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. 2020 Jul 3;11:3316. doi: 10.1038/s41467-020-17198-1

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© The Author(s) 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 3 — The solids, arbitrarily scaled, are labeled by their vertex descriptions: the icosahedral dodecahedron (555) and icosahedron (33333), the octahedral cube (444) and octahedron (3333), and the tetrahedral tetrahedron (333), where 5, 4, and 3 represent faces, respectively, 5-gons, 4-gons, and 3-gons. a Arrangement of the icosahedral solids with alignment along five-, three- and twofold rotational axes. b Arrangement of the octahedral solids with alignment along four-, three- and twofold axes. c Arrangement of the tetrahedron with its self-dual, another tetrahedron, with alignment along threefold axes with a face in front, threefold axes with a vertex in front, and twofold axes.