To pack more into the same space you have to break bigger things down into smaller things. Is this robustly true? I imagine one gallon of flour weighs more than one gallon of wheat grains.
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Fun tangentially related fact; there’s no antonym of dense
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Sparse
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Look up "bulking porosity" in mining engineering.
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Deep cut there
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There used to be an opposite: “rare”. But no one uses it in that sense anymore. And your intuition is right: dense packing requires uniform size (eg gravel) or compatible / tessalationable shape (eg shipping packages) or both.
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A good demonstration is that jar of peanuts you’ve had rattling in the cupboard for years. It has distinct layers of material: salt and peanut dust caked on the bottom, then a layer of small packed chips, then larger random chunks and then the whole nuts.
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Not according to Banach-Tarski :-) (You need a constraint like breaking into “measurable” “random” chunks...)
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