Unsolved Problems in Geometry: Unsolved Problems in Intuitive Mathematics

ISBN-13: 9780387975061 / Angielski / Twarda / 1991 / 199 str.

Hallard T. Croft; Kenneth J. Falconer; Richard K. Guy

Opis

Mathematicians and non-mathematicians alike have long been fascinated by geometrical problems, particularly those that are intuitive in the sense of being easy to state, perhaps with the aid of a simple diagram. Each section in the book describes a problem or a group of related problems. Usually the problems are capable of generalization of variation in many directions. The book can be appreciated at many levels and is intended for everyone from amateurs to research mathematicians.

Spis treści

Notation and Definitions.- Sets. 1 Geometrical transformations..- Length, Area, and volume..- A. Convexity.- Al. The equichordal point problem..- A2. Hammer’s x-ray problems..- A3. Concurrent normals..- A4. Billiard ball trajectories in convex regions..- A5. Illumination problems..- A6. The floating body problem..- A7. Division of convex bodies by lines or planes through a point..- A8. Sections through the centroid of a convex body..- A9. Sections of centro-symmetric convex bodies..- A10. What can you tell about a convex body from its shadows?.- A11. What can you tell About a convex body from its sections?.- A12. Overlapping convex bodies..- A13. Intersections of congruent surfaces..- A14. Rotating polyhedra..- A15. Inscribed and circumscribed centro-symmetric bodies..- A16. Inscribed affine copies of convex bodies..- A17. Isoperimetric inequalities and extremal problems..- A18. Volume against width..- A19. Extremal problems for elongated sets..- A20. Dido’s problem..- A21. Blaschke’s problem..- A22. Minimal bodies of constant width..- A23. Constrained is operimetric problems..- A24. Is a body Fairly round if all its sections are?.- A25. How far apart can various centers be?.- A26. Dividing up a piece of land by a short fence..- A27. Midpoints of diameters of sets of constant width..- A28. Largest convex hull of an arc of a given length..- A29. Roads on planets..- A30. The shortest curve cutting all the lines through a disk..- A31. Cones based on convex sets..- A32. Generalized ellipses..- A33. Conic sections through five points..- A34. The shape of worn stones..- A35. Geodesics..- A36. Convex sets with universal sections..- A37. Convex space-filling curves..- A38. m-convex sets..- B. Polygons, Polyhedra, and Polytopes.- Bl. Fitting one triangle inside another..- B2. Inscribing polygons in curves..- B3. Maximal regular polyhedra inscribed in regular polyhedra..- B4. Prince Rupert’s problem..- B5. Random polygons and polyhedra..- B6. Extremal problems for polygons..- B7. Longest chords of polygons..- B8. Isoperimetric inequalities for polyhedra..- B9. Inequalities for sums of edge lengths of polyhedra..- B10. Shadows of polyhedra..- B11. Dihedral angles of polyhedra..- B12. Monostatic polyhedra..- B13. Rigidity of polyhedra..- B14. Rigidity of frameworks..- B15. Counting polyhedra..- B16. The sizes of the faces of a polyhedron..- B17. Unimodality of f-vectors of polytopes..- B18. Inscribable and circumscribable polyhedra..- B19. Truncating polyhedra..- B20. Lengths of paths on polyhedra..- B21. Nets of polyhedra..- B22. Polyhedra with congruent faces..- B23. Ordering the faces of a polyhedron..- B24. The four color conjecture for toroidal polyhedra..- B25. Sequences of polygons and polyhedra..- C. Tiling and Dissection.- Cl. Conway’s fried potato problem..- C2. Squaring the square..- C3. Mrs. Perkins’s quilt..- C4. Decomposing a square or a cube into n smaller ones..- C5. Tiling with incomparable rectangles and cuboids..- C6. Cutting up squares, circles, and polygons..- C7. Dissecting a polygon into nearly equilateral triangles..- C8. Dissecting the sphere into small congruent pieces..- C9. The simplexity of the d-cube..- C10. Tiling the plane with squares..- C11. Tiling the plane with triangles..- C12. Rotational symmetries of tiles..- C13. Tilings with a constant number of neighbors..- C14. Which polygons tile the plane?.- C15. Isoperimetric problems for tilings..- C16. Polyominoes..- C17. Reptiles..- C18. Aperiodic tilings..- C19. Decomposing a sphere into circular arcs..- C20. Problems in equidecomposability..- D. Packing and Covering.- D1. Packing circles, or spreading points, in a square..- D2. Spreading points in a circle..- D3. Covering a circle with equal disks..- D4. Packing equal squares in a square..- D5. Packing unequal rectangles and squares in a square..- D6. The Rados’ problem on selecting disjoint squares..- D7. The problem of Tammes..- D8. Covering the sphere with circular caps..- D9. Variations on the penny-packing problem..- D10. Packing Balls in space..- D11. Packing and covering with congruent convex sets..- D12. Kissing numbers of convex sets..- D13. Variations on Bang’s plank theorem..- D14. Borsuk’s conjecture..- D15. Universal covers..- D16. Universal covers for several sets..- D17. Hadwiger’s covering conjecture..- D18. The worm problem..- E. Combinatorial Geometry.- El. Helly-type problems..- E2. Variations on Krasnosel’skii’s theorem..- E3. Common transversals..- E4. Variations on Radon’s theorem..- E5. Collections of disks with no three in a line..- E6. Moving disks around..- E7. Neighborly convex bodies..- E8. Separating objects..- E9. Lattice point problems..- E10. Sets covering constant numbers of lattice points..- Ell. Sets that can be moved to cover several lattice points..- E12. Sets that always cover several lattice points..- E13. Variations on Minkowski’s theorem..- E14. Positioning convex sets relative to discrete sets..- F. Finite Sets of Points.- F1. Minimum number of distinct distances..- F2. Repeated distances..- F3. Two-distance sets..- F4.Can each distance occur a different number of times?.- F5. Well-spaced sets of points..- F6. Isosceles triangles determined by a set of points..- F7. Areas of triangles determined by a set of points..- F8. Convex polygons determined by a set of points..- F9. Circles through point sets..- F10. Perpendicular Bisectors..- F11. Sets cut off by straight lines..- F12. Lines through sets of points..- F13. Angles determined by a set of points..- F14. Further problems in discrete geometry..- F15. The shortest path joining a set of points..- F16. Connecting points by arcs..- F17. Arranging points on a sphere..- G. General Geometric Problems.- G1. Magic numbers..- G2. Metrically homogeneous sets..- G3. Arcs with increasing chords..- G4. Maximal sets avoiding certain distance configurations..- G5. Moving furniture around..- G6. Questions related to the Kakeya problem..- G7. Measurable sets and lines..- G8. Determining curves from intersections with lines..- G9. Two sets which always intersect in a point..- G10. The chromatic number of the plane and of space..- G11. Geometric graphs..- G12. Euclidean Ramsey problems..- G13. Triangles with vertices in sets of a given area..- G14. Sets Containing large triangles..- G15. Similar copies of sequences..- G16. Unions of similar copies of sets..- Index of Authors Cited.- General Index.