Abstract
From Thurston’s geometrization theorem to the Banach-Tarski paradox, spatial complexity can be enigmatic, due to singularities, immersions, infinities (e.g. Alexander’s wild knots, Antoine’s necklace, Ford circles, Hilbert curves). Measuring spatial complexity in already “complex” spatial settings constitutes a major challenge for future research, as it should combine algorithmics, computation and topology within a single whole. Further, completely homogeneous maps can still be complex if their complexity is measured by CP1, because topological differentiation (i.e. division in square cells) creates spatial complexity: a completely undifferentiated square map without cells defined on it has a different spatial complexity than a square map with cells identified in it. And yet, the same allocations of numbers to cells of binary maps can produce different configurations with different spatial complexities.
It is a riddle, wrapped in a mystery,
inside an enigma; but perhaps there is a key
(Sir Winston Churchill, 1874–1965,“The Russian Enigma”, 01-20-1939)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arnold, V. I. (1992). Catastrophe theory. Berlin: Springer-Verlag.
Borsuk, K. (1933). Drei satze iiber dien-dimensionale euklidische sphare. Fundamenta Mathematicae, 20, 177–190.
Bölcskei, A., & Szilágyi,B. (2006). Visualization of curves and spheres in sol geometry. KoG-10, 26–32.
Francis, G. K., & Morin, B. (1980). Arnold shapiro’s eversion of the sphere. Mathematical Intelligencer, 2(4), 200–203.
Kahn, J., & Kalai, G. (1993). A counterexample to Borsuk’s conjecture. Bulletin of the American Mathematical Society, NS, 29, 60–62.
Kusner, R. (1987). Conformal geometry and complete minimal surfaces. Bulletin of the American Mathematical Society, New Series, 17(2), 291–295.
Lopez, R., & Munteanu, M. I. (2011). Surfaces with constant mean curvature in sol geometry. Differential Geometry and its Applications, 29, S238–S245.
Northshield, S. (2015). Ford circles and spheres. arxiv:1503.00813v1math.NT, 3 March.
Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 v1 November 11, 2002.
Perelmann, G. (2003a). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109 v1 March 10,2003, preprint.
Perelmann, G. (2003b). Finite extinction time to the solutions to the Ricci flow on certain three manifolds. arXiv:math.DG/0307245 July 17, 2003, preprint.
Phillips, A. (1966). Turning a surface inside out. Scientific American May, 112–120.
Pickover, C. (2009). The maths book. London: Sterling.
Smale, S. (1958). A classification of immersions of the two-sphere. Transactions of the American Mathematical Society, 90(2), 281–290.
Thom, R. (1989). Structural stability and morphogenesis: An outline of a general theory of models. Reading, MA: Addison-Wesley.
Thurston, W. P. (1982). Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bulletin of the American Mathematical Society, 6, 357–381.
Thurston, W. P. (1997). Three-dimensional geometry and topology (Vol. 1). Princeton NJ: Princeton University Press.
Zeeman, E. C. (1977). Catastrophe theory-selected papers –1977. Reading, MA: Addison-Wesley.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Papadimitriou, F. (2020). Enigmas of Spatial Complexity. In: Spatial Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-59671-2_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-59671-2_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-59670-5
Online ISBN: 978-3-030-59671-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)