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- W4317756875 abstract "We have shown that when any finite number n, of line segments with total combined length less than one, have their centers placed randomly inside the unit interval [0,1], the probability of obtaining a mutually disjoint placement of the segments within [0,1], is given by the expression (1−L)n where L=|L1|+⋯+|Ln|, and |Lk| denotes the length of the k-th segment, Lk. The result is established by a careful analysis of the geometry of the event, “all segments disjoint and contained within [0,1],” considered as a subset of the uniform probability space of n centers, each of which is in [0,1]; that is to say, the unit n-cube of Rn. This event has an interesting geometric structure consisting of n! disjoint, congruent, (up to a mirror image) polytopes within the unit n-cube. It is shown these event polytopes fit together perfectly to form, except for a set of measure zero, a partition of an n-dimensional cube with common edge length 1−L, and hence an n-volume given by the formula. In the case of n = 3 segments, the polytopes form one of the known tetrahedral partitions of the cube as discussed, for example in [4 Senechal, M. (1981). Which Tetrahedra Fill Space? Math. Mag. 54(5): 227–243. DOI: 10.1080/0025570X.1981.11976933.[Taylor & Francis Online] , [Google Scholar]]. In fact for all n > 0, the polytopes comprise a partition of the n-dimensional hypercube, and are therefore n-dimensional space filling." @default.
- W4317756875 created "2023-01-23" @default.
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- W4317756875 date "2023-01-01" @default.
- W4317756875 modified "2023-10-02" @default.
- W4317756875 title "Disjoint Placement Probability of Line Segments via Geometry" @default.
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- W4317756875 doi "https://doi.org/10.1080/07468342.2022.2160619" @default.
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