UoP maths seminar (2013-14)
An analytical theory of Gilbert tessellations
Speaker: James Burridge
Venue: Wednesday 15th Jan 2014, 2pm, LG2.04a (refreshments provided)
Abstract: According to Wikipedia, a Gilbert tessellation is created "when cracks begin to form at a set of points randomly spread throughout the plane. Each crack spreads in two opposite directions along a line through the initiation point, with the slope of the line chosen uniformly at random. The cracks continue spreading at uniform speed until they reach another crack, at which point they stop. A variant of the model restricts the orientations of the cracks to be axis-parallel, resulting in a random tessellation of the plane by rectangles."
Since its appearance in 1967, and until 2013, no closed form analytical properties of any Gilbert style tessellation were found. Over the last two years, we (Richard Cowan and I) have been working on a Gilbert Tessellation which we call the "half rectangular" model, having surprisingly similar properties to the full rectangular model. The "half" model differs from the "full" in that East growing cracks cannot see South, and West cannot see North. Amongst other things, we have found that the average length of a ray in this model is π /Γ(3/4)2. In this talk, I will explain the theory that leads to this result and others, and how it might be applied more broadly.