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Contributions * Preprint * Data Availability * References * < Previous * Next > Article Navigation Article Navigation Journal Article Soft cells and the geometry of seashells Gabor Domokos, Gabor Domokos Department of Morphology and Geometric Modeling, Budapest University of Technology and Economics , Budapest, 1111 , Hungary HUN-REN-BME Morphodynamics Research Group, Budapest University of Technology and Economics , Budapest, 1111 , Hungary To whom correspondence should be addressed: Email: domokos@iit.bme.hu ORCID logo https://orcid.org/0000-0002-8676-6829 Search for other works by this author on: Oxford Academic Google Scholar Alain Goriely, Alain Goriely Mathematical Institute, University of Oxford , Oxford, OX2 6GG , United Kingdom ORCID logo https://orcid.org/0000-0002-6436-8483 Search for other works by this author on: Oxford Academic Google Scholar Akos G Horvath, Akos G Horvath HUN-REN-BME Morphodynamics Research Group, Budapest University of Technology and Economics , Budapest, 1111 , Hungary Department of Algebra and Geometry, Budapest University of Technology and Economics , Budapest,1111 , Hungary ORCID logo https://orcid.org/0000-0003-2371-4818 Search for other works by this author on: Oxford Academic Google Scholar Krisztina Regos Krisztina Regos Department of Morphology and Geometric Modeling, Budapest University of Technology and Economics , Budapest, 1111 , Hungary HUN-REN-BME Morphodynamics Research Group, Budapest University of Technology and Economics , Budapest, 1111 , Hungary ORCID logo https://orcid.org/0000-0001-6866-2658 Search for other works by this author on: Oxford Academic Google Scholar Competing Interest: The authors declare no competing interest. Author Notes PNAS Nexus, Volume 3, Issue 9, September 2024, pgae311, https:// doi.org/10.1093/pnasnexus/pgae311 Published: 10 September 2024 Article history Received: 18 April 2024 Accepted: 09 July 2024 Corrected and typeset: 10 September 2024 Published: 10 September 2024 * pdfPDF * Split View * Views + Article contents + Figures & tables + Video + Audio + Supplementary Data * Cite Cite Gabor Domokos, Alain Goriely, Akos G Horvath, Krisztina Regos, Soft cells and the geometry of seashells, PNAS Nexus, Volume 3, Issue 9, September 2024, pgae311, https://doi.org/10.1093/ pnasnexus/pgae311 Select Format [Select format ] Download citation Close * Permissions Icon Permissions * Share Icon Share + Facebook + Twitter + LinkedIn + Email Navbar Search Filter [PNAS Nexus ] Mobile Enter search term [ ] Search Close Navbar Search Filter [PNAS Nexus ] Enter search term [ ] Search Advanced Search Search Menu Abstract A central problem of geometry is the tiling of space with simple structures. The classical solutions, such as triangles, squares, and hexagons in the plane and cubes and other polyhedra in three-dimensional space are built with sharp corners and flat faces. However, many tilings in Nature are characterized by shapes with curved edges, nonflat faces, and few, if any, sharp corners. An important question is then to relate prototypical sharp tilings to softer natural shapes. Here, we solve this problem by introducing a new class of shapes, the soft cells, minimizing the number of sharp corners and filling space as soft tilings. We prove that an infinite class of polyhedral tilings can be smoothly deformed into soft tilings and we construct the soft versions of all Dirichlet-Voronoi cells associated with point lattices in two and three dimensions. Remarkably, these ideal soft shapes, born out of geometry, are found abundantly in nature, from cells to shells. tessellation, Nautilus shell, Dirichlet-Voronoi cell, tip growth Significance Statement Polygonal and polyhedral tessellations, consisting of cells with flat faces and sharp corners are successful models in geology, physics, and chemistry, describing phenomena ranging from crack networks to convection cells, foams and supramolecular patterns. However, these models fail to address the diversity of highly curved geometric shapes in biology. Here, we introduce a new class of tessellations, called soft tilings where cells have highly curved faces and the number of sharp corners is minimal, imitating constraints in biological growth. We prove a theorem demonstrating that soft tilings are abundant in the combinatorial sense and we demonstrate that these geometric shapes are strikingly reflected in natural examples, ranging from biological cells to the chambers of seashells, including the Nautilus. Introduction The quest to find tilings, i.e. space-filling patterns consisting of nonoverlapping, finite domains, started more than 10,000 years ago with the advent of masonry walls. However, tilings are much older than that: they are an integral part of Nature. Here, we describe a new class of space-filling patterns called soft tilings with highly curved cells which minimize the number of sharp corners. To motivate this concept, we first briefly review simpler tilings. From Plato to Plateau: cells with flat and slightly curved faces The first geometric theory of tilings dates back to Plato (1) who claimed that the five regular polyhedra, the Platonic solids, fill space without gaps, forming the four fundamental substances: earth, air, fire, and water, while the fifth solid (the dodecahedron) is the building block of the cosmos. Plato's views were soon found to be flawed. Indeed, Aristotle claimed that only the cube and the tetrahedron can fill space without gaps. The latter statement is incorrect for the regular tetrahedron. However, there exist space-filling tetrahedral honeycombs (1) to which we will return later. Plato's idea was resurrected in the study of non-Euclidean honeycombs (2) where all Platonic solids fill space. Plato's idea and Aristotle's refinement lead, ultimately, to the fundamental concept of the solid angle which proved to be an essential tool in describing the combinatorial properties of convex tilings (3, 4) which are also polyhedral tilings, filling space by convex polyhedra, without gaps and overlaps. Polyhedral tilings provide good models for phenomena ranging from fragmentation processes (5) and the emergence of aeolian ridges on Mars (6) to supramolecular patterns in monolayers (7). However, in many cases the geometry of the natural tessellation appears to be more complex than a polyhedral tiling. Yet, these more intricate structures can still often be related to polyhedral tilings via combinatorial equivalence, which is a central concept in our work. If we imagine a tiling being constructed from rubber then, under elastic deformations, the shape of faces and edges can be largely distorted. However, if we require that material parts neither break apart nor are glued together in this process, then adjacent faces, adjacent edges, and adjacent nodes will remain adjacent, disjoint faces, disjoint edges, and disjoint nodes will remain disjoint. In this case, we call the original and the distorted tiling combinatorially equivalent (for a formal and more general definition, see Ref. (8)). Combinatorial equivalence is a fundamental tool in relating tilings with apparently different geometric features and finding such connections is one of the main goals of this study. In particular, in a tiling which is combinatorially equivalent to a polyhedral tiling, faces may not be planar and edges may not be straight. These properties are well reflected in the geometry of foams, controlled by Plateau's Laws: the Kelvin structure (9) and its improved version, the Weaire-Phelan structure (10), shown in Fig. 1. The small deviation in curvature observed in foams is the result of physical constraints, not the space-filling constraint. This idea was taken one step further in the description of the closely packed cell system of ephitelia tissue (11). This geometric model, shown in Fig. 1b3, also has cells with curved faces and edges. While these cells are still individually combinatorially equivalent to polyhedra, the tiling as a whole is not combinatorially equivalent to any polyhedral tiling, showing that in this case curvature is a result of the constraint to fill space. Examples for slightly curved polyhedric tilings. Upper row: natural examples. Lower row: geometric models. One slightly curved face highlighted on each tiling. (a1) Metal foam (source: Wikimedia Commons) (a2) Liquid foam (source: Wikimedia Commons). (a3) Ephitalia tissue (11). (b1) The Kelvin structure: a monohedric tiling. (b2) The Weaire-Phelan structure: a polyhedric tiling with two cells. Source (12, 13). (b3) Tiling with scutoids (14). Fig. 1. Examples for slightly curved polyhedric tilings. Upper row: natural examples. Lower row: geometric models. One slightly curved face highlighted on each tiling. (a1) Metal foam (source: Wikimedia Commons) (a2) Liquid foam (source: Wikimedia Commons). (a3) Ephitalia tissue (11). (b1) The Kelvin structure: a monohedric tiling. (b2) The Weaire-Phelan structure: a polyhedric tiling with two cells. Source ( 12, 13). (b3) Tiling with scutoids (14). Open in new tabDownload slide Despite having different reasons for being curved, all three aforementioned structures only differ slightly from polyhedral tilings in the sense that edges and faces meet transversely (angles both between faces and between edges are nonzero) and curvature radii along edges are very large compared to the size of the cell. This small deviation from polyhedral tilings may be the reason why such tilings are accepted as suitable models. Strikingly, in the theory of tilings no new category has been created to capture phenomena where the curvature of cells and edges may be large enough to play a central role. Large curvatures and the intuitive concept of soft tilings If we keep the combinatorial structure of faces and vertices but we allow tangencies (zero angles) between them and we also allow faces and edges where curvature radii are comparable to the cell size, we enter an entirely new domain where radically new geometric features emerge. In particular, space-filling cells with tangencies and/or large curvatures may have fewer corners than polyhedral simplices and such cells (which, depending on the number of corners we will call either softened or soft) do emerge in Nature. Figure 2 illustrates 2D examples of cells with curved boundaries which have only two corners and 3D examples will be discussed in the next section. Soft tilings in the plane. We show soft monohedric tilings which are combinatorially equivalent to monohedral tilings with regular polygons. Each row shows combinatorially equivalent soft tilings, corresponding to regular triangulation (first row), the rectangular grid (second row), and the hexagonal honeycomb (third row). If, beyond combinatorial equivalence classes, we also distinguish between sharp and soft corners then we arrive at the 12 tilings shown in the figure. In each tiling one soft cell is highlighted by solid filling. We remark that the last two rows show soft tilings which are combinatorially equivalent to Dirichlet-Voronoi mosaics on point lattices (16). Fig. 2. Soft tilings in the plane. We show soft monohedric tilings which are combinatorially equivalent to monohedral tilings with regular polygons. Each row shows combinatorially equivalent soft tilings, corresponding to regular triangulation (first row), the rectangular grid (second row), and the hexagonal honeycomb (third row). If, beyond combinatorial equivalence classes, we also distinguish between sharp and soft corners then we arrive at the 12 tilings shown in the figure. In each tiling one soft cell is highlighted by solid filling. We remark that the last two rows show soft tilings which are combinatorially equivalent to Dirichlet-Voronoi mosaics on point lattices (16). Open in new tabDownload slide To create a geometric framework for this generalization of polyhedral tilings, we introduce the concept of polyhedric tilings which includes, beyond polyhedral tilings, tilings with cells having zero internal angles, cells having strongly curved faces and/or edges; a formal definition is given in Section S1A. We do not consider here pathological tilings by assuming for the rest of the article that both polyhedral and polyhedric tilings are normal and balanced, i.e. the cell diameter has uniform upper and lower bounds and the averages of cell and nodal degrees exist (4). If a tiling consists of identical cells, we refer to the cells and to the tiling as monomorphic. In particular, if those cells are polyhedra then the tiling is monohedral and its curved generalization is called monohedric. There are infinitely many monohedral and monohedric tilings. Examples of monohedral tilings include the cubic grid and tilings with tetrahedral cells (1). An example of a monohedric tiling is the Kelvin structure of Fig. 1b1. While the combinatorial properties of tilings have been investigated in detail, less attention was paid to the smoothness of the cells. In 2D, shapes with at least C1-smoothness do not fill space (15). Since the planar sections of 3D tilings are 2D tilings and the generic planar sections of C1-smooth 3D objects are C1-smooth 2D objects, this property of 2D tilings also implies that a 3D tiling cannot consist entirely of C1-smooth tiles. Since piecewise smooth shapes, such as squares and cubes, do fill space, and smooth shapes do not, a natural question is how smooth space-filling shapes can be. The answer to this question depends on how we measure smoothness. Using the concept of polyhedric tilings, we can assign a smoothness-related measure to monohedric cells and try to minimize this measure among possible space-filling shapes. Since a monohedric cell cannot be smooth, there exists a set of nonsmooth points on its boundary. Just as the smoothness of a point has various degrees (by counting the number of existing derivatives), we can define levels of nonsmoothness by counting the codimension of smooth manifolds containing the point. The basic idea is to reduce the number of points with highest level of nonsmoothness. More precisely, we call a boundary point p of a cell a corner of the cell if no smooth curve on the boundary contains p. We denote the number of corners of a cell by v[?] and we define vmin[?] as the minimal number of corners that a cell of a monohedric tiling may have (15). Both the d=2 dimensional Euclidean plane and the d=3 dimensional Euclidean space have monohedral simplicial (triangular and tetrahedral, respectively) tilings (19) with v[?]=d+1. Therefore, for d=2,3 we have vmin[?]<=d+1 and we will show that, in fact, this inequality is strict. The quantity vmin[?] was defined using the concept of monohedric tilings, however, we can also apply the very same quantity to classify general polyhedric tilings and cells. In d dimensions, we will call cells with v[?]<=vmin[?] corners soft cells and cells with vmin[?]