New Math Model Explains How Random Structures Organize Into Predictable Patterns
Researchers have solved a decades-old puzzle about how random points naturally tessellate into rectangular grids, proving that segment lengths follow predictable statistical patterns. The findings could improve algorithms for network design, materials science, and spatial optimization problems where systems self-organize from random starting conditions.
Originaltitel: Rectangular Gilbert Tessellation
Abstract A random planar quadrangulation process is introduced as an approximation for certain cellular automata in terms of random growth of lines, called rays, from a given set of points. This model turns out to be a particular (rectangular) case of the well-known Gilbert tessellation, which originally models the growth of needle-shaped crystals from the initial random points with a Poisson distribution in a plane. From each point the rays grow on both sides of vertical and horizontal directions until they meet another ray. This process results in a rectangular tessellation of the plane. The central and still open question is the distribution of the length of the line segments in this tessellation. We derive exponential bounds for the tail of this distribution. The correlations between the segment lengths are proved to decay exponentially with the distance between their initial points. Furthermore, the sign of the correlation is investigated for some instructive examples. In the case when the initial set of points is confined in a box $$[0,N]^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , it is proved that the average number of rays reaching the border of the box has a linear order in N .