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The Topological Basis of Spatial Complexity

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Spatial Complexity

Abstract

The key topological indicators of spatial complexity are: number of boundaries, genus, dimension, knottingness, braiding, linking/writhing and immersion of the surface or object: i.e. the higher the number of boundaries and/or knots (or braids) and the higher the genus and the dimension of the surface or object, the more spatially complex it is. Some initial experimentations with spatial complexity can be made with easy-to-derive estimates by using the Gage-Hamilton-Grayson theorem, Pick’s theorem and intersections of lines with Jordan curves. Then, spatial complexity can be assessed in various ways by using concepts of the complexity of knots and braids, including knot polynomials, linking numbers and Grosberg-Nechaev complexity.

1. A point is that of which there is no part.

2. A line is length without breadth.

3. The extremities of a line are points.

“αʹ.Σημεῖόν ἐστιν, οὗ μέρος οὐθέν.

βʹ.Γραμμὴ δὲ μῆκος ἀπλατές.

γʹ.Γραμμῆς δὲ πέρατα σημεῖα.”

(Euclid, 4th century b.C.,“Elements", Book A)

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  1. Forschungsbereich Geographie, Mathematisch-Naturwissenschaftliche Fakultät, Universität Tübingen, Tübingen, Germany

    Fivos Papadimitriou

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  1. Fivos Papadimitriou
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Papadimitriou, F. (2020). The Topological Basis of Spatial Complexity. In: Spatial Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-59671-2_5

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