Title :
A balloon net discovering improved solutions in one of unsolved problems in geometry: a problem of spreading points in a unit square
Author :
Fujisawa, Kimiya ; Takefuji, Yoshiyasu
Author_Institution :
Graduate Sch. of Media & Governance, Keio Univ., Fujisawa, Japan
Abstract :
A balloon net model is introduced and demonstrated for discovering improved solutions in one of unsolved problems in geometry which is referred to as a problem of “spreading points in a square”. How should n points be arranged in a unit square so that the minimum distance between them is the greatest? Note that d(n) is the greatest possible minimum distance between n points in a unit square. Exact results are known for n⩽9 and n=14, 16, 25, and 36. Many investigators including Schaer, Meir, Kirchner, Wengerodt, Goldberg, Schluter, Valette and others have studied this geometrical problem for many years. The best known result is summarized in the book of “Unsolved Problems in Geometry” (H.T. Croft, K.J. Falconer and R.K. Guy (1991)). We have found the improved solutions for n=13 and n=15 by using the proposed algorithm
Keywords :
geometry; mathematics computing; neural nets; optimisation; balloon net model; geometry; minimum distance; motion equation; neural network; points spreading problem; unit square; Artificial neural networks; Geometry; Neurons; Nonlinear dynamical systems; Nonlinear equations; Solid modeling; Vectors;
Conference_Titel :
Neural Networks, 1995. Proceedings., IEEE International Conference on
Conference_Location :
Perth, WA
Print_ISBN :
0-7803-2768-3
DOI :
10.1109/ICNN.1995.487703
