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Three Cubes Packing for All Dimensions
Version 1
: Received: 22 April 2024 / Approved: 22 April 2024 / Online: 22 April 2024 (18:18:39 CEST)
How to cite: Adamko, P. Three Cubes Packing for All Dimensions. Preprints 2024, 2024041460. https://doi.org/10.20944/preprints202404.1460.v1 Adamko, P. Three Cubes Packing for All Dimensions. Preprints 2024, 2024041460. https://doi.org/10.20944/preprints202404.1460.v1
Abstract
Let Vn(d) denote the least number such that every collection of nd-cubes with total volume 1 in d-dimensional (Euclidean) space can be packed parallelly into some d-box of volume Vn(d). We show that V3(d)=r1−dd if d≥11 and V3(d)=1r+1rd+1r−rd+1 if 2≤d≤10, where r is the only solution of the equation 2(d−1)kd+dkd−1=1 on 22,1 and (k+1)d(1−k)d−1dk2+d+k−1=kddkd+1+dkd+kd+1 on 22,1, respectively. The maximum volume is achieved by hypercubes with edges x, y, z such that x=2rd+1−1/d, y=z=rx if d≥11, and x=rd+(1r−r)d+1−1/d, y=rx, z=(1r−r)x if 2≤d≤10. We also proved that only for dimensions less than 11 there are two different maximum packings, and for all dimensions greater than 10 the maximum packing has the two smallest cubes the same.
Keywords
packing of cubes; extreme
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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