Article thumbnail
Location of Repository

The unrestricted blocking number in convex geometry

By S. Sezgin

Abstract

Let K be a convex body in \mathbb{R}^n. We say that a set of translates \left \{ K + \underline{u}_i \right \}_{i=1}^{p} block K if any other translate of K which touches K, overlaps one of K + \underline{u}_i, i = 1, . . . , p. The smallest number of non-overlapping translates (i.e. whose interiors are disjoint) of K, all of which touch K at its boundary and which block any other translate of K from touching K is called the Blocking Number of K and denote it by B(K). This thesis explores the properties of the blocking number in general but the main purpose is to study the unrestricted blocking number B_\alpha(K), i.e., when K is blocked by translates of \alpha K, where \alpha is a fixed positive number and when the restrictions that the translates are non-overlapping or touch K are removed. We call this number the Unrestricted Blocking Number and denote it by B_\alpha(K). The original motivation for blocking number is the following famous problem: Can a rigid material sphere be brought into contact with 13 other such spheres of the same size? This problem was posed by Kepler in 1611. Although this problem was raised by Kepler, it is named after Newton since Newton and Gregory had a dispute over the solution which was eventually settled in Newton’s favour. It is called the Newton Number, N(K) of K and is defined to be the maximum number of non-overlapping translates of K which can touch K at its boundary. The well-known dispute between Sir Isaac Newton and David Gregory concerning this problem, which Newton conjectured to be 12, and Gregory thought to be 13, was ended 180 years later. In 1874, the problem was solved by Hoppe in favour of Newton, i.e., N(\beta^3) = 12. In his proof, the arrangement of 12 unit balls is not unique. This is thought to explain why the problem took 180 years to solve although it is a very natural and a very simple sounding problem. As a generalization of the Newton Number to other convex bodies the blocking number was introduced by C. Zong in 1993. “Another characteristic of mathematical thought is that it can have no success where it cannot generalize.” C. S. Pierce As quoted above, in mathematics generalizations play a very important part. In this thesis we generalize the blocking number to the Unrestricted Blocking Number. Furthermore; we also define the Blocking Number with negative copies and denote it by B_(K). The blocking number not only gives rise to a wide variety of generalizations but also it has interesting observations in nature. For instance, there is a direct relation to the distribution of holes on the surface of pollen grains with the unrestricted blocking number

Publisher: UCL (University College London)
Year: 2010
OAI identifier: oai:eprints.ucl.ac.uk.OAI2:19509
Provided by: UCL Discovery

Suggested articles

Citations

  1. (1991). asp´ ar : Covering a Sphere by Equal Circles, and the Rigidity of its Graph.
  2. (1960). c : Some Coverings of a Spherical Surface with Equal Circles (in
  3. (1991). Guy : Unsolved Problems in Geometry.
  4. (1963). Rogers : Covering a Sphere with Spheres.
  5. (1970). T´ oth : ¨ Uber eine affininvariante Masszahl bei Eipolyedern.
  6. (1969). T´ oth : Kreis¨ uberdeckungen der Sph¨ are.
  7. (1972). T´ oth : Lagerungen in der Ebene auf der Kugel und im Raum, 2nd edition.
  8. (1943). T´ oth : On Covering a Spherical Surface with Equal Spherical Caps (in Hungarian).
  9. (1930). Tammes : On the Origin of Number and Arrangement of the Places of Exit on the Surface of Pollen-grains. Recueil des travaux botaniques n´ eerlandais; 27
  10. (1991). The Observed Form of Coated Vesicles and a Mathematical Covering Problem.
  11. (1948). Theorie der Konvexen Krper.
  12. (1996). Transition from Spherical Circle Packing to Covering: Geometrical Analogues of Chemical Isomerization.
  13. Ungel´ oste und unl´ osbare Probleme der Geometrie.
  14. (1955). utte : ¨ Uberdeckung der Kugel mit h¨ ochstens acht Kreisen.
  15. (1997). Zong : A Problem of Blocking Light Rays. Geometriae Dedicata;
  16. (2000). Zong : On Generalized Kissing Numbers and Blocking Numbers. Rendiconti Del Circolo Mathematico Di Palermo; Serie II,
  17. (1995). Zong : Some Remarks Concerning Kissing Number, Blocking number and covering numbers.
  18. (1996). Zong : Strange Phenomena in Convex and Discrete Geometry.
  19. (2000). Zong : The Blocking Numbers of Convex Bodies.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.