CROATICA CHEMICA ACTA CCACAA 77 (1–2) 133¿140 (2004) ISSN-0011-1643 CCA-2910 Original Scientific Paper Regular Polytopes, Root Lattices, and Quasicrystals* R. Bruce King Department of Chemistry, University of Georgia, Athens, Georgia 30602, USA (E-mail: [email protected].) RECEIVED APRIL 1, 2003; REVISED JULY 8, 2003; ACCEPTED JULY 10, 2003 The icosahedral quasicrystals of five-fold symmetry in two, three, and four dimensions are re- lated to the corresponding regular polytopes exhibiting five-fold symmetry, namely the regular { } pentagon (H2 reflection group), the regular icosahedron 3,5 (H3 reflection group), and the { } regular four-dimensional polytope 3,3,5 (H4 reflection group). These quasicrystals exhibit- ing five-fold symmetry can be generated by projections from indecomposable root lattices with ® ® ® twice the number of dimensions, namely A4 H2, D6 H3, E8 H4. Because of the subgroup Ì Ì ® relationships H2 H3 H4, study of the projection E8 H4 provides information on all of the possible icosahedral quasicrystals. Similar projections from other indecomposable root lattices Key words can generate quasicrystals of other symmetries. Four-dimensional root lattices are sufficient for polytopes projections to two-dimensional quasicrystals of eight-fold and twelve-fold symmetries. How- root lattices ever, root lattices of at least six-dimensions (e.g., the A6 lattice) are required to generate two- quasicrystals dimensional quasicrystals of seven-fold symmetry. This might account for the absence of icosahedron seven-fold symmetry in experimentally observed quasicrystals. INTRODUCTION but with point symmetries incompatible with traditional crystallography. More specifically, ordered solid state The symmetries that cannot be exhibited by true crystal quasicrystalline structures with five-fold symmetry ex- lattices include five-fold, eight-fold, and twelve-fold sym- hibit quasiperiodicity in two dimensions and periodicity metry. For this reason crystallographers were startled by in the third. the discovery by Shechtman and coworkers1 that rapidly Quasicrystals are seen to exhibit lower order than true solidified aluminum-manganese alloys of approximate crystals but a higher order than truly amorphous materials. composition Al6Mn gave electron diffraction patterns The order in icosahedral quasicrystals, i.e., quasicrystals having apparently sharp spots and five-fold symmetry exhibiting five-fold symmetry, can formally be described axes. Subsequent work2 led to the discovery of a number in six-dimensional hyperspace in which the atoms are of related alloys, particularly aluminum-rich alloys, ex- three-dimensional subspaces. The actual icosahedral quasi- hibiting five-fold and other crystallographically forbid- crystal structures are then three-dimensional projections den symmetries. These materials are now called quasi- of this six-dimensional hyperspace.7,8,9 crystals.3,4,5,6 Such quasicrystals are seen to represent a The use of such projection models to describe quasi- new type of incommensurate crystal structure whose Fourier crystal structures requires an understanding of higher di- transform consists of a d function as for periodic crystals mensional crystal lattices from which such projections to * This paper is dedicated to Nenad Trinajsti} in recognition of his pioneering contributions to mathematical chemistry. 134 R. B. KING two or three dimensions can be made. The simplest such TABLE I. Properties of the regular (Platonic) polyhedra higher dimensional lattices are the primitive hypercubic n Polyhedron Face Vertex Number Number Number lattices Z , which are generated from all integral linear type degrees of edges of faces of vertices combinations of unit vectors along n orthogonal Carte- Tetrahedron Triangle 3 6 4 4 sian axes. However, in many cases more complicated higher {3,3} dimensional lattices are required, such as various types Octahedron Triangle 4 12 8 6 of centered hypercubic lattices. These higher dimensional {3,4} lattices correspond to the root lattices, which are gener- Cube {4,3} Square 3 12 6 8 ated from all integral linear combinations of the vectors Icosahedron Triangle 5 30 20 12 { } (roots) of so-called root systems.10,11 These root systems 3,5 form a certain class of vector stars with specific allowed Dodecahedron Pentagon 3 30 12 20 {5,3} symmetries, lengths, and angles.10,11,12 Such root sys- tems arise in different contexts such as crystallographic finite reflection groups10,12 or finite-dimensional semi- n-dimensional polytope is a connected set (»complex«) simple Lie algebras.11 The connection between root lat- of facets of each dimension less than n where the 0-di- tices and quasicrystals was first presented in a short let- mensional facets are called vertices, the 1-dimensional ter by Baake, Joseph, Kramer, and Schlottmann13 with facets are called edges, the 2-dimensional facets are particular attention to two-dimensional quasilattices hav- called faces, the 3-dimensional facets are called cells, ing rotational symmetries of orders 5, 8, 10, and 12. etc. Such a polytope is a regular polytope when the com- This paper extends this connection between root lat- ponent facets of each dimension are identical regular tices and quasicrystals to the important three-dimensional polytopes and all facets of each dimension are equiva- case of icosahedral quasicrystals with a more detailed lent (i.e., transitive as discussed below). In two-dimen- survey of the key areas of mathematics involved in this sions the regular polytopes are the infinite number of connection. In addition the existence of root lattices is regular polygons whereas in three dimensions the regu- related to the existence of regular polytopes in higher di- lar polytopes are the five regular »Platonic« polyhedra mensional spaces, which is also examined in this paper. (Figure 1 and Table I). The concept of a tessellation is useful for generating REGULAR POLYTOPES FROM REGULAR regular polytopes of low dimension as well as related lat- TESSELLATIONS tices. In this connection, embedding a network of poly- gons into a surface can be described as a tiling or tessella- The term polytope generalizes the two-dimensional con- tion of the surface.15 In a formal sense a tiling or tessella- cept of polygon and the three-dimensional concept of 14 tion of a surface is a countable family of closed sets T = polyhedron to any number of dimensions. Thus an { } T1,T2… which cover the surface without gaps or over- laps. More explicitly, the union of the sets T1,T2… (which are known as the tiles of T) is the whole surface and the interiors of the sets Ti are pairwise disjoint. In the tessella- tions of interest in this paper, the tiles are the polygons, which, in the case of tessellations corresponding to poly- hedra, are the faces of the polyhedra. Tessellations can be described in terms of their flags, where a flag is a triple (V, E, F) consisting of a vertex V, and edge E, and a face F which are mutually incident. A tiling T is called regular if its symmetry group G(T) is transitive on the flags of T. A regular tessellation consisting of q regular p-gons at each vertex can be described by the so-called Schäfli no- tation {p,q}. The Schäfli notation can be generalized to higher dimensions in the obvious way. First consider regular tessellations in the Euclidean plane, i.e., a flat surface of zero curvature. In the Euclidean plane, the angle of a regular p-gon, {p},is(1–2/p)ð; hence q equal {p}'s (of any size) will fit together around a common vertex if this angle is equal to 2ð/q, leading to the relationship Figure 1. The five regular »Platonic« polyhedra showing their Schäfli symbols and the dual pairs. Note that the tetrahedron is self-dual. (p – 2)(q –2)=4 (1) Croat. Chem. Acta 77 (1–2) 133–140 (2004) REGULAR POLYTOPES, ROOT LATTICES, AND QUASICRYSTALS 135 (ii) Dual polyhedra have the same symmetry elements and thus belong to the same symmetry point group. (iii) Dualization of the dual of a polyhedron leads to the original polyhedron. (iv) The degrees of the vertices of the polyhedron cor- respond to the number of edges in the corresponding face polygons of its dual. Thus the duals of the deltahedra are trivalent polyhedra, i.e., polyhedra in which all vertices are of degree 3. (v) The dual of a regular polyhedron {p,q} in the Schäfli notation is {q,p}. The dual pairs for the Platonic polyhedra consist of the Figure 2. The three regular tessellations of the plane, namely the cube/octahedron and dodecahedron/icosahedron (Figure 1). self-dual {4,4} checkerboard tessellation and the {6,3}«{3,6} dual pair. The tetrahedron is self-dual, i.e., the dual of a tetrahedron is another tetrahedron. Also the concept of duality can be extended from standard polyhedra embedded in the surface There are only three integral solutions of equation of a sphere to polygonal networks embedded in surfaces of (1), which lead to the three regular tessellations of the non-zero genus. In this connection the author has studied plane {4,4}, {6,3}, and {3,6} depicted in Figure 2. The the Dyck and Klein tessellations of genus 3 surfaces with tessellation {6,3} is familiar in chemistry as the struc- octagons and heptagons, respectively.17,18,19,20,21 Such Pla- ture of graphite whereas the tessellation {4,4} corre- tonic tessellations and their duals are of interest in connec- sponds to the checkerboard. tion with possible structures of zeolite-like carbon and bo- Now consider the regular tessellations of the sphere ron nitride allotropes with negative curvature. that correspond to the regular polyhedra (Figure 1). The Now consider regular polytopes of higher dimension angle of a regular spherical polygon {p} is greater than (n ³ 3). In any dimension there are always the following (1–2/p)ð and gradually increases from this value to ð three regular polytopes: when the circumradius increases from 0 to ð/2.