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- W3203068541 abstract "Abstract We regard a smooth, $$d=2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> -dimensional manifold $$mathcal {M}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>M</mml:mi> </mml:math> and its normal tiling M , the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by $$bar{v}^{star }$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mo>⋆</mml:mo> </mml:msup> </mml:math> and we prove that if M is periodic then $$bar{v}^{star } ge 2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msup> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mo>⋆</mml:mo> </mml:msup> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . We show the same result for the monohedral case by an entirely different argument. Our theory also makes a closely related prediction for non-periodic tilings. In 3 dimensions we show a monohedral construction with $$bar{v}^{star }=0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msup> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mo>⋆</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ." @default.
- W3203068541 created "2021-10-11" @default.
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- W3203068541 date "2022-06-03" @default.
- W3203068541 modified "2023-10-05" @default.
- W3203068541 title "A two-vertex theorem for normal tilings" @default.
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- W3203068541 doi "https://doi.org/10.1007/s00010-022-00888-0" @default.
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