Abstract
The Goldberg construction of symmetric cages involves pasting a patch cut out of a regular tiling onto the faces of a Platonic host polyhedron, resulting in a cage with the same symmetry as the host. For example, cutting equilateral triangular patches from a 6.6.6 tiling of hexagons and pasting them onto the full triangular faces of an icosahedron produces icosahedral fullerene cages. Here we show that pasting cutouts from a 6.6.6 tiling onto the full hexagonal and triangular faces of an Archimedean host polyhedron, the truncated tetrahedron, produces two series of tetrahedral (Td) fullerene cages. Cages in the first series have 28n2 vertices (n ≥ 1). Cages in the second (leapfrog) series have 3 × 28n2. We can transform all of the cages of the first series and the smallest cage of the second series into geometrically convex equilateral polyhedra. With tetrahedral (Td) symmetry, these new polyhedra constitute a new class of “convex equilateral polyhedra with polyhedral symmetry”. We also show that none of the other Archimedean polyhedra, six with octahedral symmetry and six with icosahedral, can host full-face cutouts from regular tilings to produce cages with the host’s polyhedral symmetry.
Highlights
Known to the ancient Greeks, the five Platonic polyhedra and 13 Archimedean polyhedra are the first two classes of convex equilateral polyhedra with polyhedral symmetry [1]
A fourth class, the “Goldberg polyhedra”, was recently described [3]. These are primarily icosahedral fullerene cages transformed into geometrically convex equilateral polyhedra—which necessarily have convex planar faces [4,5]
Preceding the work on icosahedral viruses by Caspar and Klug [6], Goldberg’s construction of of icosahedral fullerene cages employed decoration of the full equilateral triangular faces of a host icosahedral fullerene cages employed decoration of the full equilateral triangular faces of a host icosahedron (Figure triangularcutouts cutoutsfrom from a tiling of regular hexagons icosahedron (Figure left)with withequilateral equilateral triangular a tiling of regular hexagons
Summary
Known to the ancient Greeks, the five Platonic polyhedra and 13 Archimedean polyhedra are the first two classes of convex equilateral polyhedra with polyhedral symmetry (icosahedral, octahedral or tetrahedral) [1]. These geometrically icosahedral, octahedral or tetrahedral cages are equilateral, with regular which contains 7 vertices, becomes a spherical triangle with 7 vertices on the cage These geometrically small faces (5‐gons, 4‐gons or 3‐gons)cages but nonplanar. The triangular cutouts of the tiling of hexagons can be different sizes and orientations, with the span 1, 2 or 3 triangles (Figure 1b) These size and orientation variants can be expressed by indices pattern continuing across borders triangular of adjacent triangular and across the gaps (h,k) still that neatly characterize one side of thethe equilateral cutout, wherefaces h indicates steps Square cutouts in a variety of sizes and orientations from a tiling of squares can decorate the full square faces of a cube to produce octahedral cages with 4-gons and eight triangles [10]. Cages we can produce geometrically convex equilateral polyhedra with Td symmetry, creating another class of and convex equilateral polyhedra with polyhedral symmetry
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call