Lognormality and Triangles of Unit Area
Steven R. Finch
TL;DR
This paper explores methods for generating triangles of unit area using random length products, leveraging the lognormal distribution's properties, and investigates the underlying probabilistic structures.
Contribution
It introduces a novel approach to generate triangles of unit area based on the lognormal distribution's multiplicative closure, extending classical stick-breaking methods.
Findings
Lognormal distribution facilitates modeling of triangle area generation.
Proposed methods relate to the multiplicative properties of random lengths.
The study provides partial answers to the problem of uniform triangle area generation.
Abstract
To generate a triangle of unit perimeter, break a stick of length 1 in two places at random, with the condition that triangle inequalities are satisfied. Is there a similarly natural method for generating triangles of unit area? Study of a product (rather than a sum) of random lengths is facilitated by closure of multiplication within the lognormal family of distributions. Our (necessarily incomplete) answers to the question each draw upon this property.
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Taxonomy
TopicsPoint processes and geometric inequalities · Random Matrices and Applications · Bayesian Methods and Mixture Models