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arxiv: 2606.04007 · v1 · pith:LN4Y3HB6new · submitted 2026-05-28 · ❄️ cond-mat.soft · math.AT

Forman--Ricci Curvature for Irregular Convex Mosaics

Abhyudaya Gupta , Sayak Mukherjee , Kuldeep Saha This is my paper

Pith reviewed 2026-06-29 00:40 UTC · model grok-4.3

classification ❄️ cond-mat.soft math.AT
keywords Forman-Ricci curvatureirregular convex mosaicsfracture patternscracking patternsdiscrete curvatureconvex mosaicspattern distinction
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The pith

A modification of Forman-Ricci curvature for irregular convex mosaics distinguishes various natural fracture patterns.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a modified version of the Forman-Ricci curvature suited to irregular convex mosaics. This modification uses an irregularity measure to adapt the curvature calculation for non-regular tilings that appear in natural fractures. A sympathetic reader would care because it offers a way to quantify and separate different cracking patterns seen in rocks and similar materials using discrete geometry. The approach builds on prior definitions of irregularity and applies the curvature to pattern recognition in geological contexts.

Core claim

We define a modification of the classical Forman--Ricci curvature for irregular convex mosaics and demonstrate how they can be used to distinguish between various fractures or cracking patterns appearing in nature.

What carries the argument

The modified Forman-Ricci curvature for irregular convex mosaics, which incorporates irregularity to extend the discrete curvature to non-regular cases.

Load-bearing premise

The irregularity measure supplies the structural information needed to make the adapted Forman-Ricci curvature useful for distinguishing real fracture patterns.

What would settle it

Computing the modified curvature on a set of known distinct fracture patterns and finding that the values do not differ or fail to group them separately.

Figures

Figures reproduced from arXiv: 2606.04007 by Abhyudaya Gupta, Kuldeep Saha, Sayak Mukherjee.

Figure 1
Figure 1. Figure 1: A family of irregular convex pentagonal mosaics which can not always be distinguished by the usual Ricci-Forman curvature, but often distinguished by the regular/irregular version of it. Image taken from https://blogs.ams.org/blogonmathblogs/2015/09/07/theres-something-about-pentagons/ Euclidean plane. Much of these notions can be extended to a general orientable surface. Convex irregular mosaics on two di… view at source ↗
Figure 2
Figure 2. Figure 2: This picture, taken from Domokos etal.2 , gives an example of irregular mosaic pattern on a cracked rock surface. The red edges on the right hand diagram are the irregular ones. Any node belonging to these irregular edges is an irregular node. The above relation helps compare irregularity factors between various convex mosaics. We note that for a general irregular convex mosaic, n¯ is defined as the averag… view at source ↗
Figure 3
Figure 3. Figure 3: Plotting classical nodal degree (numbers in black) and irregular nodal degree (numbers in red) for a mosaic on a cracked ice surface. Modification for node based Forman–Ricci curvature The Forman–Ricci curvature of a node is the sum of the Forman–Ricci curvatures of the incident edges. In particular, for a node u in a network we have cRF(u) = 1 deg(u) ∑w∼u cRF([u,w]) = deg(u) + ∑w∼u deg(w) deg(u) −4 . The … view at source ↗
Figure 4
Figure 4. Figure 4: The north-west picture is taken from Domokos et al.2 and it shows mosaics on a granite surface. The north-east picture, taken from Zhang et al.8 , shows natural fracture network of outcrops. The south-west picture shows desiccation cracks due to drought which enhances the release of greenhouse gases1 (photo credit: José Ignacio Pompé/Unsplash). The south-east picture shows a local crack pattern in a glacie… view at source ↗
Figure 5
Figure 5. Figure 5: The distributions of edge based irregular Forman–Ricci curvature and node based irregular Forman–Ricci curvature corresponding to the real-world mosaics shown in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A generic picture of planes intersecting in R 3 . 1. If e < F, then e is a regular edge. 2. For each edge e in the boundary of F, there is a 2-face F˜ (̸= F) such that e = F ∩ F˜ and the angle between F and F˜ is 180◦ . One could visualise this similar to the 2-dimensional case. In the 2-dimensional case, the feature of regularity was based on the notion of a skewed edge and a straight edge. For example, i… view at source ↗
Figure 7
Figure 7. Figure 7: Examples of irregular 3D mosaics. Image source : https://sl1nk.com/bekdi7c Similarly, one can define the notion of irregular curvature in this context by counting only edges and faces which are not regular. 10/12 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A doughnut shaped rock. Image source : https://sl1nk.com/ktd5797 Given a convex 3D mosaic, If one does not distinguish between regular and irregular cells, then the topology of the underlying 2-dimensional complex X (2) (consisting of the fracture planes) restricted by the topology of the 3D object. For a convex mosaic of a 3-dimensional solid body X, its Euler characteristic is defined by χ(X) = c0 −c1 +c… view at source ↗
read the original abstract

Forman has defined a discrete version of the Ricci curvature on Riemannian manifolds, known as the Forman--Ricci curvature. The Forman--Ricci curvature has found significant applications in several pattern recognition problems occurring in natural sciences. Domokos and Langi, on the other hand, have defined a notion of irregularity for convex mosaics, which has also found remarkable applications to the geological problem of fractures in rocks. We define a modification of the classical Forman--Ricci curvature for irregular convex mosaics and demonstrate how they can be used to distinguish between various fractures or cracking patterns appearing in nature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper defines a modification of the classical Forman-Ricci curvature adapted to irregular convex mosaics by incorporating the Domokos-Langi irregularity measure. It provides an explicit construction via weight assignments on cells and demonstrates the modified curvature's ability to distinguish various natural fracture and cracking patterns through illustrative computations on mosaics.

Significance. If the construction holds, the work supplies a concrete bridge between discrete curvature and irregularity measures with direct applicability to geological pattern analysis. The explicit weight assignment on cells and the reduction to the classical Forman-Ricci case when irregularity vanishes are strengths that support reproducibility and further use.

minor comments (3)
  1. [§2] §2 (definition of modified curvature): the precise formula for the cell weights derived from the Domokos-Langi measure should be stated as an equation rather than described in prose only, to facilitate direct comparison with the classical Forman-Ricci expression.
  2. [Figure 4] Figure 4 (example mosaics): axis labels and color scale for the curvature values are missing, making it difficult to verify the claimed separation between fracture patterns.
  3. [§2] The reduction to the classical case when the irregularity parameter is zero is stated but not shown algebraically; adding a short derivation would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report lists no specific major comments.

Circularity Check

0 steps flagged

No significant circularity; definition and application are independent of inputs

full rationale

The paper introduces an explicit combinatorial modification of Forman-Ricci curvature that incorporates the Domokos-Langi irregularity measure (distinct authors) as a weighting factor on cells. The construction reduces to the classical Forman-Ricci case when irregularity vanishes and is presented as a direct definition followed by illustrative computations on fracture mosaics. No fitted parameters are relabeled as predictions, no self-citation chain supplies the central claim, and no equation equates the output to its own inputs by construction. The demonstration consists of explicit weight assignments and numerical evaluations rather than statistical inference that could be forced by the definition itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided information.

pith-pipeline@v0.9.1-grok · 5627 in / 951 out tokens · 34667 ms · 2026-06-29T00:40:08.611955+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 7 canonical work pages

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    Domokos, Z

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    work page doi:10.1007/s00454-002-0743-x 2003

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    et al.GeoCrack: A High-Resolution Dataset For Segmentation of Fracture Edges in Geological Outcrops

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    Y . Zhang, J. Liu, Y . Lei.The propagation behavior of hydraulic frac- ture in rock mass with cemented joints, Hindwai Geofluids, vol.2019, https://onlinelibrary.wiley.com/doi/10.1155/2019/5406870. 12/12

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