{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,12]],"date-time":"2024-06-12T13:14:14Z","timestamp":1718198054746},"reference-count":17,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2016,8,20]],"date-time":"2016-08-20T00:00:00Z","timestamp":1471651200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"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 \u2265 1). Cages in the second (leapfrog) series have 3 \u00d7 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 \u201cconvex equilateral polyhedra with polyhedral symmetry\u201d. 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\u2019s polyhedral symmetry.<\/jats:p>","DOI":"10.3390\/sym8080082","type":"journal-article","created":{"date-parts":[[2016,8,22]],"date-time":"2016-08-22T14:40:33Z","timestamp":1471876833000},"page":"82","source":"Crossref","is-referenced-by-count":1,"title":["Decoration of the Truncated Tetrahedron\u2014An Archimedean Polyhedron\u2014To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry"],"prefix":"10.3390","volume":"8","author":[{"given":"Stan","family":"Schein","sequence":"first","affiliation":[{"name":"California NanoSystems Institute, Mailcode 722710, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA"},{"name":"Brain Research Institute, Mailcode 951761, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA"},{"name":"Department of Psychology, Mailcode 951563, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA"}]},{"given":"Alexander","family":"Yeh","sequence":"additional","affiliation":[{"name":"Department of Chemistry and Biochemistry, Mailcode 951569, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA"}]},{"given":"Kris","family":"Coolsaet","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium"}]},{"given":"James","family":"Gayed","sequence":"additional","affiliation":[{"name":"Department of Psychology, Mailcode 951563, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA"}]}],"member":"1968","published-online":{"date-parts":[[2016,8,20]]},"reference":[{"key":"ref_1","unstructured":"Cromwell, P.R. 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