Self-similar shapes under Errera division rule on the cone

This study provides a general construction method of cell shape invariant by the Errera rule of division on a cone and provides analytical bounds for the apical angle of the cone on which these cells are connected and thus biologically meaningful. This idealized model highlights how the curvature of the tissue can influence Errera rule.

💡 Research Summary

The paper investigates how the curvature of a plant meristem, idealized as a cone, influences cell division governed by the Errera rule, which states that a cell divides into two equal‑area daughters while minimizing the length of the new wall. By modeling the meristem as a cone with planar angular span γ (related to the apical angle Θ through γ = 2π sin(Θ/2)), the author translates the classic planar “fencing problem” into a non‑Euclidean setting.

Key geometric objects are constant‑curvature arcs (CCAs) on the cone, each described by a radius R, a distance D from the cone apex, and a central angle φ. An N‑sided closed contour composed of N such arcs, called a CCAOC, is required to have orthogonal corners (adjacent arcs intersect at π/2). The division process is represented by a “halving solution” (HS), an arc that splits the area exactly in half and is either orthogonal to the contour or does not touch it.

The central result (Theorem 1) shows that a self‑similar N‑sided CCAOC can exist only if the radii and distances of all arcs share a common ratio ρ and if the indexing of arcs follows an arithmetic progression σ on the cyclic group ℤ/Nℤ with a step k that is coprime to N. Consequently, the scaling factor between mother and daughter cells must be √2, reflecting the equal‑area requirement. The number of distinct self‑similar configurations for a given N equals Euler’s totient function φ(N), i.e., the number of integers k (1 ≤ k < N) that are coprime to N.

Angle differences between consecutive arcs, denoted δφ_j, can take only two values. Let k′ be the modular inverse of k (k · k′ ≡ 1 (mod N)). Then N − k′ arcs have one angle δφ_{σ(0)} and k′ arcs have the other angle δφ_{σ(−1)}. These angles are given explicitly by arccos expressions involving ρ and powers of √2, with a sign ε that distinguishes internal from external tangency of neighboring arcs.

The relationship between the cone’s angular span γ and the geometric parameters is derived in three regimes: ρ = 0 (no real γ), ρ = 1 (γ expressed as a linear combination of π with coefficients depending on ε and k′), and 0 < ρ < 1 (γ given by more intricate trigonometric formulas). Because γ = 2π sin(Θ/2), the results translate directly into bounds on the apical angle Θ. Small Θ (sharp cones) severely restrict admissible N, whereas larger Θ (flatter cones) allow many more self‑similar configurations, reproducing the planar case when Θ → π.

The proof proceeds by constructing a homothety‑rotation h that maps the mother contour P onto its daughter P₁₂. The homothety factor √2 forces the radii and distances to obey R_{σ(j)} = (√2)^j R_{σ(0)} and D_{σ(j)} = (√2)^j D_{σ(0)}. By tracking how h permutes the arcs, the author shows that σ must be an arithmetic progression with step k, and that k must be invertible modulo N. Lemma‑based calculations then yield the cosine relations for δφ_j and the final expressions for γ.

Biologically, the model predicts that regions of high curvature (small Θ) in a meristem will display fewer regular division patterns, consistent with observed irregularities near growth tips or at junctions between meristem and primordia. Conversely, flatter regions behave like the classical planar case, supporting the prevalence of regular tessellations in those zones.

In conclusion, the paper provides a rigorous, purely geometric framework for understanding how conical curvature constrains Errera‑type cell division, offering explicit analytical bounds on the cone’s apical angle for the existence of self‑similar cell shapes. The work bridges a gap between idealized planar models and realistic curved tissues, and it lays a foundation for future experimental validation and extension to more complex three‑dimensional organ geometries.