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Crystals with various point group symmetries belong to different crystal classes, which are synonymous terms. Despite being in the same class, crystals may have distinct shapes, like cubes and octahedra. There are 32 three-dimensional point groups, all of which are systematically divided into seven crystal systems.The basic cubic crystal system, exemplified by NaCl, features orthogonal vectors (α = β = �� = 90°) of equal lengths (a = b = c). When specific...
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Ribbon crystals.

Jakob Bohr1, Steen Markvorsen

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Summary
This summary is machine-generated.

Twisting straight ribbons spontaneously forms a crystal-like pattern, resembling tessellated triangles. This ribbon structure optimizes end-to-end length, demonstrating long-range order from incompatible geometries.

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Area of Science:

  • Geometry and Topology
  • Materials Science
  • Complex Systems

Background:

  • Straight ribbons exhibit complex patterns when twisted.
  • Previous models like the Frenet-Serret frame have limitations with singular points.

Purpose of the Study:

  • To describe the spontaneous formation of repetitive, crystal-like patterns in twisted ribbons.
  • To develop a new mathematical framework for analyzing developable ribbons.
  • To explain the underlying physical principles governing ribbon pattern formation.

Main Methods:

  • Utilized a ruled construction defining ribbons by two generating functions.
  • Developed a singular-point-free differentiable frame (the ribbon frame).
  • Modeled the observed pattern using planar triangles and cylindrical arcs.

Main Results:

  • A spontaneous, tessellated, crystal-like pattern emerges from twisting ribbons.
  • The ribbon structure arises from maximizing the ribbon's end-to-end length.
  • Demonstrated the emergence of long-range order from incompatible intrinsic geometries.

Conclusions:

  • The study provides a novel description of developable ribbons and their geometric properties.
  • Ribbon twisting offers a physical model for understanding pattern formation and self-organization.
  • The findings shed light on the interplay between intrinsic geometry and emergent order.