1. Patterns: From Chaos to Order -- 1.1. The search for regularity -- 1.2. Some regular structures -- 1.3. The mathematical study of regularity -- 2. A Pattern Formalism -- 2.1. The principle of atomism -- 2.2. The combinatory principle -- 2.3. The principle of observability -- 2.4. The principle of realism -- 3. Algebra of Regular Structures -- 3.1. Generator coordinates -- 3.2. Configuration coordinates -- 3.3. Connectors -- 3.4. Configuration homomorphisms -- 3.5. Configuration categories -- 3.6. Set operations in C (R) -- 3.7. Operations on images -- 3.8. Homomorphisms for given global regularity -- 3.9. Representations by image isomorphisms -- 4. Some Topology of Image Algebras -- 4.1. A topology for configurations -- 4.2. A topology for images -- 4.3. Some examples -- 5. Metric Pattern Theory -- 5.1. Regularity controlled probabilities -- 5.2. Conditioning by regularity -- 5.3. Frozen patterns: finite G and n -- 5.4. Frozen patterns: infinite G and finite n. -- 5.5. Quadratic energy function -- 5.6. Frozen patterns: infinite G and n -- 5.7. Asymptotically minimum energy -- 5.8. Asymptotics for large configurations -- 5.9. Spectral density matrix for ? = LINEAR(?) -- 5.10. Factorization of the spectral density matrix -- 5.11. Representation of the random configurations -- 5.12. Spectral density matrix for ? = LATTICE(?) -- 5.13. Factorization of the spectral density matrix in two dimensions -- 5.14. Representations of the random configurations in the two dimensional case -- 5.15. Laws of large numbers in pattern theory -- 5.16. Random dynamics for configurations -- 6. Patterns of Scientific Hypotheses -- 6.1. Hypotheses as regular structures -- 6.2. Patterns of statistical hypotheses -- 6.3. Generators for statistical hypotheses -- 6.4. Examples of configurations -- 6.5. Hypotheses as images -- 6.6. Image algebras of hypotheses -- 6.7. Conclusions -- 7. Synthesis of Social Patterns of Domination -- 7.1. Patterns in mathematical sociology -- 7.2. Domination regularity -- 7.3. Configuration dynamics -- 7.4. System in equilibrium -- 7.5. Large configurations - simulation results -- 7.6. Large configurations - analytical result -- 7.7. Further problems and extensions -- 7.8. Appendix -- 8. Taxonomic Patterns -- 8.1. A logic for taxonomic patterns -- 8.2. Logic of taxonomic affinity patterns -- 8.3. Synthesis of taxonomic affinity patterns -- 8.4. Analysis of affinity patterns -- 9. Patterns in Mathematical Semantics -- 9.1. Introduction -- 9.2. Introducing mathematical semantics -- 9.3. Formalization through regular structures -- 9.4. Two special image algebras -- 9.5. The choice of language type for the study -- 9.6. Semantic maps -- 9.7. Special semantic maps -- 9.8. Learning semantics -- 9.9. Abduction of semantic maps -- Outlook -- Notes.