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

Volume

5

fYear

1995

fDate

Nov/Dec 1995

Firstpage

2208

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;

fLanguage

English

Publisher

ieee

Conference_Titel

Neural Networks, 1995. Proceedings., IEEE International Conference on

Conference_Location

Perth, WA

Print_ISBN

0-7803-2768-3

Type

conf

DOI

10.1109/ICNN.1995.487703

Filename

487703