Seed-Geometry: Multidisciplinary Insights

• Seed-Geometry is an interdisciplinary framework that models seed growth and arrangement using differential geometry, discrete packing, and stochastic methods.
• It applies precise mathematical constructs—like Ricci flow, Voronoi analysis, and metric evolution—to reveal functional and genetic impacts of seed morphology.
• The approach integrates automated theorem proving and computational techniques for 3D reconstruction, advancing research in biology, physics, and computer science.

Seed-Geometry encompasses a spectrum of formal, computational, mathematical, and biological frameworks for representing, analyzing, and generating geometric structures associated with seeds and seed-like entities. Approaches under this umbrella address biological growth and morphogenesis, packing and phyllotaxis, combinatorial automata, probabilistic genealogies, computational geometry engines for theorem proving, and more. The topic is highly interdisciplinary, unifying Riemannian flows, metric geometry, pattern formation, stochastic modeling, and automated algebraic reasoning.

1. Differential-Geometric and Biological Foundations

A central model for seed growth interprets an expanding plant organ (e.g., a seed, leaf, or petal) as a time-dependent Riemannian manifold with a metric tensor that evolves under the influence of material transport and local curvature (Pulwicki et al., 2010). For an $n$-dimensional domain, the squared distance element is

$ds^2 = g_{ik} dx^i dx^k$

with $g_{ik}$ a time-dependent metric. Growth dynamics are controlled by

• Evolution of the metric tensor, governed by Ricci flow, nonlinear stress, and strain-rate terms:

$$\partial_t g_{ik} = \kappa R_{ik} + \kappa_1 v_i v_k + \kappa_2 (\nabla_i v_k + \nabla_k v_i)$$

• Velocity field of material transport coupled to the geometry via Christoffel symbols.
• Material density, obeying continuity:

$\partial_t \rho = S(x, t) - \nabla_i (\rho v^i)$

In 1D—relevant for seed elongation—the metric reduces to a scale factor $f(x,t)$; intrinsic curvature vanishes, and a pseudo-curvature term replaces the Ricci tensor. The model connects with the Kardar–Parisi–Zhang and Burgers’ equations, capturing sigmoidal seed growth curves, spatial localization (“elongation zones”), and dynamic slowing of flow as maturation approaches.

This bidirectional coupling—geometry regulating flow, and flow retroacting on geometry—provides a quantitative theory for how intrinsic geometry may dictate and record developmental programs within seeds.

2. Discrete Geometry, Phyllotaxis, and Packing

Seed geometry is also analyzed in terms of discrete arrangements and their efficiency of packing. In the canonical sunflower spiral, seeds are modeled in the complex plane as

$$\varphi_\theta(n) = \sqrt{n} \, e^{2\pi i n \theta}$$

where the divergence angle $2\pi\theta$ (golden ratio for optimal spacing) generates near-uniform distribution and minimal overlap (Bacher, 2013).

Local neighborhoods linearized around a seed correspond to planar lattices, which have a moduli space identification with points on the modular curve $\mathrm{PSL}_2(\mathbb{Z})\backslash \mathbb{H}$. The geodesic

$$\gamma_\theta(t) = \frac{4i\pi t}{1 + 4i\pi\{\theta\} t}$$

tracks the “type” of local lattice as one moves radially outward. When $\theta$ is irrational with bounded continued fraction coefficients (as in the golden ratio), Voronoi cells of the seeds converge asymptotically to constant area, a feature essential for robust natural packing and efficient resource utilization.

This geometric construction connects phyllotactic spirals—empirically prominent in seeds, cones, and reproductive organs—to deep properties in hyperbolic geometry and Diophantine approximation.

3. Seed Hull Morphometrics and Adaptation

Morphometric studies of actual biological seeds, such as Setaria spp., reveal that seed “geometry” is a functional adaptation (Donnelly et al., 2014). Elongation, rugosity, and elevated surface-to-volume ratio (S:V) are quantitatively linked to enhanced antenna-like function for environmental signal acquisition:

• Increasing S:V allows more effective water and oxygen film accumulation;
• More elongate (high S:V) shapes are associated with weedy taxa requiring environmental sensitivity;
• Rounder shapes with lower S:V ratios, characteristic of domesticated forms, coordinate with rapid, uniform germination and decreased environmental signal dependency.

These effects are substantiated by ellipsoid and Fourier-reconstructed 3D models: $$V = \frac{4}{3}\pi abc,\qquad SA \approx 4\pi \left[\frac{(ab)^p + (ac)^p + (bc)^p}{3}\right]^{1/p},\qquad \text{(Legendre)}$$ with S:V calculated as $SA/V$. Functional geometry thus acts as a mechanism for niche separation, efficiently tuning germination response beyond mere phylogenetic constraints.

4. Population Genetics: Seed-Bank Coalescent and Stochastic Geometry

Seed-geometry in population genetics refers to the genealogical structure induced by dormancy, as in the seed-bank coalescent (Blath et al., 2014), and spatially structured models with seed-banks (Pederzani et al., 2016, Greven et al., 2020). Here, a generational process interleaves “active” individuals with dormant “seeds” that can remain unmerged for many generations. The resulting genealogy exhibits:

• Marked partitions distinguishing active/dormant blocks;
• Blocked coalescence for dormant lineages, leading to slower convergence to the most recent common ancestor (scaling as $\log\log n$);
• Higher genetic variability, consistent with empirical studies.

Spatial models add migration kernels and colony structure, leading to explicit, often Fourier-space, formulae for identity-by-descent probabilities: $$\hat{\Psi}(\theta, \eta) = [1 - \Psi_{0,0}^{(4)}/N] \cdot (\mathbb{I} - \hat{B}(\theta, \eta))^{-1} \cdot \hat{\Gamma}(\theta, \eta)$$ The spatial second moment and its scaling with model parameters directly relate dormancy and geometry to the persistence and spread of genetic identity.

5. Automated Theorem Proving and Symbolic Geometry Engines

Seed-Geometry, as an automated geometry reasoning engine integrated with the Seed-Prover system (Chen et al., 31 Jul 2025), formalizes geometric constructions in theorems. It addresses the lack of geometry support in formal proof systems (e.g., Lean) by:

• Translating geometric configurations into a canonical internal form;
• Employing forward-chaining with composite constructions (e.g., isogonal conjugates, exsimilitude/insimilitude centers);
• Utilizing a specialized DSL to group primitive steps for more efficient reasoning;
• Combining C++ backend acceleration and LLM policy models for fast, scalable, search-based auxiliary construction discovery.

In practice, Seed-Geometry demonstrated high efficacy on challenging olympiad geometry benchmarks, solving $43$ IMO-AG problems (2000–2024) and outperforming prior engines on IMOSL-AG. Its integration supplies proof assistants with streamlined auxiliary construction generation, creating a hybrid neural-symbolic pipeline for formal mathematical reasoning.

6. Computational and Information-Theoretic Perspectives

Seed-geometry underlies multiple algorithmic frameworks:

• Subset seed automata for sequence similarity searches rely on a geometric state-space encoding of seed prefixes and their matches (Kucherov et al., 2014), achieving efficient pattern recognition by exploiting the geometric structure of non-match positions.
• In 3D representation learning, “seed-point-driven” strategies for 3D object generation start from sparse geometric seed sets, which are mapped to dense latent codes for detailed reconstructions (Yan et al., 23 Feb 2025). This approach ensures multi-view geometric consistency and supports direct, intuitive editing at the seed point level.

Such methods leverage geometric abstraction for tractable algorithm design, both for computational biology (mapping, alignment) and for controllable geometry generation in computer vision.

7. Physical and Astrophysical Models: Seed Fields and Geometry

In magnetohydrodynamics, “seed geometry” refers to the initial spatial and spectral properties of seed magnetic fields prior to nonlinear amplification by turbulent dynamos (Seta et al., 2020). Despite potentially wide variation in initial configuration (e.g., uniform, power-law, parabolic spectra), dynamo action results in statistical erasure of the original seed geometry, converging to universal scaling laws (Kazantsev spectrum) and coherence lengths.

Additional mechanisms for magnetic seed generation, such as plasma heat flux in curved spacetime, generate seed fields via the coupling of thermodynamic gradients, spacetime curvature, and fluid motion (Villarroel-Sepúlveda et al., 20 Nov 2024). For axisymmetric, radially dependent accretion disks, the heat flux dominates seed field generation exactly where fluid vorticity vanishes, with geometry fixed by the interplay of temperature, density, and velocity gradients. These results inform both the origin of astrophysical fields and the limits of information retained from initial seed configurations.

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The concept of Seed-Geometry thus serves as a unifying framework across the physical, biological, mathematical, and computational sciences, capturing both the generative laws and inferential challenges inherent to the structure, function, and information encoded by seeds and seed-like constructs.