Apollonian Packings in Seven and Eight Dimensions
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In an earlier work, we proposed a generalization for the Apollonian packing in arbitrary dimensions and showed that the resulting object in four, five, and six dimensions have properties consistent with the Apollonian circle and sphere packings in two and three dimensions. In this work, we investigate the generalization in seven and eight dimensions and show that they too have many of the properties of those in lower dimensions. In particular, the hyperspheres are tangent or do not intersect; they fill the hyperspace; the object includes a maximal cluster of mutually tangent hyperspheres; and there exists a perspective where all hyperspheres in the object have integer curvatures.
Apollonius; Apollonian; Circle packing; Sphere packing; Hexlet; Soddy; K3 surface; Ample cone; Lattice; Crystalography
Applied Mathematics | Physical Sciences and Mathematics
Apollonian Packings in Seven and Eight Dimensions.