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Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry

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California NanoSystems Institute, Mailcode 722710, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA
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Brain Research Institute, Mailcode 951761, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA
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Department of Psychology, Mailcode 951563, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA
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Department of Chemistry and Biochemistry, Mailcode 951569, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA
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Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium
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Department of Psychology, Mailcode 951563, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA
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Author to whom correspondence should be addressed.
Academic Editor: Egon Schulte
Symmetry 2016, 8(8), 82; https://doi.org/10.3390/sym8080082
Received: 27 July 2016 / Revised: 12 August 2016 / Accepted: 16 August 2016 / Published: 20 August 2016
(This article belongs to the Special Issue Polyhedral Structures)
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. View Full-Text
Keywords: Goldberg polyhedra; cages; fullerenes; tilings; cutouts Goldberg polyhedra; cages; fullerenes; tilings; cutouts
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    Description: Figure S1: Vertex numbers for the first series of new Td polyhedra with 28, 112, 252, 448 and 700 vertices and for the other (leapfrog) series with 84 vertices. Table S1: Coordinates of vertices in the new Td polyhedra, the first series with 28, 112, 252, 448 and 700 vertices, the second (leapfrog) series with 84 vertices. Vertex numbering is shown in Figure S1. Edge lengths are 5 units. Table S2: Internal angles in faces in the new polyhedra. Vertex numbering is shown in Figure S1. The data come directly from Spartan, which provides three angles per vertex, but approximately one angle in each face is duplicated, leaving one other missing. However, the missing angles can be found by taking advantage of the symmetry of the polyhedron and in particular its Schlegel representation in Figure S1. Table S3: Dihedral angles across edges in the new polyhedra. Vertex numbering is shown in Figure S1.
MDPI and ACS Style

Schein, S.; Yeh, A.J.; Coolsaet, K.; Gayed, J.M. Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry. Symmetry 2016, 8, 82. https://doi.org/10.3390/sym8080082

AMA Style

Schein S, Yeh AJ, Coolsaet K, Gayed JM. Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry. Symmetry. 2016; 8(8):82. https://doi.org/10.3390/sym8080082

Chicago/Turabian Style

Schein, Stan, Alexander J. Yeh, Kris Coolsaet, and James M. Gayed. 2016. "Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry" Symmetry 8, no. 8: 82. https://doi.org/10.3390/sym8080082

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