Disjoint Placement Probability of Line Segments via Geometry. (1st January 2023)

• Record Type: Journal Article
• Authors: Ennis, Christopher; Shier, John
• Is Part Of: College mathematics journal. Volume 54:Number 1(2023)
• Page Start: 44
• Page End: 53
• Publication Date: 2023-01-01
• Subjects: Mathematics -- Study and teaching -- Periodicals; Mathematics -- Periodicals; Mathematics; Mathematics -- Study and teaching; Periodicals; 510.071
• DOI: 10.1080/07468342.2022.2160619
• Languages: English
• ISSNs: 0746-8342

Abstract

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 = | L 1 | + ⋯ + | L n |, and | L k | 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 R n . 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 ]. In fact for all n > 0, the polytopes comprise a partition of the n -dimensional hypercube, and are therefore n -dimensional space filling .