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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 52: Convex and discrete geometry

Introduction

History

Applications and related fields

Subfields

• 52A: General convexity, especially 52A55: Spherical geometry.
• 52B: Polytopes and polyhedra
• 52C: Discrete geometry

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Klee, Victor: "What is a convex set?", Amer. Math. Monthly 78 1971 616--631. MR44#3202

Miyazaki, Koji: "An adventure in multidimensional space. The art and geometry of polygons, polyhedra, and polytopes", John Wiley & Sons, Inc., New York, 1986. 112 pp. ISBN 0-471-81648-5 MR87m:00001

Tilings: There are two books by Grünbaum, Branko and Shephard, G. C. entitled "Tilings and patterns", both published by W. H. Freeman and Company, New York: the first (1987. 700 pp. ISBN 0-7167-1193-1) followed by one subtitled "An introduction" (1989. 446 pp. ISBN 0-7167-1998-3) (The two authors have a long history of collaboration in this area.)

Software and tables

Other web sites with this focus

• UTK archives page.
• Here are the AMS and Goettingen resource pages for area 52.

Selected Topics at this site

• What is the Brouwer Fixed-Point Theorem?
• Helley's theorem: If several given convex sets cover R^n then n+1 convex sets cover R^n
• Loewner's theorem: there is a unique minimal-volume ellipsoid containing any given bounded set in R^n
• What is the maximum number of pieces formed with N slices of the cake?
• Number of regions formed joining chords of equidistant points on a circle.
• How many shapes formed from glueing N squares edge-to-edge? (n-ominos)
• How many shapes formed from glueing N blocks face-to-face?
• Kepler's conjecture: the densest packing of balls in R^3 is the one used to stack fruit.
• Densest sphere packings relation to distributing points on spheres
• How many spheres can be packed into a rectangular box?
• Optimal packings of {circles, squares,...} in {squares, ...} [Dave Boll]
• Optimal distribution of points in a box?
• Optimal packings of small circles in a larger one
• Table of best known packings of squares into squares
• Packing rectangles to demonstrate famous infinite sums
• Optimal lattice packings in R^n
• Convex shapes with worst packing density
• Tiling 3-space using tetrahedra and square pyramids
• Some discussions about the Penrose tilings of the plane (aperiodic tilings with as few as 2 distinct shapes).
• Penrose tilings and others.
• Quasiperiodic tilings of Euclidean space (e.g. Penrose tiles) [Chris Hillman]
• Types of pentagons which tessellate the plane
• Decomposing a square as a union of distinct squares [See also Guy's Unsolved Problems in Number Theory for a picture.]
• Tiling the plane with noncongruent equilateral triangles
• What are Wang dominoes/tiles?
• Find the largest square with exactly 3 interior lattice points
• Quick proof of the isoperimetric inequality (that other closed curves enclose less area than a circle of the same length).
• The isoperimetric quotient: area versus circumference
• Isoperimetric inequality (circle has maximum area for fixed perimeter): definitions, pointers, proofs
• Largest quadrilateral with given sides
• Literature survey on Reauleaux triangles.
• What's the volume of the cone on a region?
• What is the largest box contained in a general 3-dimensional shape?
• Into how many pieces may a torus be dissected with three planar cuts?
• The worm problem -- minimal convex set containing all length-1 curves
• Optimal search procedure for boundary of an infinite strip (= shortest path of width 1)
• Finding a global linearly independent complement to convex sets
• Bounding volume from areas of projections
• Does every curve contain the vertices of some square? (open)
• Apollonian gasket (packing the plane with non-congruent circles)
• The Kissing numbers (numbers of congruent spheres which touch each other in lattices)
• Spectrum of a matrix of distances like Cayley-Menger matrix