Shapson-Coe, A. et al. A petavoxel fragment of human cerebral cortex reconstructed at nanoscale resolution. Science 384, eadk4858 (2024).
The MICrONS Consortium. Functional connectomics spanning multiple areas of mouse visual cortex. Nature 640, 435–447 (2025).
Winding, M. et al. The connectome of an insect brain. Science 379, eadd9330 (2023).
Wilson, N. M., Ortiz, A. K. & Johnson, A. B. The vascular model repository: a public resource of medical imaging data and blood flow simulation results. J. Med. Devices 7, 040923 (2013).
Witten, E. Non-commutative geometry and string field theory. Nucl. Phys. B 268, 253–294 (1986).
Carlip, S. Quadratic differentials and closed string vertices. Phys. Lett. B 214, 187–192 (1988).
Saadi, M. & Zwiebach, B. Closed string field theory from polyhedra. Ann. Phys. 192, 213–227 (1989).
Cajal, S. R. Y., Azoulay, D. L., Swanson, N. & Swanson, L. W. Histology Of The Nervous System: Of Man And Vertebrates (Oxford Univ. Press, 1995).
Murray, C. D. The physiological principle of minimum work. Proc. Natl Acad. Sci. USA 12, 207–214 (1926).
Dehmamy, N., Milanlouei, S. & Barabási, A.-L. A structural transition in physical networks. Nature 563, 676–680 (2018).
Liu, Y., Dehmamy, N. & Barabási, A.-L. Isotopy and energy of physical networks. Nat. Phys. 17, 216–222 (2021).
Budd, J. M. L. et al. Neocortical axon arbors trade-off material and conduction delay conservation. PLoS Comput. Biol. 6, e1000711 (2010).
Markov, N. T. et al. Cortical high-density counterstream architectures. Science 342, 1238406 (2013).
Wang, Z., Zhao, M. & Yu, Q.-X. Modeling of branching structures of plants. J. Theor. Biol. 209, 383–394 (2001).
Durand, M. Architecture of optimal transport networks. Phys. Rev. E 73, 016116 (2006).
Bontorin, S., Cencetti, G., Gallotti, R., Lepri, B. & De Domenico, M. Emergence of complex network topologies from flow-weighted optimization of network efficiency. Phys. Rev. X 14, 021050 (2024).
Banavar, J. R., Maritan, A. & Rinaldo, A. Size and form in efficient transportation networks. Nature 399, 130–132 (1999).
D’Souza, R. M., Borgs, C., Chayes, J. T., Berger, N. & Kleinberg, R. D. Emergence of tempered preferential attachment from optimization. Proc. Natl Acad. Sci. USA 104, 6112–6117 (2007).
Latty, T. et al. Structure and formation of ant transportation networks. J. R. Soc. Interface 8, 1298–1306 (2011).
Sexton, Z. A. et al. Rapid model-guided design of organ-scale synthetic vasculature for biomanufacturing. Science 388, 1198–1204 (2025).
Chklovskii, D. & Stevens, C. Wiring optimization in the brain. In Advances in Neural Information Processing Systems 12: Proc. 1999 Conference 103–107 (MIT Press, 1999).
Chklovskii, D. B., Schikorski, T. & Stevens, C. F. Wiring optimization in cortical circuits. Neuron 34, 341–347 (2002).
Kim, Y., Sinclair, R., Chindapol, N., Kaandorp, J. A. & Schutter, E. D. Geometric theory predicts bifurcations in minimal wiring cost trees in biology are flat. PLoS Comput. Biol. 8, e1002474 (2012).
Hwang, F. K., Richards, D. S. & Winter, P. The Steiner Tree Problem 1st edn (Elsevier, 1992).
Rosenthal, A. Computing the reliability of complex networks. SIAM J. Appl. Math. 32, 384–393 (1977).
Winter, P. Steiner problem in networks: a survey. Networks 17, 129–167 (1987).
Amirghasemi, M. et al. in Frontiers in Nature-Inspired Industrial Optimization 1st edn (eds Khosravy, M., Gupta, N. & Patel, N.) 33–48 (Springer, 2022).
Cherniak, C. Local optimization of neuron arbors. Biol. Cybern. 66, 503–510 (1992).
Zamir, M. Optimality principles in arterial branching. J. Theor. Biol. 62, 227–251 (1976).
Corals – 3D digitization. https://3d.si.edu/corals.
Scheffer, L. K. et al. A connectome and analysis of the adult Drosophila central brain. eLife 9, e57443 (2020).
Gonzalez de Tanago, J. et al. Estimation of above-ground biomass of large tropical trees with terrestrial LiDAR. Methods Ecol. Evol. 9, 223–234 (2018).
Pan, H., Hétroy-Wheeler, F., Charlaix, J. & Colliaux, D. ARABIDOPSIS 3D+T dataset. Zenodo https://doi.org/10.5281/zenodo.5205561 (2021).
Percheron, G. Quantitative analysis of dendritic branching. I. Simple formulae for the quantitative analysis of dendritic branching. Neurosci. Lett. 14, 287–293 (1979).
Percheron, G. Quantitative analysis of dendritic branching. II. Fundamental dendritic numbers as a tool for the study of neuronal groups. Neurosci. Lett. 14, 295–302 (1979).
Miyawaki, S., Tawhai, M. H., Hoffman, E. A., Wenzel, S. E. & Lin, C.-L. Automatic construction of subject-specific human airway geometry including trifurcations based on a CT-segmented airway skeleton and surface. Biomech. Model. Mechanobiol. 16, 583–596 (2017).
Schreiner, W. & Buxbaum, P. Computer-optimization of vascular trees. IEEE Trans. Biomed. Eng. 40, 482–491 (1993).
Jessen, E., Steinbach, M. C., Debbaut, C. & Schillinger, D. Rigorous mathematical optimization of synthetic hepatic vascular trees. J. R. Soc. Interface 19, 20220087 (2022).
Keelan, J., Chung, E. M. L. & Hague, J. P. Simulated annealing approach to vascular structure with application to the coronary arteries. R. Soc. Open Sci. 3, 150431 (2016).
Bobenko, A. I., Sullivan, J. M., Schröder, P. & Ziegler, G. M. (eds) Discrete Differential Geometry (Birkhäuser, 2008).
Bianconi, G. & Rahmede, C. Complex quantum network manifolds in dimension d > 2 are scale-free. Sci. Rep. 5, 13979 (2015).
Bianconi, G., Rahmede, C. & Wu, Z. Complex quantum network geometries: evolution and phase transitions. Phys. Rev. E 92, 022815 (2015).
Bianconi, G. & Rahmede, C. Network geometry with flavor: from complexity to quantum geometry. Phys. Rev. E 93, 032315 (2016).
Gromov, M. Partial Differential Relations 1st edn (Springer, 1986).
Tong, D. Lectures on string theory. University of Cambridge http://www.damtp.cam.ac.uk/user/tong/string.html (2009).
Lynch, J. P. Steep, cheap and deep: an ideotype to optimize water and n acquisition by maize root systems. Ann. Bot. 112, 347–357 (2013).
Harris, S. D. Branching of fungal hyphae: regulation, mechanisms and comparison with other branching systems. Mycologia 100, 823–832 (2008).
Barabási, D. L. & Barabási, A.-L. A genetic model of the connectome. Neuron 105, 435–445 (2020).
West, G. B., Brown, J. H. & Enquist, B. J. A general model for the origin of allometric scaling laws in biology. Science 276, 122–126 (1997).
Thompson, D. W. On Growth and Form (Cambridge Univ. Press, 1992).
West, G. Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies (Penguin Press, 2017).
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D. U. Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006).
Pósfai, M. et al. Impact of physicality on network structure. Nat. Phys. 20, 142–149 (2024).
Glover, C. & Barabási, A.-L. Measuring entanglement in physical networks. Phys. Rev. Lett. 133, 077401 (2024).
Bonamassa, I. et al. Logarithmic kinetics and bundling in physical networks. Preprint at https://arxiv.org/abs/2401.02579 (2024).
Cimini, G. et al. The statistical physics of real-world networks. Nat. Rev. Phys. 1, 58–71 (2019).