This book delivers stimulating input for a broad range of researchers, from geographers and ecologists to psychologists interested in spatial perception and physicists researching in complex systems.
How can one decide whether one surface or spatial object is more complex than another?
What does it require to measure the spatial complexity of small maps, and why does this matter for nature, science and technology?
Drawing from algorithmics, geometry, topology, probability and informatics, and with examples from everyday life, the reader is invited to cross the borders into the bewildering realm of spatial complexity, as it emerges from the study of geographic maps, landscapes, surfaces, knots, 3D and 4D objects.
The mathematical and cartographic experiments described in this book lead to hypotheses and enigmas with ramifications in aesthetics and epistemology.
Author(s): Fivos Papadimitriou
Publisher: Springer
Year: 2020
Language: English
Pages: 298
City: Cham
Preface
Acknowledgements
Contents
Symbols
Part IIntroducing Spatial Complexity
1 What is Spatial Complexity?
1.1 Definition and Disambiguation
1.2 Disorder, Asymmetry, Inequality
1.3 Spatial Complexity in Three Dimensions
1.4 Computational Complexity Classes
1.5 Perceiving and Creating Spatial Complexity
References
2 Spatial Complexity in Nature, Science and Technology
2.1 Spatial Complexity in Cosmology
2.2 Spatial Complexity in Geography and Ecology
2.3 Spatial Complexity in Physics and Electronics
2.4 Spatial Complexity in the Life Sciences
References
Part IIThe Mathematical Basis of Spatial Complexity
3 The Geometric Basis of Spatial Complexity
3.1 Orthogonality
3.2 Intersections
3.3 Curvature and Non-Euclidean Geometries
3.4 Spatial Combinatorics and Polyominoes
References
4 The Probabilistic Basis of Spatial Complexity
4.1 Spatial Entropy Versus Complexity
4.2 Spatial Randomness and Algorithmic Complexity
References
5 The Topological Basis of Spatial Complexity
5.1 Boundaries
5.2 Knotting
5.3 Braiding
5.4 Writhing and Linking
5.5 Genus
5.6 Dimensions
5.7 Orientability and Immersions
References
6 The Algorithmic Basis of Spatial Complexity
6.1 The Language of Space
6.2 Metrics of Spatial Complexity
6.3 Extrema of Spatial Complexities CP1 and CP2
References
7 Exploring Spatial Complexity in 3d
7.1 Simplicial Complexes, Betti Groups and Matveev Complexity
7.2 Evaluation on Möbius Bands, Tori and Multiply-Connected Surfaces
7.3 Spatial Complexity of Simple 3d Solids and Voxels
7.4 Spatial Complexity of Reeb Graphs
References
8 Spatial Complexity in 4-and-Higher-Dimensional Spaces
8.1 Hypercubes, Tesseracts, Doxels, Clifford Tori
8.2 Manifolds, Gropes and Exotic Spheres
References
Part IIINumbers Behind Spatial Complexity
9 Squares, Cats and Mazes: The Art and Magic of Spatial Complexity
9.1 Square Partitions
9.2 Squares, Minimalism and Art
9.3 Mazes, Labyrinths and Spatial Games
9.4 The Arnold Cat Map
References
10 Entering the “Spatium Numerorum”: Creating Spatial Complexity from Numbers
10.1 Calculating Spatial Partitions
10.2 Entropy Class
10.3 Generic Maps and Symmetry
10.4 Calculating Binary Map Configurations
References
11 The Spatial Complexity of 3 × 3 Binary Maps
11.1 Parameters of Spatial Complexity of Binary Maps
11.2 The Spatial Complexity of 3 × 3 Binary Maps
11.3 Patchiness, Adjacency and Clumpiness
11.4 Generic 3 × 3 Binary Maps and Their Multiplicities
11.5 Spatial Analysis at “King’s Neighborhood”
References
12 Complexity of Binary Maps of Primes and Transcendentals
12.1 Numbers Defining Spaces
12.2 Complexity of Binary Square Maps of Primes and Composites
12.3 Square Maps from Transcendental Numbers
References
Part IVUnderstanding Spatial Complexity
13 Enigmas of Spatial Complexity
13.1 Geometrization, Singularities and Immersions
13.2 Spatial Complexity and Infinity
13.3 Same Numbers—Different Spatial Complexities
References
14 Taming Spatial Complexity
14.1 Taming Spatial Randomness?
14.2 Taming Spatial Complexity with Symmetries?
14.3 Taming Spatial Complexity with Grothendieck’s Inequality?
14.4 Taming Spatial Complexity with a “Sudoku Method”
References
Part VEpistemological, Psychological, Geophilosophical and Aesthetic Perspectives on Spatial Complexity
15 Spatial Complexity, Psychology and Qualitative Complexity
15.1 Gestalt Psychology and Spatial Complexity
15.2 Scale-Dependence of Spatial Complexity
15.3 Perception of Spatial Randomness and Spatial Complexity
15.4 Qualitative Complexity Versus Spatial Complexity
References
16 Spatial Complexity, Visual Complexity and Aesthetics
16.1 Visual Complexity
16.2 Visual Complexity Versus Spatial Complexity of 3d Objects
16.3 Landscape Aesthetics and Spatial Complexity
16.4 Spatial Complexity and Aesthetic Evaluations
References
17 Geophilosophy and Epistemology of Spatial Complexity
17.1 From Difference to Complexity
17.2 Curvature and Subjectivity
17.3 A Philosophical Ladder to Spatial Complexity
17.4 Experimenting in the Spatium Numerorum
17.5 Large-Scale Spatial Complexity: Limits to Spatial Analysis?
References
18 Spatial Complexity and the Future
18.1 Key Determinants of Spatial Complexity
18.2 Spatial Complexity, “Com-Possible” Worlds and Quantum Computation
18.3 Large-Scale Spatial Complexity, Spatial Computing and Planetary Futures
18.4 Mens Spatii: Towards an Observatory of Spatial Complexity
References
Index
