Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local Equivalence between Discrete Curvature and Twist Transformations

Article


Fang, Fang, Clawson, Richard and Irwin, Klee. 2018. "Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local Equivalence between Discrete Curvature and Twist Transformations." Mathematics. 6, pp. 1-19. https://doi.org/10.3390/math6060089
Article Title

Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local Equivalence between Discrete Curvature and Twist Transformations

ERA Journal ID213646
Article CategoryArticle
AuthorsFang, Fang (Author), Clawson, Richard (Author) and Irwin, Klee (Author)
Journal TitleMathematics
Journal Citation6, pp. 1-19
Article Number89
Number of Pages19
Year2018
PublisherMDPI AG
Place of PublicationSwitzerland
ISSN2227-7390
Digital Object Identifier (DOI)https://doi.org/10.3390/math6060089
Web Address (URL)https://www.mdpi.com/2227-7390/6/6/89
Abstract

In geometrically frustrated clusters of polyhedra, gaps between faces can be closed without distorting the polyhedra by the long established method of discrete curvature, which consists of curving the space into a fourth dimension, resulting in a dihedral angle at the joint between polyhedra in 4D. An alternative method—the twist method—has been recently suggested for a particular case, whereby the gaps are closed by twisting the cluster in 3D, resulting in an angular offset of the faces at the joint between adjacent polyhedral. In this paper, we show the general applicability of the twist method, for local clusters, and present the surprising result that both the required angle of the twist transformation and the consequent angle at the joint are the same, respectively, as the angle of bending to 4D in the discrete curvature and its resulting dihedral angle. The twist is therefore not only isomorphic, but isogonic (in terms of the rotation angles) to discrete curvature. Our results apply to local clusters, but in the discussion we offer some justification for the conjecture that the isomorphism between twist and discrete curvature can be extended globally. Furthermore, we present examples for tetrahedral clusters with three-, four-, and fivefold symmetry.

Keywordsquasicrystals; geometric frustration; space packing; tetrahedral packing; discrete curvature; twist transformation
ANZSRC Field of Research 2020499999. Other mathematical sciences not elsewhere classified
Byline AffiliationsQuantum Gravity Research, United States
University of Southern Queensland
Institution of OriginUniversity of Southern Queensland
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