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Soft cells and the geometry of seashells

Gábor Domokos, Alain Goriely, Ákos G. Horváth, Krisztina Regős

TL;DR

This work introduces soft tilings and soft cells as space-filling geometries with minimal sharp corners and highly curved boundaries, linking sharp polyhedral tilings to softer natural shapes. It develops a rigorous framework including the edge-bending algorithm and a Hamiltonian-circuit condition on dual vertex polyhedra to guarantee the existence of soft tilings combinatorially equivalent to given tilings, and extends these ideas to 3D via $z$-cells. The authors demonstrate that soft cells appear in nature, notably in chambered seashells and Nautilus chambers, and provide geometric models that reproduce these structures, connecting mathematics to biological growth. They also propose a softness measure based on rolling radius and show practical pathways to maximize softness, with implications for both natural patterns and architectural designs. Overall, the paper establishes a broad, constructive bridge between classical polyhedral tilings and smooth, curved tilings that better reflect natural space-filling forms.

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, non-flat 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 \textit{soft cells}, minimizing the number of sharp corners and filling space as \emph{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.

Soft cells and the geometry of seashells

TL;DR

This work introduces soft tilings and soft cells as space-filling geometries with minimal sharp corners and highly curved boundaries, linking sharp polyhedral tilings to softer natural shapes. It develops a rigorous framework including the edge-bending algorithm and a Hamiltonian-circuit condition on dual vertex polyhedra to guarantee the existence of soft tilings combinatorially equivalent to given tilings, and extends these ideas to 3D via -cells. The authors demonstrate that soft cells appear in nature, notably in chambered seashells and Nautilus chambers, and provide geometric models that reproduce these structures, connecting mathematics to biological growth. They also propose a softness measure based on rolling radius and show practical pathways to maximize softness, with implications for both natural patterns and architectural designs. Overall, the paper establishes a broad, constructive bridge between classical polyhedral tilings and smooth, curved tilings that better reflect natural space-filling forms.

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, non-flat 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 \textit{soft cells}, minimizing the number of sharp corners and filling space as \emph{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.
Paper Structure (18 sections, 1 theorem, 1 equation, 7 figures)

This paper contains 18 sections, 1 theorem, 1 equation, 7 figures.

Table of Contents

  1. Introduction
  2. From Plato to Plateau: cells with flat and slightly curved faces
  3. Large curvatures and the intuitive concept of soft tilings
  4. Soft tilings and soft cells
  5. Soft tilings in two dimensions
  6. Soft tilings in three dimensions
  7. The edge bending algorithm and the main result
  8. Soft, softer, softest
  9. Soft cells in nature
  10. The soft geometry of chambered seashells
  11. The chamber of the Nautilus
  12. Summary and outlook
  13. Open questions
  14. Materials and Methods
  15. Construction of $z$-cells
  16. ...and 3 more sections

Key Result

Theorem 1

Let $M$ be a balanced, normal convex tiling and let $\mathcal{V}(M)$ denote the set of the duals of vertex polyhedra in $M$. If every polyhedron $P \in \mathcal{V}(M)$ has a Hamiltonian circuit, then there exists a soft polyhedric tiling $M'$ which is combinatorially equivalent to $M$.

Figures (7)

  • Figure 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 scutoid_ephitalia_image. (b1) The Kelvin structure: a monohedric tiling. (b2) The Weaire-Phelan structure: a polyhedric tiling with 2 cells. Source Bitsche_2005daxner_2006. (b3) Tiling with scutoids scutoid_gomez.
  • Figure 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 in red color. We remark that the last two rows show soft tilings which are combinatorially equivalent to Dirichlet-Voronoi mosaics on point lattices ghorvath_dirichlet.
  • Figure 3: Soft tilings in the plane: examples from nature and architecture. Column (1): examples of monohedric, soft tilings with $v^{*}=2$ cells.(Remark that (a1),(c1) and (d1) are monohedric normal tilings, combinatorially equivalent to tilings with regular polygons, shown in Figure \ref{['fig:2']}, while (b1) is not a normal tiling.) Columns (2) and (3): examples in Nature where these patterns emerge. (a2) Betsiboka River estuary in northwestern Madagascar. (b2) Zebra stripes. (c2) Cross section of see shell. (d2) Geometric model of tip growth in algae algae_tipgrowth (a3) Smooth muscle tissue. (b3) Cross section of onion. (c3) Wheat awn (d3) meridian section of blood cell secomb1991red. Column (4):works of architect Zaha Hadid. (a4) Galaxy Soho, Beijing. (b4) Football stadium, Quatar. (c4) Heydar Aliev Center, Baku (d4) Design for condominium in Surfside, Florida (2023).
  • Figure 4: Genesis of soft 3D cells. First row, panels a,c,e,g: non-soft cells. Second row, panels b,d,f,h: soft cells, with softness value indicated below each cell. Insets in white circles show the dual of the vertex polyhedron with Hamiltonian circuit indicated in red. II. Individual panels: (a1) Half cylinder: non-soft, non-spacefilling cell. (b1) Cylinder. Maximal softness ($\sigma=1$) if $h=0$ (two-sided circular disc). (c1) Space-filling $z$-cell, resembling the Nautilus chamber. (c2-c4) Monohedral cells of prismatic tilings. (c2) Goldberg tetrahedron. (c3-c4): Dirichlet-Voronoi lattice $z$-cells. (c3) cube (c4) hexagonal prism. (d1-d4): Soft versions of (c1-c4) as monohedric soft $z$-cells. (e1-e3) Dirichlet-Voronoi lattice non-$z$ cells. (e1) Elongated dodecahedron. (e2): Truncated octahedron. (e3): Rombododecahedron.(f1-f3): Soft versions of (c1-c3) as monohedric, soft, non-$z$-cells. (g1) Non-spacefilling non-soft cell: Icosahedron. (h1) Non-spacefilling soft cell: sphere.
  • Figure 5: Soft cells in chambered shells. Upper row: Ammonite shell chambers (Cadoceras) (A1) All chambers, reconstructed from micro CT dataset Lemanis. (for details see Supplementary Information). (A2) Individual chamber with smooth upper and lower contours $b_1,b_2$ and base contour $b_0$. (A3) Soft, non-spacefilling $z$-cell as model of Cadoceras chamber. (A4) Seilacher's paper model for septa: cylinder with base $b_0$ intersected by two smooth surfaces along curves $b_1,b_2$. Seilacher_1975Bottom row: the chamber of the extant Spirula spirula shell. (B1) All chambers, reconstructed from micro CT dataset Lemanis. (B2) Individual chamber with smooth upper and lower contours $b_1,b_2$ and base contour $b_0$. (B3) Soft, non-spacefilling $z$-cell as model of the Spirula chamber. (B4) Seilacher's balloon model: inflated balloons glued along the curves $b_1,b_2$ to the wall of cylinder with base $b_0$. Seilacher_1975.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Conjecture 1