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Combinatorial geometries
164
Book announcements
Folding strips of triangles. Kaleidocycles. The isoaxis. Further exploration. Information on construction materials. E. The Rhombic Dodecahedron: Its Relation to the Cube and the Octahedron, ARTHUR L. LOEB. The pyramids. Construction of the polyhedra. Juxtaposition of plyhe&. Space-fillers. A note on materials (contributed by JACK GRAY). Part Il. Lecturesfiom the Shaping Space Conference. 3. Regular and Semiregular Polyhedra, H.S.M. COXETER.4. Milestones in the History of Polyhedra, JOSEPHMALKEVITCH.5. Polyhedra and Crystal Structures, CHIJNGCHIEH. 6. Polyhedra: Surfaces or Solids?, ARTHUR L. LOEB. 7. Spatial Perception and Creativity, JANOS BARACS.8. Why Study Polyhedra?, JEANPEDERSEN.Jennifer’s puzzle. Jennifer’s instructions. How to make the puzzle pieces. Assembling the polyhedra. Putting the puzzle together. Variations and modifications. Part III. Roles of Polyhedra in Science. 9. Form, Function, and Functioning, GEORGE FLECK. Does form explain function? Science looks to geometry for models. Plato’s ideas. Spheres and polyhedra as models for matter. Spheres and whirlpools as models for atoms and molecules. Polyhedra as models for atoms, molecules, and viruses. Modeling condensed matter. Packing of spheres of various sorts. Polyhedra as models for plant structures. Does form explain dynamic functioning? Science looks to geometry for mechanistic models. Plant growth and polyhedral transformations. Polyhedral models for self-assembly of viruses. Robotics and motions of polyhedra. Polyhedron theory accommodates changing expectations. 10. Polyhedral Molecular Geometries, ISIVAN HARGITI-AI AND MAGDOLNA HARG~AI. Boron hydride cages. Polycyclic hydrocarbons. Structures with a central atom. Regularities in nonbonded distances. The VSEPR model. Consequences of intramolecular motion. Part IV. Theory of Polyhedra. 11. Introduction to Polyhedron Theory, MARJO~UESENFXHAL.What is a polyhedron? Why a theory of polyhedra? Polyhedral themes. A word of warning. Notes. 12. Combinatorial Prototiles, ECKIN SCMJLTE. Nontiles. Constructions of monotypic tilings. 13. Duality of Polyhedra, BRANKO GR~~VBAUM AND G.C. SHEPHARD.14. Polyhedral Analogues of the Platonic Solids, J.M. WILLS.Platonohedra. Construction of the platonohedra. Regular polyhedra. The flag diagram. 15. Uniform Polyhedra from Diophantine Equations, BARRY MONSON.Interlude. Uniform polyhedra with icosahedral symmetry. 16. TONS Decompositions of Regular Polytopes in CSpace, THOMASF. BANCHOFF.Decompositions. The cube and its associated polyhedra. The hypercube and its associated polytopes. Fold-out decomposition of the hypercube and 24-ceh. Cartesian and torus coordinates. Coordinates for polyhedra and polytopes. The Hopf mapping. The Hopf decomposition of the hypercube. Torus decomposition of the 24-ceU. Conclusion. 17. Convex Polyhedra, Dirichlet Tessellations, and Spider Webs, PETER ASH, ETHAN BOLKER,HENRYCRAPO,AND WALTERWHITELEY.Cell decompositions and reciprocal figures. Spider webs and projections. The main result. Realizations of abstract graphs. Infinite plane examples. Notes. 18. Unsolved Problems. A. Can Neighborly Polyhedra Be Realized Geometrically?, JOHN REAY. Historical Background. Unsolved Problems. B. How Many Faces Does a Polytope Have?, MARGARET BAYER.C. Problems on the Realizability and Rigidity of Polyhedra, WALTER WHITELEY. Realizability. Inhnitesimal rigidity. Unique realizability. Dihedral angles and locally unique realixabiMy_ Concluding remarks on the design of spatial polyhedra. D. Problems Concerning Polyhedral 2-Manifokls, PETER GRI%%ANN. E. Extending the Conway Criterion, DORISSCHAITXHNEIDER.Part V. Further Steps. 19. Polyhedra and Curricuhrm? MARJORIE SENECZHAL AND GEORGE FLECK.
DORIS SCHATISCHNEIDER.
Combinatorial Geometries, Edited by Neil white (Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney l!B7) 212 pages. 1. Coordinatizations, Neil White. 1.1 Introduction and basic definitions. 1.2 Equivalence of coordinatixations and canonical forms. 1.3 Matroid operations. 1.4 Non-coordinatixable geometries. 1.5 Necessary and sufficient conditions for coordinatixation. 1.6 Brackets. 1.7 Coordinatixation over algebraic extensions. 1.8 Characteristic sets. 1.9 Coordinatixations over transcendental extensions. 1.10 Algebraic representation. 2. Binary Matroidr, J.C. Fournier. 2.1 Definition and basic properties. 2.2 Characterizations of binary matroids. 2.3 Related characterizations. 2.4 Spaces of circuits of binary matroids. 2.5 Coordinatixing matrices of binary matroids. 2.6 Special classes of binary matroids; graphic matroids. 2.7 Appendix on modular pairs of circuits in a matroid. 3. (In&nodular Matroids. Neil White. 3.1 Equvalent conditions for unimodularity. 3.2 Tutte’s Homotopy Theorem
Book announcements
16.5
and excluded minor characterization. 3.3 Applications of unimodularity. 4. Introduction to Matching Theory, Richard A. Brualdi. 4.1 Matchings on matroids. 4.2 Matching matroids. 4.3 Applications. 5. Transversal Matroids. Richard A. Brualdi. 5.1 Introduction. 5.2 Presentations. 5.3 Duals of transversal matroids. 5.4 Other properties and generalizations. 6. Simplicial Matroids, Raul Cordovil and Brent LinLtrcm. 6.1 Introduction. 6.2 Orthogonal full simplicial matroids. 6.3 Binary and unimodular full simplicial geometries. 6.4 Uniquely coordinatixable full simplicial matroids. 6.5 Matroids on the bases of matroids. 6.6 Spemer’s lemma for geometries. 6.7 Other results. 7. The Mijbius Function and the Characteristic Polynomial, Thomas Zaslavsky. 7.1 The Mobius function. 7.2 The characteristic polynomial. 7.3 The beta invariant. 7.4 Tutte-Grothendieck invariance. 7.5 Examples. 7.6 The critical problem. 8. Whitney Numbers, Martin Aigner. 8.1 Introduction. 8.2 The characteristic and rank polynomials. 8.3 The Mobius algebra. 8.4 The Whitney numbers of the first kind. 8.5 The Whitney numbers of the second kind. 8.6 Comments. 9. Matroids in Combinatorial Optimization, Ulrich Faigle. 9.1 The greedy algorithm and rnatroid polyhedra. 9.2 Intersections and unions of matroids. 9.3 Integral matroids. 9.4 Submodular systems. 9.5 Submodular flows.
Book announcements
Folding strips of triangles. Kaleidocycles. The isoaxis. Further exploration. Information on construction materials. E. The Rhombic Dodecahedron: Its Relation to the Cube and the Octahedron, ARTHUR L. LOEB. The pyramids. Construction of the polyhedra. Juxtaposition of plyhe&. Space-fillers. A note on materials (contributed by JACK GRAY). Part Il. Lecturesfiom the Shaping Space Conference. 3. Regular and Semiregular Polyhedra, H.S.M. COXETER.4. Milestones in the History of Polyhedra, JOSEPHMALKEVITCH.5. Polyhedra and Crystal Structures, CHIJNGCHIEH. 6. Polyhedra: Surfaces or Solids?, ARTHUR L. LOEB. 7. Spatial Perception and Creativity, JANOS BARACS.8. Why Study Polyhedra?, JEANPEDERSEN.Jennifer’s puzzle. Jennifer’s instructions. How to make the puzzle pieces. Assembling the polyhedra. Putting the puzzle together. Variations and modifications. Part III. Roles of Polyhedra in Science. 9. Form, Function, and Functioning, GEORGE FLECK. Does form explain function? Science looks to geometry for models. Plato’s ideas. Spheres and polyhedra as models for matter. Spheres and whirlpools as models for atoms and molecules. Polyhedra as models for atoms, molecules, and viruses. Modeling condensed matter. Packing of spheres of various sorts. Polyhedra as models for plant structures. Does form explain dynamic functioning? Science looks to geometry for mechanistic models. Plant growth and polyhedral transformations. Polyhedral models for self-assembly of viruses. Robotics and motions of polyhedra. Polyhedron theory accommodates changing expectations. 10. Polyhedral Molecular Geometries, ISIVAN HARGITI-AI AND MAGDOLNA HARG~AI. Boron hydride cages. Polycyclic hydrocarbons. Structures with a central atom. Regularities in nonbonded distances. The VSEPR model. Consequences of intramolecular motion. Part IV. Theory of Polyhedra. 11. Introduction to Polyhedron Theory, MARJO~UESENFXHAL.What is a polyhedron? Why a theory of polyhedra? Polyhedral themes. A word of warning. Notes. 12. Combinatorial Prototiles, ECKIN SCMJLTE. Nontiles. Constructions of monotypic tilings. 13. Duality of Polyhedra, BRANKO GR~~VBAUM AND G.C. SHEPHARD.14. Polyhedral Analogues of the Platonic Solids, J.M. WILLS.Platonohedra. Construction of the platonohedra. Regular polyhedra. The flag diagram. 15. Uniform Polyhedra from Diophantine Equations, BARRY MONSON.Interlude. Uniform polyhedra with icosahedral symmetry. 16. TONS Decompositions of Regular Polytopes in CSpace, THOMASF. BANCHOFF.Decompositions. The cube and its associated polyhedra. The hypercube and its associated polytopes. Fold-out decomposition of the hypercube and 24-ceh. Cartesian and torus coordinates. Coordinates for polyhedra and polytopes. The Hopf mapping. The Hopf decomposition of the hypercube. Torus decomposition of the 24-ceU. Conclusion. 17. Convex Polyhedra, Dirichlet Tessellations, and Spider Webs, PETER ASH, ETHAN BOLKER,HENRYCRAPO,AND WALTERWHITELEY.Cell decompositions and reciprocal figures. Spider webs and projections. The main result. Realizations of abstract graphs. Infinite plane examples. Notes. 18. Unsolved Problems. A. Can Neighborly Polyhedra Be Realized Geometrically?, JOHN REAY. Historical Background. Unsolved Problems. B. How Many Faces Does a Polytope Have?, MARGARET BAYER.C. Problems on the Realizability and Rigidity of Polyhedra, WALTER WHITELEY. Realizability. Inhnitesimal rigidity. Unique realizability. Dihedral angles and locally unique realixabiMy_ Concluding remarks on the design of spatial polyhedra. D. Problems Concerning Polyhedral 2-Manifokls, PETER GRI%%ANN. E. Extending the Conway Criterion, DORISSCHAITXHNEIDER.Part V. Further Steps. 19. Polyhedra and Curricuhrm? MARJORIE SENECZHAL AND GEORGE FLECK.
DORIS SCHATISCHNEIDER.
Combinatorial Geometries, Edited by Neil white (Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney l!B7) 212 pages. 1. Coordinatizations, Neil White. 1.1 Introduction and basic definitions. 1.2 Equivalence of coordinatixations and canonical forms. 1.3 Matroid operations. 1.4 Non-coordinatixable geometries. 1.5 Necessary and sufficient conditions for coordinatixation. 1.6 Brackets. 1.7 Coordinatixation over algebraic extensions. 1.8 Characteristic sets. 1.9 Coordinatixations over transcendental extensions. 1.10 Algebraic representation. 2. Binary Matroidr, J.C. Fournier. 2.1 Definition and basic properties. 2.2 Characterizations of binary matroids. 2.3 Related characterizations. 2.4 Spaces of circuits of binary matroids. 2.5 Coordinatixing matrices of binary matroids. 2.6 Special classes of binary matroids; graphic matroids. 2.7 Appendix on modular pairs of circuits in a matroid. 3. (In&nodular Matroids. Neil White. 3.1 Equvalent conditions for unimodularity. 3.2 Tutte’s Homotopy Theorem
Book announcements
16.5
and excluded minor characterization. 3.3 Applications of unimodularity. 4. Introduction to Matching Theory, Richard A. Brualdi. 4.1 Matchings on matroids. 4.2 Matching matroids. 4.3 Applications. 5. Transversal Matroids. Richard A. Brualdi. 5.1 Introduction. 5.2 Presentations. 5.3 Duals of transversal matroids. 5.4 Other properties and generalizations. 6. Simplicial Matroids, Raul Cordovil and Brent LinLtrcm. 6.1 Introduction. 6.2 Orthogonal full simplicial matroids. 6.3 Binary and unimodular full simplicial geometries. 6.4 Uniquely coordinatixable full simplicial matroids. 6.5 Matroids on the bases of matroids. 6.6 Spemer’s lemma for geometries. 6.7 Other results. 7. The Mijbius Function and the Characteristic Polynomial, Thomas Zaslavsky. 7.1 The Mobius function. 7.2 The characteristic polynomial. 7.3 The beta invariant. 7.4 Tutte-Grothendieck invariance. 7.5 Examples. 7.6 The critical problem. 8. Whitney Numbers, Martin Aigner. 8.1 Introduction. 8.2 The characteristic and rank polynomials. 8.3 The Mobius algebra. 8.4 The Whitney numbers of the first kind. 8.5 The Whitney numbers of the second kind. 8.6 Comments. 9. Matroids in Combinatorial Optimization, Ulrich Faigle. 9.1 The greedy algorithm and rnatroid polyhedra. 9.2 Intersections and unions of matroids. 9.3 Integral matroids. 9.4 Submodular systems. 9.5 Submodular flows.