Variations in the orientation and cross-sectional shape of filamentous microfossils provide quantitative measures for characterizing them and probing their native mechanical structure. Here, we determine the tangent correlation length, which is the characteristic length scale for the variation in direction of a sinuous curve, for both a suite of Precambrian filamentous microfossils and six strains of modern filamentous cyanobacteria, all with diameters of a few microns. Among 1.9–2-Ga microfossils, Gunflintia grandis, Gunflintia minuta and Eomycetopsis filiformis possess, respectively, correlation lengths of 360 ± 40 µm, 670 ± 40 µm and 700 ± 100 µm in two dimensions. Hundreds of times larger than the filament diameters, these values lie in the same range as the cyanobacteria Geitlerinema and Pseudanabaena, but are smaller than several strains of Oscillatoria. In contrast, the 2-Ga microfossil trichome Halythrix, is found to have a short correlation length of 29 ± 4 µm in two dimensions. Micron-wide pyritic replacement filaments observed in 3.23-Ga volcanogenic deposits also display a modest correlation length of 100 ± 15 µm in two dimensions. Sequences of species in two genera of our modern cyanobacteria possess tangent correlation lengths that rise as a power of the filament diameter DD3.3 ± 1 for Oscillatoria and D5.1 ± 1 for Geitlerinema. These results can be compared with power-law scaling of D3 for hollow tubes and D4 for solid cylinders that is expected from continuum mechanics. Extrapolating the observed scaling behavior to smaller filament diameters, the measured correlation length of the pyrite filaments is consistent with modern Geitlerinema whereas that of Halythrix lies not far from modern Oscillatoria, suggesting that there may be structural similarities among these genera.

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