Neighborly translational tessellations of the n-torus

Abstract The concept of an n -NTT (neighborly translational tessellation of the n -torus) is introduced as a tessellation where every pair of tiles are translates of each other, and share precisely one of their facets. An n -NTT with cubic tiles is studied for each $$n \in \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , and particular attention is given to a 4-NTT whose tiles are isometric 24-cells. We also use this concept to describe a tessellation of $$\mathbb {E}^4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:math> with isometric tiles with fractal boundary, as well as a NTT of an infinite-dimensional space.