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Notes

  1. The file extension .txt enables the file to be read in any text editor. To use the file in OpenSCAD, remove the extension to make it a .scad file.

References

  1. H. S. M. Coxeter, P. Du Val, H. T. Flather, and J. F. Petrie. The fifty-nine icosahedra. University of Toronto Studies (Mathematical Series) 6 (1938), 1–26.

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  2. H. M. Cundy and A. P. Rollett. Mathematical Models, 3rd ed. Tarquin Publications, 1981.

  3. J. D. Ede. Rhombic triacontahedra. Math. Gazette 41 (1957), 189–194.

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  4. George Hart. Stellations of the rhombic triacontahedron. Available online at https://www.georgehart.com/virtual-polyhedra/srtc-info.html.

  5. P. W. Messer. Stellations of the rhombic triacontahedron and beyond. Structural Topology 21 (1995) 25–46.

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  6. G. S. Pawley. The 227 triacontahedra. Geometriae Dedicata 4 (1975) 221–232.

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  7. P. Freire. Stellations of the rhombic triacontahedron. Available online at https://www3.mpifr-bonn.mpg.de/staff/pfreire/polyhedra/rhombic_triacontahedron.htm.

  8. E. Pegg. Triacontahedron stellations. Available online at https://demonstrations.Wolfram.com/TriacontahedronStellations.

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Acknowledgments

I wish to thank the referee for a very diligent effort, especially in finding details on the web that had eluded me. Consequently, I have added one journal reference and three web links to the reference list. George Hart’s web page [4] states that there are \(358\,833\,072\) stellations of the triacontahedron, where he has waived the rule I introduced in 1975 that forbids reentrant polyhedra; these were allowed by Coxeter et al. [1] for the icosahedron. This huge number was produced by J. A. Gingrich and reported personally to P. W. Messer [5], who gives no detail as to how it was determined, though Gingrich later reduced the number to \(155\,014\,690\) by a stricter interpretation of the rules. The classic stellations of the triacontahedron involve 29 building blocks, and using the first 15, we can get a small number of reentrant polyhedra. Using the next 14 would give a number somewhat less than \(14!=8\,7178\,291\,200\), which is a large number indeed. In this work, I have refrained from making a list of rules, that being a temptation for enumeration, and it leaves a good number yet to be discovered, to be constructed rather than listed. I thought that these new stellations might already be out there on the web, so I was both surprised and delighted when the referee’s findings in fact contrasted with this work. Paulo Freire [7] and Ed Pegg, Jr. [8], have websites, and links to them have been added to the current reference list. Freire’s work shows these stellations constructed in a totally different way, focusing on the intersection lines rather than the facets. Pegg’s work shows a few fully symmetric classic stellations that can be manipulated orientationally online, and seeing this could well inspire a reader to become active with OpenSCAD and work with the files here on offer.

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Authors and Affiliations

  1. Emeritus Professor of Computational Physics, University of Edinburgh, Edinburgh, Scotland, UK

    G. Stuart Pawley

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  1. G. Stuart Pawley
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Correspondence to G. Stuart Pawley.

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—To the memory of John Conway

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Supplementary Information

Appendix

Appendix

A number of supplementary files are available for anyone to download (see the section “Supplementary Information” below). The first, the_227.pdf, gives the facets for all the established 226 stellations of the triacontahedron. I have introduced an arbitrary numbering for convenience. There is also a corrected diagram of the building blocks for these stellations. The names for these stellations consist of the index letters for the visible building blocks. When one stellation is selected, the second file, all_227.scad.txt,Footnote 1 can be edited to make its diagram, where one should choose the left- or right-handed program line for each block as appropriate, or both for full symmetry. This file begins with a table of 226 numbered stellations. There are 35 blocks available, so \(35! \approx 10^{40}\) programs can be produced, with the added attraction that blocks can be chosen to be somewhat transparent.

A PDF file with a number of templates is given for those keen and impatient, temp_triacontahedra.pdf, and for those wishing to search for more stellations of the new type, galaxy.scad.txt is available. Here again, you edit by activating a program line by removing the comment directive, selecting the areas for your chosen facet. To get the program software to accept a file, it must be given the file extension .scad.

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Pawley, G.S. New Stellations. Math Intelligencer 43, 108–114 (2021). https://doi.org/10.1007/s00283-021-10070-0

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