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This study visualizes 2-periodic self-dual tilings, including all uninodal and binodal types, and selected trinodal structures. Many of these complex mathematical and physical tiling patterns are presented here for the first time.

Keywords:
netsself-dual tilings

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Area of Science:

  • Combinatorial Tiling Theory
  • Mathematical Physics
  • Crystal Chemistry

Background:

  • 2-Periodic self-dual tilings are significant across diverse scientific disciplines.
  • Previous work has identified 1 uninodal, 5 binodal, and 62 trinodal self-dual tilings through systematic enumeration.
  • A gap exists in the visualization and detailed description of these identified structures.

Purpose of the Study:

  • To visually present all known uninodal and binodal 2-periodic self-dual tilings.
  • To illustrate selected examples of trinodal 2-periodic self-dual tilings.
  • To introduce novel tiling structures to the scientific community.

Main Methods:

  • Systematic enumeration using combinatorial tiling theory.
  • Illustrative representation of identified tiling structures.
  • Descriptive analysis of tiling properties.

Main Results:

  • Comprehensive visualization of all uninodal 2-periodic self-dual tilings.
  • Complete illustration of all binodal 2-periodic self-dual tilings.
  • Selected detailed descriptions of trinodal 2-periodic self-dual tilings.

Conclusions:

  • This work provides the first detailed illustrations for most of the described uninodal, binodal, and trinodal self-dual tilings.
  • The presented visualizations enhance the understanding of these complex structures in fields like crystal chemistry and mathematical physics.
  • This research expands the accessible knowledge base of self-dual tiling configurations.