Capecchi / Buscema / Contucci Applications of Mathematics in Models, Artificial Neural Networks and Arts

Mathematics and Society
1. Auflage 2010
ISBN: 978-90-481-8581-8
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 617 Seiten

The book shows a very original organization addressing in a non traditional way, but with a systematic approach, to who has an interest in using mathematics in the social sciences. The book is divided in four parts: (a) a historical part, written by Vittorio Capecchi which helps us understand the changes in the relationship between mathematics and sociology by analyzing the mathematical models of Paul F. Lazarsfeld, the model of simulation and artificial societies, models of artificial neural network and considering all the changes in scientific paradigms considered; (b) a part coordinated by Pier Luigi Contucci on mathematical models that consider the relationship between the mathematical models that come from physics and linguistics to arrive at the study of society and those which are born within sociology and economics; (c) a part coordinated by Massimo Buscema analyzing models of artificial neural networks; (d) a part coordinated by Bruno D'Amore which considers the relationship between mathematics and art. The title of the book 'Mathematics and Society' was chosen because the mathematical applications exposed in the book allow you to address two major issues: (a) the general theme of technological innovation and quality of life (among the essays are on display mathematical applications to the problems of combating pollution and crime, applications to mathematical problems of immigration, mathematical applications to the problems of medical diagnosis, etc.) (b) the general theme of technical innovation and creativity, for example the art and mathematics section which connects to the theme of creative cities. The book is very original because it is not addressed only to those who are passionate about mathematical applications in social science but also to those who, in different societies, are: (a) involved in technological innovation to improve the quality of life; (b) involved in the wider distribution of technological innovation in different areas of creativity (as in the project 'Creative Cities Network' of UNESCO).

VITTORIO CAPECCHIVittorio Capecchi ha avuto una laurea in economics e una specializzazione in mathematical sociology alla Columbia University con Paul F. Lazarsfeld ed è in seguito a questo incontro che nel 1967 ha fondato ed è diventato editor della rivista 'Quality and Quantity'. Dopo questa rivista ha fondato ed è diventato editor nel 1971 della rivista 'Inchiesta', una rivista di economia e sociologia che ha cercato di diffondere la pratica dell'inchiesta fuori e all'interno dell'università. I suoi interessi vanno in due direzioni. Una prima direzione è quella della metodologia della ricerca e Capecchi ha studiato sia le applicazioni della matematica alla sociologia ma anche i cambiamenti nei paradigmi della ricerca (paradigma della oggettività, paradigma della ricerca azione/co-ricerca, paradigma della feminist methodology). Una seconda direzione è quella dello sviluppo e cambiamento della innovazione tecnologica. PIERLUIGI CONTUCCIPierluigi Contucci is a Professor at University of Bologna, Department of Mathematics. He has taught and done scientific research at Princeton University and University of California. In 2000 he was been awarded the Schloessman prize from the Max Plank Society. His research interests are in Statistical Mechanics and applications to the social and economical sciences. MASSIMO BUSCEMAMassimo Buscema (1955). Professor and Computer Scientist, expert in Neural Networks and Adaptive Artificial Systems. Founder and Director of Semeion - Research Centre of Science and Communication (a Scientific Organization Recognized by Italian Ministry of Research in 1991). Professor and Director of the Department of Science of Communications at the University of Charleston (West Virginia-USA 1981-1985) and Professor of 'Computers and Linguistics' at the University of Perugia (1984-1985). Member on the Editorial Board of various international journals. He has designed, constructed developed new models and algorithms of Artificial Intelligence. Author of scientific publications on theoretical aspects of Natural Computation, with over 150 scientific articles, essays, and 18 books on the same subject. Inventor of 14 international patents. He programmed over 20 software packages in Artificial Neural Networks and Evolutionary Systems. Scientific Director of research projects on the application of artificial intelligence systems in biomedical field and security field.BRUNO D'AMOREBruno D'Amore is a graduate in mathematics, philosophy and pedagogy; PhD in Mathematics Education; full professor in Mathematics Education at the Bologna University; co-director of Research Doctoral School at the Distrital University of Bogotà (Colombia). Is investigation field is Mathematics Education.CV complete: www.dm.unibo.it/rsddm

Autoren/Hrsg.

Capecchi, Vittorio

Buscema, Massimo

Contucci, Pierluigi

D'Amore, Bruno

Inhaltsverzeichnis

1;Preface;5
2;Contents;9
3;Contributors;12
4;1 Mathematics and Sociology;15
4.1;1.1 Theory, Methodology and Mathematics: the Choices of Paul F. Lazarsfeld;17
4.1.1;1.1.1 Types of Sociological Research: Some Suggestions;20
4.1.2;1.1.2 Die Arbeitslosen von Marienthal (1932);23
4.1.3;1.1.3 Personal Influence (1955);24
4.1.4;1.1.4 The Academic Mind (1958);25
4.1.5;1.1.5 Other Dimensions for the Classification of Sociological Researches;27
4.2;1.2 Mathematics and Sociology from Statistical Models to Artificial Societies and Social Simulation;28
4.2.1;1.2.1 The Paradigm of Action Research/Co-research;29
4.2.2;1.2.2 The Paradigm of Feminist Methodology;32
4.2.3;1.2.3 The Relation Between Mathematics and Sociology;35
4.2.3.1;1.2.3.1 Logic for Sociological Theory;37
4.2.3.2;1.2.3.2 Mathematics for the Relation Between Individual and Collective Properties;46
4.2.3.3;1.2.3.3 Mathematics for Less Visible Relations;49
4.2.3.4;1.2.3.4 Mathematics for the Social Sciences;51
4.2.3.5;1.2.3.5 Mathematical Models;52
4.2.3.6;1.2.3.6 (Ia) Statistical Models;53
4.2.3.7;1.2.3.7 (Ib) Models of Simulation;56
4.2.3.8;1.2.3.8 (IIa) Models of Process;61
4.2.3.9;1.2.3.9 (IIb) Models of Structure;62
4.2.3.10;1.2.3.10 (IIc) Rational Choice Models, Agent-Based Models, Artificial Society Models;65
4.3;1.3 Possibilities for Sociology of Artificial Neural Networks;71
4.3.1;1.3.1 Sociology and Artificial Neural Networks;72
4.3.2;1.3.2 Semeion Research Institute's Applications to Sociology;76
4.3.3;1.3.3 Possibilities of ANNs to the Relation Between Mathematics and Society;82
4.4;Bibliography;83
5;Part I Mathematics and Models;93
5.1;2 Equilibria of Culture Contact Derived from In-Group and Out-Group Attitudes;94
5.1.1;2.1 Introduction;94
5.1.2;2.2 General Framework;95
5.1.3;2.3 Culture Contact in Immigration;96
5.1.4;2.4 Discussion;100
5.1.5;Appendix;100
5.1.6;Bibliography;101
5.2;3 Society from the Statistical Mechanics Perspective;102
5.2.1;3.1 Introduction;102
5.2.2;3.2 The Physics of Spin Systems;103
5.2.2.1;3.2.1 An Example;104
5.2.3;3.3 The Spin Model in a Magnetic Field;106
5.2.4;3.4 Social Models of Binary Choices;107
5.2.4.1;3.4.1 A Yes or No Answer to a Referendum Question;108
5.2.4.2;3.4.2 Cross-cultural Interactions;108
5.2.5;3.5 Hypotheses and Assumptions in Social Psychology;109
5.2.6;Bibliography;111
5.3;4 Objects, Words and Actions: Some Reasons Why Embodied Models are Badly Needed in Cognitive Psychology;112
5.3.1;4.1 Introduction;112
5.3.2;4.2 Theoretical Framework: Embodied Theories of Cognition;113
5.3.3;4.3 Embodied Models Are Necessary to Reproduce Experimental Results on Object Vision and Action;114
5.3.3.1;4.3.1 An Example of an Experiment with a Possible Model;115
5.3.4;4.4 Embodied Models Are Necessary to Reproduce Experimental Results on Language Grounding;118
5.3.4.1;4.4.1 Examples of Possible Models;120
5.3.5;4.5 Embodied Models Can Help to Formulate Clearer Predictions;121
5.3.6;4.6 Tentative Conclusions and Open Issues;123
5.3.7;Bibliography;124
5.4;5 Shared Culture Needs Large Social Networks;126
5.4.1;5.1 Introduction;126
5.4.2;5.2 Models;127
5.4.3;5.3 Results;129
5.4.3.1;5.3.1 Fast-Changing Social Networks or Slowly Changing Traits;129
5.4.3.2;5.3.2 Fixed Social Networks or Fast-Changing Traits;129
5.4.3.3;5.3.3 Biased Traits and External Agencies;131
5.4.4;5.4 Discussion;131
5.4.5;Appendix;132
5.4.5.1;Model Analysis;132
5.4.6;Bibliography;134
5.5;6 Mathematical Models of Financial Markets;136
5.5.1;6.1 Introduction;136
5.5.2;6.2 Models for Financial Market Predictions;137
5.5.3;6.3 Exiting from the Trend;138
5.5.4;6.4 Cointegration and Pair Trading;140
5.5.5;6.5 Final Remarks;142
5.5.6;Bibliography;142
5.6;7 Tackling Climate Change Through Energy Efficiency: Mathematical Models to Offer Evidence-Based Recommendations for Public Policy;144
5.6.1;7.1 Introduction;144
5.6.2;7.2 Behavioural Models: Beyond Rational Man and Homo Economicus ;146
5.6.2.1;7.2.1 Bounded Rationality and Emotion: Discrete Choice Analysis;147
5.6.2.1.1;7.2.1.1 Theory;148
5.6.2.1.2;7.2.1.2 Empirical Estimation;150
5.6.2.1.3;7.2.1.3 Applications;150
5.6.2.2;7.2.2 Imitation and Social Interactions: Statistical Mechanics;151
5.6.3;7.3 Application Energy Efficiency and Climate Change;153
5.6.3.1;7.3.1 Case Study -- United Kingdom;155
5.6.4;7.4 Beyond Climate Change: Why Focus on Behavioural and Cultural Change to Achieve Policy Goals?;156
5.6.5;7.5 Discussion;157
5.6.6;Bibliography;158
5.7;8 An Application of the Multilevel Regression Technique to Validate a Social Stratification Scale;160
5.7.1;8.1 Introduction to the Stratification Scale;160
5.7.2;8.2 Validation of the Ordering of the Categories;163
5.7.3;8.3 The Multilevel Technique Applied to the Stratification Scale;166
5.7.4;8.4 Results of the Analysis;168
5.7.5;8.5 Conclusions;173
5.7.6;Bibliography;174
5.8;9 The Academic Mind Revisited: Contextual Analysis via Multilevel Modeling;175
5.8.1;9.1 Lazarsfelds Contextual Analysis;176
5.8.1.1;9.1.1 Apprehension and Its Correlates;177
5.8.1.2;9.1.2 Assessing Permissiveness;179
5.8.1.3;9.1.3 Institutional Incidents;181
5.8.2;9.2 Developing Multilevel Statistical Models;184
5.8.2.1;9.2.1 Step 1: Exploring the Data;185
5.8.2.2;9.2.2 Step 2: The Baseline Unconditional Means Model;186
5.8.2.3;9.2.3 Step 3: The Level-1 Model;191
5.8.2.4;9.2.4 Step 4: Adding a Level-2 Covariate;192
5.8.2.5;9.2.5 Step 5: The Combined Multilevel Model;193
5.8.2.6;9.2.6 Step 6: Estimate the Parameters of the Combined Model;193
5.8.2.7;9.2.7 Step 7: Testing the Preferred Model Against Alternative Models;194
5.8.2.8;9.2.8 Step 8: Replicate the Analysis Using Alternative Statistical Procedures;195
5.8.2.9;9.2.9 Step 9: Replicate the Analysis with Other Data;202
5.8.3;9.3 Conclusion;203
5.8.4;Bibliography;204
6;Part II Mathematics and Neural Networks;206
6.1;10 The General Philosophy of the Artificial Adaptive Systems;207
6.1.1;10.1 Artificial Adaptive Systems;208
6.1.2;10.2 A Brief Introduction to Artificial Neural Networks;212
6.1.2.1;10.2.1 Architecture;212
6.1.2.1.1;10.2.1.1 The Nodes;213
6.1.2.1.2;10.2.1.2 The Connections;214
6.1.2.1.3;10.2.1.3 The Signal Flow;215
6.1.2.2;10.2.2 The Learning;215
6.1.2.3;10.2.3 Artificial Neural Networks Typology;216
6.1.2.3.1;10.2.3.1 Supervised ANNs;216
6.1.2.3.2;10.2.3.2 Dynamic Associative Memories;217
6.1.2.3.3;10.2.3.3 Autopoietic ANNs;219
6.1.3;10.3 A Brief Introduction to Evolutionary Algorithms;220
6.1.3.1;10.3.1 Genetic Algorithm;221
6.1.3.2;10.3.2 Natural Evolution and Artificial Evolution;222
6.1.3.2.1;10.3.2.1 Natural Evolution;222
6.1.3.2.2;10.3.2.2 Artificial Evolution;224
6.1.3.2.3;10.3.2.3 The Holland Model;224
6.1.3.2.4;10.3.2.4 Types of Encoding of Artificial Genome;226
6.1.3.3;10.3.3 Other Evolutionary Algorithms;227
6.1.3.3.1;10.3.3.1 The Evolutionary Programming;227
6.1.3.3.2;10.3.3.2 The Evolutionary Strategies;227
6.1.3.3.3;10.3.3.3 The Classifying Systems;228
6.1.3.3.4;10.3.3.4 The Genetic Programming;228
6.1.3.3.5;10.3.3.5 Comparison with Other Techniques;229
6.1.3.4;10.3.4 The Genetic Doping Evolutionary Algorithm;230
6.1.3.4.1;10.3.4.1 Theory;231
6.1.3.4.2;10.3.4.2 The Criteria of Vulnerability;232
6.1.3.4.3;10.3.4.3 Criteria of Connectivity;232
6.1.3.4.4;10.3.4.4 The Criteria of the Last Chance;232
6.1.3.4.5;10.3.4.5 The New Crossover;233
6.1.4;References;236
6.2;11 Auto-contractive Maps, the H Function, and the Maximally Regular Graph (MRG): A New Methodology for Data Mining;237
6.2.1;11.1 Introduction and Motivation;238
6.2.2;11.2 The Auto-contractive Map;239
6.2.3;11.3 Experimenting with the Auto-contractive Map;245
6.2.4;11.4 Auto-CMs: A Theoretical Discussion;252
6.2.4.1;11.4.1 The Contractive Factor;254
6.2.4.2;11.4.2 Auto-CM and Minimum Spanning Tree;255
6.2.4.3;11.4.3 Other Algorithms for MST;257
6.2.4.4;11.4.4 Some Qualitative Features of MST Optimization;260
6.2.5;11.5 Graph Complexity: The H Function;261
6.2.5.1;11.5.1 Graph and MST Complexity;263
6.2.6;11.6 The Delta H Function;267
6.2.6.1;11.6.1 Auto-CM, MST, and the Delta H Function: An Example;268
6.2.7;11.7 Auto-CM and Maximally Regular Graph (MRG);275
6.2.7.1;11.7.1 Maximally Regular Graph: An Example;278
6.2.8;11.8 Conclusions;281
6.2.9;Bibliography;282
6.2.9.1;Contractive Maps;282
6.2.9.2;MST, Graphs, and Physical Networks;282
6.2.9.3;Theory of Probability and Bayesian Networks;283
6.2.9.4;Euclidean Distance;283
6.2.9.5;Back-Propagation Networks;283
6.2.9.6;Research Software;285
6.3;12 An Artificial Intelligent Systems Approach to Unscrambling Power Networks in Italy's Business Environment;286
6.3.1;12.1 Introduction;287
6.3.2;12.2 Interlocking Directorates: A Tentative (Partial) Chronology of the Literature;288
6.3.3;12.3 Context Matters: Dealing with Complexity Through the Reverse Approach;292
6.3.4;12.4 AutoCM: A New Methodological Foundation for Fundamental Network Analysis;295
6.3.5;12.5 Power Structure Dynamics in the Italian Business Environment;298
6.3.6;12.6 Power Structure Dynamics in the Italian Business Environment: March 2006November 2007;305
6.3.7;12.7 Conclusions;315
6.3.8;Bibliography;317
6.4;13 MultiMeta-SOM;321
6.4.1;13.1 The Multi-SOMMeta-SOM System;321
6.4.1.1;13.1.1 The SOM Networks;322
6.4.1.2;13.1.2 SOM: Architecture;322
6.4.1.3;13.1.3 SOM: Base Algorithm;322
6.4.1.3.1;13.1.3.1 Initialization Stage;323
6.4.1.3.2;13.1.3.2 Cyclic Calibration Stage;323
6.4.1.3.3;13.1.3.3 Topology of Neighbourhood;324
6.4.1.4;13.1.4 Correction of the Codebook;325
6.4.2;13.2 The LVQ Learning Vector Quantization;327
6.4.2.1;13.2.1 LVQ: Architecture;328
6.4.2.2;13.2.2 LVQ: Base Algorithm;328
6.4.2.2.1;13.2.2.1 Initialization Phase;329
6.4.2.2.2;13.2.2.2 Learning Cyclic Phase;330
6.4.2.2.3;13.2.2.3 Correction of the Codebook (Matrix of the Wij Weights);330
6.4.3;13.3 The Multi-SOMMeta-SOM System;330
6.4.3.1;13.3.1 Multi-SOM: Classification and Mapping of the Input in Separate Classes;331
6.4.3.2;13.3.2 Architecture;331
6.4.3.3;13.3.3 Basic Facts;331
6.4.3.4;13.3.4 Base Algorithm;333
6.4.3.4.1;13.3.4.1 Initialization Phase;333
6.4.3.4.2;13.3.4.2 Training Cyclic Phase;334
6.4.3.4.3;13.3.4.3 Testing Phase;335
6.4.3.5;13.3.5 Conclusions;336
6.4.4;13.4 From the Multi-SOM to the Meta-SOM;336
6.4.5;13.5 Meta-SOM;338
6.4.6;13.6 An Example: The Classification of Digits;338
6.4.6.1;13.6.1 The DB;338
6.4.6.2;13.6.2 The Training--Testing Samples;340
6.4.6.3;13.6.3 Structure of the Multi-SOM;342
6.4.6.4;13.6.4 The Multi-SOM Training;343
6.4.6.5;13.6.5 The Multi-SOM Testing;344
6.4.7;13.7 From the Multi-SOM Codebooks to the Meta-SOM Training;346
6.4.7.1;13.7.1 Set-Up of the Meta-SOM Network;346
6.4.7.1.1;13.7.1.1 Training of the Meta-SOM Network;347
6.4.7.2;13.7.2 Recall in Multi--Meta-SOM System;347
6.4.8;Bibliography;355
6.5;14 How to Perform Data Mining: The Persons Arrested Dataset;356
6.5.1;14.1 Data Description;357
6.5.2;14.2 Explorative Analysis Using Self-Organizing Maps;367
6.5.3;14.3 Explorative Analysis Using Auto-Contractive Maps;388
6.5.3.1;14.3.1 Learning Equations;388
6.5.3.2;14.3.2 Auto-CM: Theoretical Consideration;393
6.5.3.3;14.3.3 Auto-CM and Minimum Spanning Tree;396
6.5.3.4;14.3.4 The Graph Complexity: The H Function;397
6.5.3.5;14.3.5 Auto-CM and the Maximally Regular Graph;398
6.5.3.6;14.3.6 Application of Auto-CM to the Persons Dataset;399
6.5.4;14.4 Comparison of Data Mining Techniques;399
6.5.4.1;14.4.1 The Intersection Index;406
6.5.4.2;14.4.2 The Evidence and the Singularity Indexes;407
6.5.4.3;14.4.3 The Models Fusion Methodology (MFM);409
6.5.5;14.5 Conclusions;417
6.5.6;Bibliography;420
6.6;15 Medicine and Mathematics of Complex Systems: An Emerging Revolution;422
6.6.1;15.1 Introduction: Some Milestones of Mathematics in Medicine;423
6.6.1.1;15.1.1 Pierre Louis and the Numerical Method;423
6.6.1.2;15.1.2 John Snow and the 1854 Golden Square Cholera Epidemic;424
6.6.2;15.2 The Progress with Acute Disease and the Languish with Chronic Diseases;426
6.6.3;15.3 The Increasing Complexity of Clinical Data;427
6.6.4;15.4 Nonlinear Dynamics in Human Physiology;428
6.6.5;15.5 Examples of Applications of Chaos Theory to Medical Settings;430
6.6.6;15.6 Artificial Adaptive Systems and Medical Decision Support;431
6.6.7;15.7 Conclusions;435
6.6.8;Bibliography;436
6.7;16 J-Net System: A New Paradigm for Artificial Neural Networks Applied to Diagnostic Imaging;438
6.7.1;16.1 Introduction;438
6.7.2;16.2 General Overview on ACM Families: Basic Notions;439
6.7.3;16.3 J-NET: A New ACM System;441
6.7.4;16.4 J-Net: Medical Applications;448
6.7.5;16.5 Discussion;457
6.7.5.1;16.5.1 A Medical Perspective;457
6.7.5.2;16.5.2 A Mathematical Perspective;460
6.7.6;Bibliography;461
6.8;17 Digital Image Processing in Medical Applications, April 22, 2008;463
6.8.1;17.1 Introduction;463
6.8.2;17.2 Methods;466
6.8.2.1;17.2.1 Region Growing;466
6.8.2.2;17.2.2 Active Contour Model;466
6.8.2.3;17.2.3 Features Extraction: Gray-Level Co-occurrence Matrix;467
6.8.2.4;17.2.4 Feed-Forward Neural Network;469
6.8.3;17.3 Application;470
6.8.3.1;17.3.1 RG Application: Internal Lung Volume Segmentation;470
6.8.3.2;17.3.2 ACM Application: Anatomic Lung Contour Selection;473
6.8.3.3;17.3.3 Feed-Forward Neural Network Application: Classification of Massive Lesions in Breast Mammographic Images;476
6.8.4;Bibliography;478
7;Part III Mathematics and Art;480
7.1;18 Mathematics, Art, and Interpretation: A Hermeneutic Perspective;481
7.1.1;18.1 Introduction;482
7.1.2;18.2 Artefacts and Tools;482
7.1.3;18.3 A Semiotic Reflection, Following Peirce;483
7.1.4;18.4 Rorty and the Conversation;485
7.1.5;18.5 Final Reflections;486
7.1.6;Bibliography;487
7.2;19 Point, Line and Surface, Following Hilbert and Kandinsky;489
7.3;20 Figurative Arts and Mathematics: Pipes, Horses, Triangles and Meanings;494
7.3.1;20.1 Meaning and Its Representation: The Case of Mathematics;494
7.3.2;20.2 The Case of Figurative Art: Pipes and Horses;495
7.3.3;20.3 Gottlob Frege and Meaning in Mathematics;498
7.3.4;20.4 Horses and Meanings Before and After Frege;499
7.3.5;20.5 Art and Meaning After Magritte;500
7.3.6;20.6 Ternary Schemes of Meaning;501
7.3.7;20.7 Binary Schemes of Meaning;502
7.3.8;20.8 The Complex and Problematic Nature of Conceptual Meaning and of Its Representations;504
7.3.9;Bibliography;505
7.4;21 The Idea of Space in Art, Technology, and Mathematics;507
7.4.1;21.1 Introduction: Visual Mathematics;507
7.4.2;21.2 Space Is Mathematics;509
7.4.3;21.3 Fundamental Elements;512
7.4.4;21.4 Topology;513
7.4.5;21.5 International Architecture Exhibition 2004;515
7.4.6;21.6 Toward a Virtual Architecture;517
7.4.7;21.7 Final Observations;519
7.4.8;Bibliography;520
7.5;22 Mathematical Structures and Sense of Beauty;521
7.5.1;22.1 Introduction;521
7.5.2;22.2 Evolutionary Beauty;525
7.5.3;22.3 Socio-cultural Beauty;528
7.5.4;22.4 Mathematical Formalized Beauty;530
7.5.5;22.5 Conclusion;535
7.5.6;Bibliography;536
7.6;23 Visual Impact and Mathematical Learning;538
7.6.1;Bibliography;546
7.7;24 Art by Numbers;547
7.7.1;Bibliography;553
7.8;25 My Way of Playing with the Computer: Suggestions for a Personal Experience in Vector Graphics;554
7.8.1;25.1 Compositions of Circles;561
7.8.2;25.2 Complex Mapping;563
7.8.3;25.3 Moiré Patterns;565
7.8.4;25.4 Tessellations;566
7.8.5;25.5 Linear Fractals and Branching;570
7.8.6;25.6 Cellular Automata;573
7.8.7;25.7 Contour Lines;574
7.8.8;25.8 Polyhedra;574
7.8.9;25.9 A Sample of Other 3D Figures;577
7.8.10;25.10 Final Considerations;581
7.8.11;Bibliography;584
7.9;26 Four-Dimensional Ideas;586
7.9.1;26.1 Flatland;588
7.9.2;26.2 The Sphere;589
7.9.3;26.3 The Hypersphere;591
7.9.4;26.4 Tiling and Polyhedra;593
7.9.5;26.5 The 120-cell or Hyperdodecahedron;595
7.9.6;26.6 Tori;597
7.9.7;Bibliography;598
7.10;27 From Art to Mathematics in the Paintings of Theo van Doesburg;599
7.10.1;27.1 Introduction;599
7.10.2;27.2 Mathematics and Art;600
7.10.3;27.3 From a Painting to Mathematics;600
7.10.4;27.4 The Gnomon;601
7.10.5;27.5 The Symmetry;604
7.10.6;Bibliography;608
8;Author Index;609