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- W190066204 abstract "A family of translates of the unit cube [ 0 , 1 ) d + T = { [ 0 , 1 ) d + t : t ∈ T } , T ⊂ R d , is called a cube tiling of R d if cubes from this family are pairwise disjoint and ⋃ t ∈ T [ 0 , 1 ) d + t = R d . A non-empty set B = B 1 × ⋯ × B d ⊆ R d is a block if there is a family of pairwise disjoint unit cubes [ 0 , 1 ) d + S , S ⊂ R d , such that B = ⋃ t ∈ S [ 0 , 1 ) d + t and for every t , t ′ ∈ S there is i ∈ { 1 , … , d } such that t i − t i ′ ∈ Z ∖ { 0 } . A cube tiling of R d is blockable if there is a finite family of disjoint blocks B , | B | > 1 , with the property that every cube from the tiling is contained in exactly one block of the family B . We construct a cube tiling T of R 4 which, in contrast to cube tilings of R 3 , is not blockable. We give a new proof of the theorem saying that every cube tiling of R 3 is blockable." @default.
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- W190066204 date "2013-01-01" @default.
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- W190066204 title "On the structure of cube tilings of <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.gif overflow=scroll><mml:msup><mml:mrow><mml:mi mathvariant=double-struck>R</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si2.gif overflow=scroll><mml:msup><mml:mrow><mml:mi mathvariant=double-struck>R</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:…" @default.
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- W190066204 doi "https://doi.org/10.1016/j.jcta.2012.06.007" @default.
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