Graph Theoretic Analyses of Tessellations of Five Aperiodic Polykite Unitiles

John R. Jungck () and Purba Biswas
Additional contact information
John R. Jungck: Departments of Biological Sciences and Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
Purba Biswas: Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA

Mathematics, 2025, vol. 13, issue 18, 1-19

Abstract: Aperiodic tessellations of polykite unitiles, such as hats and turtles, and the recently introduced hares, red squirrels, and gray squirrels, have attracted significant interest due to their structural and combinatorial properties. Our primary objective here is to learn how we could build a self-assembling polyhedron that would have an aperiodic tessellation of its surface using only a single type of polykite unitile. Such a structure would be analogous to some viral capsids that have been reported to have a quasicrystal configuration of capsomeres. We report on our use of a graph–theoretic approach to examine the adjacency and symmetry constraints of these unitiles in tessellations because by using graph theory rather than the usual geometric description of polykite unitiles, we are able (1) to identify which particular vertices and/or edges join one another in aperiodic tessellations; (2) to take advantage of being scale invariant; and (3) to use the deformability of shapes in moving from the plane to the sphere. We systematically classify their connectivity patterns and structural characteristics by utilizing Hamiltonian cycles of vertex degrees along the perimeters of the unitiles. In addition, we applied Blumeyer’s 2 × 2 classification framework to investigate the influence of chirality and periodicity, while Heesch numbers of corona structures provide further insights into tiling patterns. Furthermore, we analyzed the distribution of polykite unitiles with Voronoi tessellations and their Delaunay triangulations. The results of this study contribute to a better understanding of self-assembling structures with potential applications in biomimetic materials, nanotechnology, and synthetic biology.

Keywords: aperiodic tessellations; graph theory; Hamiltonian cycles; polykites; chirality; periodicity; Heesch number; corona; surround number; Voronoi tessellations; Delaunay triangulations; Pitteway violations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/18/2982/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/18/2982/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:18:p:2982-:d:1749772

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().