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Generating Closed Shapes from Regular Tilings

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William O. J. Boo and Daniell L. Mattern

Department of Chemistry and Biochemistry, University of Mississippi, University, MS 38677

J. Chem. Educ., 2002, 79 (8), p 1017

DOI: 10.1021/ed079p1017

Publication Date (Web): August 1, 2002

Abstract

Closed geometrical shapes may be obtained from a hexagonal tiling by substituting 12 pentagons for hexagons and fusing the appropriate edges. Such geometrical shapes describe the fullerenes, demonstrating that this mathematical rule has utility in structural chemistry. Closure from a hexagonal tiling may also be obtained by substituting six squares; or four triangles; or combinations of pentagons (one point each), squares (two points each), and triangles (three points each) so that the total is 12 points. Similar recipes exist for obtaining closed shapes from tetragonal or trigonal tilings. Structures thus obtained may evolve into additional structures having the same symmetry in a manner similar to the evolution of the Archimedean solids from the Platonic solids. Important examples include the deltahedra, a group of eight convex polyhedra consisting of all triangles and belonging to high-symmetry point groups. These polyhedra evolve into additional high symmetry shapes, many of which have already found use in describing the structures of molecules, ions, and clusters.

Keywords (Audience):

Upper-Division Undergraduate

Keywords (Subject):

Group Theory / Symmetry

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This article has been cited by 1 ACS Journal articles (1 most recent appear below).

Concomitant Ordering and Symmetry Lowering
William O. J. Boo and Daniell L. Mattern
Journal of Chemical Education2008 85 (5), 710
- Concomitant Ordering and Symmetry Lowering
  William O. J. Boo and Daniell L. Mattern
  Journal of Chemical Education2008 85 (5), 710
  Examples of concomitant ordering include magnetic ordering, Jahn–Teller cooperative ordering, electronic ordering, ionic ordering, and ordering of partially-filled sites. Concomitant ordering sets in when a crystal is cooled and always lowers the degree ...
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