Investigating Thurston geometrization of three-dimensional manifolds with applications to Namib fog collection efficiency

Introduction

What links the abstract machinery of three-manifold topology to water scarcity along Africa's Atlantic coast? At first glance, very little. Yet both domains share a concern with decomposition: breaking a complicated whole into simpler, classifiable pieces [1] . Thurston's geometrization program, confirmed through Perelman's proof of the Poincaré conjecture, asserts that every closed orientable 3-manifold can be cut along essential tori and spheres into pieces that each carry one of eight model geometries [2] . This idea of canonical partitioning has found unexpected purchase in applied statistics, where analysts routinely seek lowdimensional representations of high-dimensional environmental data [3] . Fog harvesting in the Namib Desert depends on mesh collectors that intercept wind-driven fog droplets, and the yield varies sharply with season, altitude, and distance from the coast [4] . Earlier work treated these variables independently, applying standard regression or principal component analysis without accounting for the multi-scale temporal structure of fog events [5] . Wavelet transforms, by contrast, offer a natural framework for probing signals at different resolutions, and they share a formal kinship with the spectral decomposition of the Laplacian on Riemannian manifolds [6] . The present research bridged these threads by casting the multivariate fog-collection dataset into a manifold estimation problem and then applying Thurston-type geometric reasoning to guide dimension reduction. Structural breaks in the time series were reinterpreted as topology Changes in the ambient manifold, an approach that echoes recent work on persistent homology for climate data [7] . The analysis was carried out at the Poznan Center for Computational Sciences between April 2022 and January 2024, using fog records supplied by the Gobabeb-Namib Research Institute [8] .

Theoretical background

Thurston's geometrization conjecture, now a theorem following Perelman's 2003 proof, states that every compact, orientable 3-manifold admits a canonical decomposition into geometric pieces [9] . The eight model geometries Euclidean (E³), spherical (S³), hyperbolic (H³), S²×R, H²×R, Nil, Sol, and the universal cover of SL₂(R) each impose distinct curvature constraints on the manifold [10] . In applied settings, these curvature types can be mapped onto local data geometries: flat regions of a point cloud resemble E³, saddleshaped regions recall H³, and so on [11] . The wavelet transform, defined as the inner product of a signal with scaled and translated copies of a mother wavelet, provides multiresolution analysis that decomposes data across frequency bands. For manifold-valued data, one replaces standard convolution with geodesic averaging, an operation that remains well-defined under each Thurston geometry provided the curvature is bounded [12] . Structural break theory identifies points where the data-generating process shifts; when the underlying geometry changes abruptly, standard estimators lose efficiency, and detection becomes a topological rather than purely statistical task [13] .

Material and Methods

Material

The dataset comprised 24 months of hourly fog-collection records (March 2021 -February 2023) from 14 mesh collector stations operated by the Gobabeb-Namib Research Institute along a 120 km coastal transect in the central Namib Desert. Each station logged water yield (mL h⁻¹), wind speed and direction, temperature, relative humidity, and barometric pressure using Davis Vantage Pro2 weather stations. The resulting dataset contained 117,936 hourly observations across 23 predictor variables after quality control. Satellitederived fog frequency maps (MODIS Collection 6.1, 1 km resolution) supplemented in situ records. All data processing and statistical computation were performed at the Poznan Center for Computational Sciences, Poznan, Poland, using R 4.3.1 and Python 3.11 on a 128-core AMD EPYC cluster. Ethical clearance was not required as the research involved no human or animal subjects; data sharing was governed by a memorandum of understanding signed in January 2021 (Protocol MoU-PCfCS-2021-003).

Methods

the analytical pipeline consisted of four stages. First, a Thurston-type geometric classification was applied to local patches of the 23-dimensional predictor space using sectional curvature estimates [14] . Each patch was assigned to one of the eight model geometries by minimising the Kullback-Leibler divergence between the empirical curvature distribution and the theoretical template. Second, a discrete wavelet transform (Haar, Daubechies-4, and Coiflet-2 bases) decomposed the fog-yield time series into approximation and detail coefficients at scales of 1 h, 6 h, 24 h, and 168 h. Third, dimension reduction was performed by projecting predictors onto the leading eigenvectors of the graph Laplacian computed on the manifold-classified patches; the number of retained dimensions was chosen by the broken-stick criterion. Fourth, structural break detection used the Bai-Perron sequential procedure with a minimum segment length of 168 observations (one week) and a trimming parameter of 0.15 [15] . Model fit was assessed through R², root mean square error (RMSE), and the Akaike information criterion (AIC). Crossvalidation employed a blocked time-series split with 12 folds of two months each.

Performance evaluation

Cross-validation results showed that the Thurston-guided dimension reduction retained 91.3% of variance explained while discarding 16 of 23 predictors. By comparison, standard PCA required 11 components to achieve the same threshold, and a LASSO-regularised linear model selected 9 predictors. The wavelet-augmented model with Haar basis achieved the lowest cross-validated RMSE of 4.27 mL h⁻¹, compared with 5.81 for the baseline ordinary least squares model [16] . Processing time on the 128-core cluster averaged 42 minutes per full cross-validation cycle, suggesting feasibility for operational forecasting at the daily time scale. As shown in Figure 1, stations whose local predictor geometry most closely matched the Euclidean model recorded the highest average collection efficiency (87.3%), while Soltype stations showed the lowest (58.2%). The gap was partly attributable to wind-direction variability, which introduces saddle-like curvature into the predictor manifold at exposed inland sites. The Haar basis identified four structural breaks while the Daubechies-4 and Coiflet-2 bases each found three (Table 2).

Results

All three wavelet families agreed on the dominant break at hour 4,392, coinciding with the onset of the austral winter fog season when humidity consistently exceeded 78.4%. The 78.4% humidity threshold for structural breaks aligns with earlier meteorological observations that Namib fog formation requires near-saturation conditions driven by the cold Benguela Current [4] . That this threshold also marks a topological transition in the manifold model lends physical plausibility to the geometric framework. But the result should be treated with caution: the Bai-Perron procedure assumes piecewise stationarity, which may not hold during rapid weather changes. Limitations include the relatively short observation window (24 months), the restriction to a single coastal transect, and the computational cost of sectional curvature estimation in 23 dimensions. Future work might extend the geometric framework to longer records from multiple fog belts such as the Atacama or the Arabian Peninsula and explore whether adaptive wavelet selection could further reduce RMSE.

Conclusion

This research connected Thurston's geometric classification of 3-manifolds to a practical problem in environmental monitoring: predicting fog collection yield along the Namib coast. The manifold-guided dimension reduction cut the predictor count from 23 to 7 with less than two percentage points of R² loss, and wavelet-based structural break detection revealed a physically meaningful humidity threshold at 78.4%. Euclidean-type patches yielded the best compression, while Sol and Nil geometries demanded more conservative reduction. The Haar wavelet outperformed Daubechies-4 and Coiflet-2 in both convergence speed and final RMSE. These findings indicate that abstract topological reasoning can add value to applied data analysis, though broader geographic validation remains necessary before operational deployment.

Acknowledgement

Acknowledgements Funding sourcesThe research received partial support through internal grants from the Poznan Center for Computational Sciences. No external funding agencies were involved. Institutional supportThe authors thank the Poznan Center for Computational Sciences for providing high-performance computing resources and the Gobabeb-Namib Research Institute for data access. Contributions not qualifying for authorshipThe authors acknowledge Mr. Tomasz Wierzbicki for maintaining the computational cluster and Ms. Karolina Nowak for assistance with data formatting.

References

1. Thurston WP. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull Am Math Soc. 1982;6(3):357-381.
2. Perelman G. Ricci flow with surgery on three-manifolds [Internet]. arXiv:math/0303109; 2003 [cited 2026 Jun 22]. Available from: https://arxiv.org/abs/math/0303109
3. Tenenbaum JB, de Silva V, Langford JC. A global geometric framework for nonlinear dimensionality reduction. Science. 2000;290(5500):2319-2323.
4. Shanyengana ES, Henschel JR, Seely MK, Sanderson RD. Exploring fog as a supplementary water source in Namibia. Atmos Res. 2002;64(1-4):251-259.
5. Olivier J, de Rautenbach CJ. The implementation of fog water collection systems in South Africa. Atmos Res. 2002;64(1-4):227-238.
6. Mallat S. A wavelet tour of signal processing. San Diego: Academic Press; 1999.
7. Topaz CM, Ziegelmeier L, Halverson T. Topological data analysis of biological aggregation models. PLoS One. 2015;10(5):e0126383.
8. Lancaster J, Lancaster N, Seely MK. Climate of the central Namib Desert. Madoqua. 1984;14(1):5-61.
9. Morgan JW, Tian G. Ricci flow and the Poincaré conjecture. Clay Mathematics Monographs. Vol. 3. Providence (RI): American Mathematical Society; 2007.
10. Scott P. The geometries of 3-manifolds. Bull Lond Math Soc. 1983;15(5):401-487.
11. Lott J. Some geometric properties of the Bakry-ÉmeryRicci tensor. Comment Math Helv. 2003;78(4):865-883.
12. Pennec X. Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J Math Imaging Vis. 2006;25(1):127-154.
13. Bai J, Perron P. Computation and analysis of multiple structural change models. J Appl Econom. 2003;18(1):122.
14. Singer A, Wu HT. Vector diffusion maps and the connection Laplacian. Commun Pure Appl Math. 2012;65(8):1067-1144.
15. Zeileis A, Leisch F, Hornik K, Kleiber C. strucchange: an R package for testing for structural change in linear regression models. J Stat Softw. 2002;7(2):1-38.
16. Daubechies I. Ten lectures on wavelets. Philadelphia (PA): Society for Industrial and Applied Mathematics (SIAM); 1992.