Rational tilings by 𝑛-dimensional crosses

Abstract

Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold e 1 comma ellipsis comma bold e Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">e</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">e</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {e}}_1}, \ldots ,{{\mathbf {e}}_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and whose centers are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i bold e Subscript j"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">e</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">i{{\mathbf {e}}_j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-slanted-equals StartAbsoluteValue i EndAbsoluteValue less-than-or-slanted-equals k"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>i</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0 \leqslant |i| \leqslant k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional Euclidean space. The union of these cubes is called a cross. This cross consists of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 k n plus 1"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2kn + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> cubes; a central cube together with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 n"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> arms of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A family of translates of a cross whose union is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional Euclidean space and whose interiors are disjoint is a tiling. Denote the set of translation vectors by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">L</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {L}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If the vector set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">L</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {L}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a vector lattice, then we say that the tiling is a lattice tiling. If every vector of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">L</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {L}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has rational coordinates, then we say that the tiling is a rational tiling, and, similarly, if every vector of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">L</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {L}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has integer coordinates, then we say that the tiling is an integer tiling. Is there a noninteger tiling by crosses? In this paper we shall prove that if there is an integer lattice tiling by crosses, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 k n plus 1"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2kn + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not a prime, and if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than k"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p &gt; k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every prime divisor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 k n plus 1"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2kn + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then there is a rational noninteger lattice tiling by crosses and there is an integer nonlattice tiling by crosses. We will illustrate this in the case of a cross with arms of length 2 in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="55"> <mml:semantics> <mml:mn>55</mml:mn> <mml:annotation encoding="application/x-tex">55</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional Euclidean space. Throughout, the techniques are algebraic.

Locations

• Proceedings of the American Mathematical Society - View - PDF