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Fragmenting any Parallelepiped into a Signed Tiling.

Joseph Doolittle1, Alex McDonough2

  • 1Institute of Geometry, TU Graz, Graz, Austria.

Discrete & Computational Geometry
|September 22, 2025
PubMed
Summary
This summary is machine-generated.

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Researchers developed a new method for tiling space using parallelepipeds, allowing for negative volumes and creating signed tilings. This advancement ensures a consistent net count of positive and negative volume tiles across space.

Area of Science:

  • Geometry
  • Tiling Theory
  • Higher-Dimensional Spaces

Background:

  • Parallelepipeds traditionally tile space through translation.
  • Previous work on sandpile groups introduced a method to fragment parallelepipeds into smaller tiles.
  • This earlier construction was limited to cases where determinant expansion components were non-negative.

Purpose of the Study:

  • To generalize the parallelepiped tiling construction to all cases, removing the non-negative condition.
  • To introduce and define the concept of 'signed tiling' using tiles with positive and negative volumes.
  • To prove properties of these generalized signed tilings.

Main Methods:

  • Extending a novel construction for fragmenting parallelepipeds.
  • Introducing tiles with negative volumes, interpreted as cancellations.
Keywords:
Determinant expansionParallelepipedPeriodic tilingSigned tiling

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  • Proving that the net number of signed tiles remains constant as a point moves through space.

    • Main Results:

    • A generalized construction works for all parallelepipeds, not just those with non-negative determinant components.
    • The concept of signed tiling is established, where negative volumes cancel positive ones.
    • It is proven that every point in space is covered by exactly one more positive-volume tile than negative-volume tiles.

    Conclusions:

    • The study successfully extends parallelepiped tiling to include negative volumes, defining a signed tiling.
    • A key invariant property of these signed tilings is demonstrated: the net count of tiles is constant.
    • The underlying geometric structure of these generalized tilings remains an open area for further investigation.