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is smaller than that for any other convex body. That is, according to the conjecture, the ball is the convex solid which forces the largest fraction of space to remain empty in its optimal packing structure. This conjecture is therefore related to the
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columns that Ulam communicated this conjecture to him in 1972. Though the original reference to the conjecture states only that Ulam "suspected" the ball to be the worst case for packing, the statement has been subsequently taken as a conjecture.
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bodies, the ball constitutes a local maximum of the fraction of empty space forced. That is, any point-symmetric solid that does not deviate too much from a ball can be packed with greater efficiency than can balls.
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The analog of Ulam's packing conjecture in two dimensions would say that no convex shape forces more than ≈9.31% of the plane to remain uncovered, since that is the fraction of empty space left uncovered in the
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force the largest fraction of the plane to remain uncovered. In dimensions above four (excluding 8 and 24), the situation is complicated by the fact that the analogs of the
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Numerical experiments with a large variety of convex solids have resulted in each case in the construction of packings that leave less empty space than is left by
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Is there any three-dimensional convex body with lower packing density than the sphere?
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The optimal packing of spheres, leaving an average empty space of ≈25.95%
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Kallus, Yoav (2014), "The 3-ball is a local pessimum for packing",
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This conjecture was attributed posthumously to Ulam by
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give counter-examples. It is conjectured that regular
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291:Kallus, Yoav (2015), "Pessimal packing shapes",
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165:New Mathematical Diversions (Revised Edition)
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183:de Graaf, Joost; van Roij, René;
49:highest possible packing density
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105:close-packing of equal spheres
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362:Unsolved problems in geometry
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119:Analogs in other dimensions
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269:10.1016/j.aim.2014.07.015
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65:packing congruent spheres
41:Ulam's packing conjecture
161:Gardner, Martin (1995),
126:densest packing of disks
294:Geometry & Topology
245:Advances in Mathematics
190:Physical Review Letters
128:. However, the regular
318:10.2140/gt.2015.19.343
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55:in three-dimensional
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99:Supporting arguments
213:2011PhRvL.107o5501D
185:Dijkstra, Marjolein
91:Mathematical Games
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142:Kepler conjecture
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352:Conjectures
301:: 343–363,
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341:Categories
148:References
308:1305.0289
259:1212.2551
204:1107.0603
138:heptagons
229:22107298
357:Spheres
327:3318753
278:3250288
209:Bibcode
130:octagon
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80:Origin
72:about
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