a roof. In this section we present a new design for a cupola using a surface of revolution but, contrary to common sense, the rotational axis will be horizontal. In this case, we feel it is better to refer to the surface as a ‘roof’ rather than a cupola. Thus, we turn our attention to the singular minimal surfaces given in Theorem 2 whose rotation axis is included in the plane z= 0. First, we need to change the coordinates in the proof of Theorem 2 because here we assumed that the rotation axis was the z-axis and the direction ~v was (a1,0,0) or (0, a2,0). Without loss of generality, we suppose ~v = (1,0,0) and consider the positively oriented rigid motion M:R3→R3determined by the transformation M:   (1,0,0) 7→ (0,0,1) (0,1,0) 7→ (0,−1,0) (0,0,1) 7→ (1,0,0). The surface of revolution in (7) changes to M◦X(r, θ) = (u, −rsin θ, r cos θ). On the other hand, we know that usatisfies (10). We make a new change of variables interchanging the roles of uand r. Then the parametrization of the surface is (12) X(r, θ) = (r, −u(r) sin θ, u(r) cos θ), and (10) is now (13) u′′ 1 + u′2=2 u. Notice that this equation looks like the equation (3) of the catenary with the only difference being that now the numerator in the right-hand side is 2. For this reason, we call the solutions of (13) 2-catenaries. For non-constant solutions, we multiply by u′and integrating, we find (14) u′=±√c2u4−1, c 6= 0. In particular, differentiating with respect to r, and using (14) (15) u′′(r) = ±2c2u3u′ √c2u4−1= 2c2u(r)3. The differential equation (14) is known in the literature as an Emden-Fowler type equation ([28]). The generating curve u=u(x) is contained in the xz-plane after we replace the variable rwith x. The properties of uare the following: 20 RAFAEL L ´ OPEZ (1) The function uhas only one critical point. Without loss of generality, we can assume that this point is x= 0. At x= 0, uhas a global minimum. The value of uin x= 0 is 1/√cby (14). (2) The function uis symmetric about the z-line. The maximal domain of uis a bounded interval (−a, a) and limx→±au(x) = ∞. (3) The function uis convex thanks to (15). If we were to build the roof rotating the curve u=u(x) around the x-axis, the projection of the roof would be included in the horizontal strip Ω = {(x, y, 0) ∈R3:−a≤ x≤a}and its walls, or its skeleton structure, would be near vertical at far away points. We plot numerical solutions of (13) using Mathematica. For this, we consider initial conditions u(0) = 1, u′(0) = 0. The maximal domain (−a, a) occurs for the value a≈1.31102. The surface is tangent to the vertical planes of equations x=−aand x=a. When we rotate about the x-axis, the lower half of the surface is located in the halfspace z < 0 which cannot be considered. In Figure 9, left, we plot the generating curve (thick) and the corresponding rotations of this curve for angles θ∈(−π/2, π/2) (thin). In Figure 9, middle, we invert with respect to the horizontal plane having equation z= 3 and on the right, we show the roof modeled by the surface. Both vertical walls are supporting the roof. Notice that all points of the surface are saddle points. Figure 9. Left: the 2-catenary (bold) and its rotations about the xaxis. Middle: the same curves after inverting. Right: the rotational cupola. WHAT IS THE SHAPE OF A CUPOLA? 21 In the last surface in Figure 9, the roof does not cover the entire corridor (the strip Ω). We have reduced the size of the roof along the y-axis as Figure 10 shows. Figure 10. Rotational cupolas by cutting along vertical planes parallel to the xz-plane. The values for yare varying in the interval [−3,3] (left), [−2,2] (middle) and [−1,1] (right). 5. Outlook and Conclusions. Motivated by the shape of a catenary, we have deduced the differential equation governing surfaces suspended by their own weight and discussed some of their properties. Singular minimal surfaces can be models for cupolas, at least under the simple hypothesis of compression. Hence, light structures can be constructed in architecture imitating the shape of these surfaces. In reality, these surfaces may be difficult to produce on a large scale. However, it is remarkable that there are singular minimal surfaces that are surfaces of revolution about a horizontal axis. Thanks to these surfaces, we have presented a novel structural shape of a roof in Figures 9 and 10 which can be regarded in the context of the so-called “funicular shape” in architecture. The two families of parametric curves in this surface show the visual design of a skeleton that opens up towards its border, increasing its beauty. And, as has been justified, its shape is ‘natural’ in the sense that loads and tensions act tangentially on the roof, giving solidity and stability to the construction. Gaud´ı used principles from the natural sciences in his architecture, generating interest in the design of structures by observing the effect of weight. The idea to create architectonic structures inspired by natural shapes is now expanding ([5, 13]), and the 22 RAFAEL L ´ OPEZ designs of singular minimal surfaces give stability in these constructions. Finally, in the future, it would be desirable to investigate the implementation of methods of discrete differential geometry which can produce this type of roof model in practice. Acknowledgment. The author wishes to thank the anonymous referees for a careful reading of the manuscript, providing many useful remarks and corrections. These suggestions greatly helped to improve the final version. In particular, one of the referees pointed out the reference [21] of Nitsche for the Example 1. The author also thanks to Bennet Palmer, Alvaro P´ampano and Anthony Gruber who revised the initial draft. This work has been partially supported by the grant no. PID2020-117868GB-I00 Ministerio de Ciencia e Innovaci´on. References [1] Alexandrov, A. D. (1962). 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Math. 142: 1771– 1795. doi.org/10.1353/ajm.2020.0044 [21] Nitsche, J. C. C. (1986). A nonexistence theorem for the two-dimensional analogue of the catenary. Analysis 6: 143–156. doi.org/10.1524/anly.1986.6.23.143 [22] Oppenheim, I. J., Gunaratnam, D. J., Allen, R. H. (1989). Limit state analysis of masonry domes. J. Structural Eng. 115: 868–882. doi.org/10.1061/(ASCE)0733-9445(1989)115:4(868) [23] Osserman, R. (2010). Mathematics of the gateway arch. Notices AMS. 57: 220–229. [24] Otto, F. (1962/1966). Zugbeanspruchte Konstruktionen. Bd. I, II. Berlin, Frankfurt, Wien: Ullstein. [25] Frei Otto: Spanning the Future. (2020). Documentary film. Dir. Joshua Hassel. https://www.youtube.com/watch?v=P5hKnOyg43k [26] Paradiso, M., Rapallini, M., Tempesta, G. (2003). Masonry domes. Comparison between some solutions under no-tension hypothesis. Proceedings of the First International Congress on Construction History, Madrid: Instituto Juan de Herrera, Escuela T´ecnica Superior de Arquitectura. pp. 1571-1581. [27] Poisson, S. D. (1812). M´emoire sur les surfaces ´elastiques. M´emoires de l’Institut de France, 1814/1816, Vol. 9: 167–226. [28] Polyanin, A. D., Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations. Boca Raton: Chapman & Hall/CRC. [29] Pottmann, H., Asperl, A., Hofer, M., Kilian, A. (2007). Architectural Geometry. Exton: Bentley Institute Press. [30] Todhunter, I., Pearson, K. (1986). A History of Elasticity and Strength of Materials. Cambridge: Cambridge Univ. Press, Vol. 1. [31] Volterra, V. (1884/1885). Sulla deformazione delle superficie flessibili ed inestensibili. Atti della R. Accad. Dei Lincei, Rendiconti, Series 4, Vol.1: 274–278. Rafael L´opez works in classical differential geometry, in particular, surfaces with prescribed mean curvature. He is a Professor of Mathematics at the University of Granada where he received his Ph.D. in 1996. Rafael enjoys performing mathematics outreach 24 RAFAEL L ´ OPEZ activities in schools using soap bubbles and, in his spare time, he likes trekking and cycling in Sierra Nevada. Departamento de Geometr´ ıa y Topolog´ ıa, Universidad de Granada. Granada, Spain Email address:[email protected]