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%I #4 Mar 30 2012 18:40:49
%S 4,6,7,9,10,11,13,14,15,17,18,19,21,22,23,24
%N Floor of the volumes of the first sixteen Lobell polyhedra.
%C This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angled objects, until it is a disjoint union of Lobell polyhedra, a class of polyhedra which generalizes the dodecahedron. Furthermore, these combinatorial operations are shown to have geometric realizations which are volume decreasing. This allows for an organization of the volumes of right-angled hyperbolic polyhedra and allows, in particular, the determination of the polyhedra with smallest and second-smallest volumes.
%H Taiyo Inoue, <a href="http://arxiv.org/abs/0809.2111">Organizing Volumes of Right-Angled Hyperbolic Polyhedra</a>, arXiv:0809.2111 [math.GT]
%F a(n) = Floor[vol(L(n))].
%e n..|.vol(L(n))
%e ==============
%e 5..|.4.306....
%e 6..|.6.023....
%e 7..|.7.563....
%e 8..|.9.019....
%e 9..|10.426....
%e 10.|11.801....
%e 11.|13.156....
%e 12.|14.494....
%e 13.|15.822....
%e 14.|17.140....
%e 15.|18.452....
%e 16.|19.758....
%e 17.|21.059....
%e 18.|22.356....
%e 19.|23.651....
%e 20.|24.943....
%e ==============
%K nonn
%O 5,1
%A _Jonathan Vos Post_, Sep 14 2008
%E Replaced link to cached arXiv URL by the permanent version - _R. J. Mathar_, Mar 01 2010
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