The initial purposes of this 1983 text were to develop mathematical topics relevant to the study of the incidence and symmetry structures of geometrical objects and to expand the reader's geometric intuition. The two fundamental mathematical topics employed in this endeavor are graph theory and the theory of transformation groups. Part I, Incidence, starts with two sections on the basics of graph theory and continues with a variety of specific applications of graph theory. Following this, the text becomes more theoretical; here graph theory is used to study surfaces other than the plane and the sphere. Part II, Symmetry, starts with a section on rigid motions or symmetries of the plane, which is followed by another on the classification of planar patterns. Additionally, an overview of symmetry in three-dimensional space is provided, along with a reconciliation of graph theory and group theory in a study of enumeration problems in geometry.

### Contents

Forward; Preface; Part I. Incidence: Introduction; Section 1. Incidence and Graph Theory: 1. Topological transformations; 2. Basic graph theory; 3. Directed graphs; 4. Traversability; 5. Distance; Section II. Incidence in the Plane: 6. Maps; 7. Planar graphs; 8. Euler's formula; 9. Polyhedra; Section III. Further Applications of Graph Theory: 10. Bracing structures; 11. Optimal route design; 12. Mean distance; 13. Triangulations and organization graphs; Section IV. Topology of Surfaces: 14. Surfaces; 15. Maps on surfaces; 16. Tesselations of the plane; 17. Compact surfaces; Part II. Symmetry: Introduction; Section V. Symmetry and Group Theory: 18. Planar isometries; 19. Basic group theory; 20. Reflections on the plane; 21. The isometry group of the plane; Section VI. Symmetry in the Plane: 22. Discrete groups; 23. The circular groups; 24. The frieze groups; 25. The wallpaper groups; Section VII. Symmetry in Space: 26. Space isometries; 27. Discrete space groups; 28. The layer groups; 29. the rod groups; Section VIII. Symmetry and Enumeration: 30. A combinational approach to symmetry; 31. Graph symmetry; 32. Enumeration; 33. Fundamental architectural arrangements revisited; Bibliography; Indices.