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Packing lines, planes, etc.: Packings in Grassmannian spaces. (English) Zbl 0864.51012
Exp. Math. 5, No. 2, 139-159 (1996); editor’s note ibid. 6, No. 2, 175 (1997).
Summary: We address the question: How should \(N\) \(n\)-dimensional subspaces of \(m\)-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of \(N\), \(n\), \(m\) are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe \(n\)-dimensional subspaces of \(m\)-space as points on a sphere in dimension \({1\over 2}(m-1)(m+2)\), which provides a (usually) lower-dimensional representation than the Plücker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multi-dimensional data via Asimov’s grand tour method.

MSC:
51M20 Polyhedra and polytopes; regular figures, division of spaces
14M15 Grassmannians, Schubert varieties, flag manifolds
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
Software:
XGobi
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