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Deltahedron
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<h2 id="strictly-convex-deltahedron">Strictly convex deltahedron</h2> <p>A polyhedron is said to be <i>convex</i> if a line between any two of its vertices lies either within its interior or on its boundary, and additionally, if no two faces are <a href="/facts/Coplanarity/dSTa1bcb">coplanar</a> (lying in the same plane) and no two edges are <a href="/facts/Collinearity/SxbVBJYR">collinear</a> (segments of the same line), it can be considered as being strictly convex.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>Of the eight <a href="/facts/Convex_set/vdAuJRJl">convex</a> deltahedra, three are <a href="/facts/Platonic_solid/jraREXFD">Platonic solids</a> and five are <a href="/facts/Johnson_solid/8SR7CyNz">Johnson solids</a>. They are:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <ul><li><a href="/facts/Regular_tetrahedron/7mkSrl6j">regular tetrahedron</a>, a pyramid with four equilateral triangles, one of which can be considered the base.</li> <li><a href="/facts/Triangular_bipyramid/fU4kIBTV">triangular bipyramid</a>, <a href="/facts/Regular_octahedron/G38q92xW">regular octahedron</a>, and <a href="/facts/Pentagonal_bipyramid/Mt3bUSPI">pentagonal bipyramid</a>; <a href="/facts/Bipyramid/MkzrCA0W">bipyramids</a> with six, eight, and ten equilateral triangles, respectively. They are constructed by identical pyramids base-to-base.</li> <li><a href="/facts/Gyroelongated_square_bipyramid/fIP8gBx4">gyroelongated square bipyramid</a> and <a href="/facts/Regular_icosahedron/1U93vmXV">regular icosahedron</a> are constructed by attaching two pyramids onto a square antiprism or pentagonal antiprism, respectively, such that they have sixteen and twenty triangular faces.</li> <li><a href="/facts/Triaugmented_triangular_prism/SAFVrVey">triaugmented triangular prism</a>, constructed by attaching three square pyramids onto the square face of a triangular prism, such that it has fourteen triangular faces.</li> <li><a href="/facts/Snub_disphenoid/kdLUzB07">snub disphenoid</a>, with twelve triangular faces, constructed by involving two regular hexagons in the following order: these hexagons may form a bipyramid in <a href="/facts/Degeneracy_(mathematics)/BqhqlXPH">degeneracy</a>, separating them into two parts along a coinciding diagonal, pressing inward on the end of diagonal, rotating one of them in 90°, and rejoining them together.</li></ul> <p>The number of possible convex deltahedrons was given by Rausenberger (1915), using the fact that multiplying the number of faces by three results in each edge is shared by two faces, by which substituting this to <a href="/facts/Euler%2527s_polyhedron_formula/JCjzWUr2">Euler's polyhedron formula</a>. In addition, it may show that a polyhedron with eighteen equilateral triangles is mathematically possible, although it is impossible to construct it geometrically. Rausenberger named these solids as the <i>convex pseudoregular polyhedra</i>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>Summarizing the examples above, the deltahedra can be conclusively defined as the class of polyhedra whose faces are <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangles</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Another definition by Bernal (1964) is similar to the previous one, in which he was interested in the shapes of holes left in irregular close-packed arrangements of spheres. It is stated as a convex polyhedron with equilateral triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres. Because of this restriction, some polyhedrons may not be included as a deltahedron: the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and regular icosahedron (because it has interior room for another sphere).<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>Most convex deltahedra can be found in the study of <a href="/facts/Chemistry/SrNqkGVm">chemistry</a>. For example, they are categorized as the <i>closo</i> polyhedron in the study of <a href="/facts/Polyhedral_skeletal_electron_pair_theory/hnVKFea2">polyhedral skeletal electron pair theory</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Other applications of deltahedra—excluding the regular icosahedron—are the visualization of an <a href="/facts/Atom_cluster/EsuLgxJU">atom cluster</a> surrounding a central atom as a polyhedron in the study of <a href="/facts/Chemical_compounds/37FXqqtv">chemical compounds</a>: regular tetrahedron represents the <a href="/facts/Tetrahedral_molecular_geometry/583AL1sP">tetrahedral molecular geometry</a>, triangular bipyramid represents <a href="/facts/Trigonal_bipyramidal_molecular_geometry/fTayDtlN">trigonal bipyramidal molecular geometry</a>, regular octahedron represents the <a href="/facts/Octahedral_molecular_geometry/bNXLVvks">octahedral molecular geometry</a>, pentagonal bipyramid represents the <a href="/facts/Pentagonal_bipyramidal_molecular_geometry/tQ4fA46l">pentagonal bipyramidal molecular geometry</a>, gyroelongated square bipyramid represents the <a href="/facts/Bicapped_square_antiprismatic_molecular_geometry/0dIRzC1z">bicapped square antiprismatic molecular geometry</a>, triaugmented triangular prism represents the <a href="/facts/Tricapped_trigonal_prismatic_molecular_geometry/klxVT8AG">tricapped trigonal prismatic molecular geometry</a>, and snub disphenoid represents the <a href="/facts/Dodecahedral_molecular_geometry/qFCra4Sw">dodecahedral molecular geometry</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h2 id="non-convex-deltahedron">Non-convex deltahedron</h2> <p>A <i>non-convex deltahedron</i> is a deltahedron that does not possess convexity, thus it has either coplanar faces or collinear edges. There are infinitely many non-convex deltahedra.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> Some examples are <a href="/facts/Stella_octangula/oNkHNB3C">stella octangula</a>, the third stellation of a regular icosahedron, and <a href="/facts/Boerdijk%25E2%2580%2593Coxeter_helix/aMuw8hxN">Boerdijk–Coxeter helix</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>There are subclasses of non-convex deltahedra. Cundy (1952) shows that they may be discovered by finding the number of varying vertex's <i>types</i>. A set of vertices is considered the same type as long as there are subgroups of the polyhedron's same group <a href="/facts/Transitive_group/Vh9VtUUw">transitive</a> on the set. Cundy shows that the <a href="/facts/Great_icosahedron/jaAk3a5Q">great icosahedron</a> is the only non-convex deltahedron with a single type of vertex. There are seventeen non-convex deltahedra with two types of vertex, and soon the other eleven deltahedra were later added by Olshevsky,<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> Other subclasses are the <a href="/facts/Isohedral_figure/ukLVQlK2">isohedral</a> deltahedron that was later discovered by both McNeill and Shephard (2000),<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> and the <i>spiral deltahedron</i> constructed by the strips of equilateral triangles was discovered by Trigg (1978).<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> <h3>Footnotes</h3> <h3>Works cited</h3> <ul><li><a href="/facts/John_Desmond_Bernal/Cpg8NvuW">Bernal, J. D.</a> (1964), "The Bakerian Lecture, 1962. The Structure of Liquids", <i><a href="/facts/Proceedings_of_the_Royal_Society_of_London/qyn2m2ja">Proceedings of the Royal Society of London</a></i>, Series A, Mathematical and Physical Sciences, 280 (1382): 299–322, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1964RSPSA.280..299B">1964RSPSA.280..299B</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1098%2Frspa.1964.0147">10.1098/rspa.1964.0147</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2415872">2415872</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:178710030">178710030</a>.</li> <li>Boissonnat, J. D.; Yvinec, M. (June 1989), "Probing a scene of non convex polyhedra", <i>Proceedings of the fifth annual symposium on Computational geometry - SCG '89</i>, pp. 237–246, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1145%2F73833.73860">10.1145/73833.73860</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-89791-318-3.</li> <li>Burdett, Jeremy K.; Hoffmann, Roald; Fay, Robert C. (1978), "Eight-Coordination", <i><a href="/facts/Inorganic_Chemistry_(journal)/yEeLo9Na">Inorganic Chemistry</a></i>, 17 (9): 2553–2568, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1021%2Fic50187a041">10.1021/ic50187a041</a>.</li> <li>Cromwell, Peter R. (1997), <a href="https://archive.org/details/polyhedra0000crom"><i>Polyhedra</i></a>, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>.</li> <li><a href="/facts/Martyn_Cundy/7UQhSW1X">Cundy, H. Martyn</a> (1952), "Deltahedra", <i><a href="/facts/Mathematical_Gazette/2PoBAS9b">Mathematical Gazette</a></i>, 36 (318): 263–266, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F3608204">10.2307/3608204</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/3608204">3608204</a>.</li> <li>———; Rollett, A. (1989), "3.11. Deltahedra", <a href="/facts/Mathematical_Models_(Cundy_and_Rollett)/IjzP47y2"><i>Mathematical Models</i></a> (3rd ed.), Stradbroke, England: Tarquin Pub., pp. 142–144.</li> <li><a href="/facts/David_Eppstein/hIjTU6nC">Eppstein, D.</a> (2021), "On Polyhedral Realization with Isosceles Triangles", <i>Graphs and Combinatorics</i>, 37 (4), Springer: 1247–1269, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2009.00116">2009.00116</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00373-021-02314-9">10.1007/s00373-021-02314-9</a></li> <li>Foulds, L. R.; Robinson, D. F. (1979), "Construction properties of combinatorial deltahedra", <i>Discrete Applied Mathematics</i>, 1 (1–2): 75–87, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0166-218X%2879%2990015-5">10.1016/0166-218X(79)90015-5</a>.</li> <li><a href="/facts/Hans_Freudenthal/l8V4uA0N">Freudenthal, H.</a>; <a href="/facts/Bartel_Leendert_van_der_Waerden/rITfcujK">van der Waerden, B. L.</a> (1947), "On an assertion of Euclid", <i><a href="/facts/Simon_Stevin_(journal)/1nl2jYTB">Simon Stevin</a></i>, 25: 115–121, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0021687">0021687</a>.</li> <li><a href="/facts/Martin_Gardner/HGfEnPDH">Gardner, Martin</a> (1992), <i>Fractal Music, Hypercards, and More: Mathematical Recreations from Scientific American</i>, New York: W. H. Freeman, pp. 40, 53, and 58–60.</li> <li>Gillespie, Ronald J.; Hargittai, István (2013), <i>The VSEPR Model of Molecular Geometry</i>, <a href="/facts/Dover_Publications/RwjvlSFv">Dover Publications</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-48615-4.</li> <li>Kepert, David L. (1982), "Polyhedra", <i>Inorganic Chemistry Concepts</i>, vol. 6, Springer, pp. 7–21, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-68046-5_2">10.1007/978-3-642-68046-5_2</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-68048-9.</li> <li>Kharas, K. C. C.; Dahl, L. F. (1988), "Ligand-Stabilized Metal Clusters: Structure, Bonding, Fluxionarity, and the Metallic State", in Prigogine, I.; Rice, S. A. (eds.), <a href="https://books.google.com/books?id=ur7Nqe4ueBYC"><i>Evolution of Size Effects in Chemical Dynamics Part 2: Advances in Chemical Physics Volume LXX</i></a>, <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">John Wiley & Sons</a>, p. 8, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-470-14180-9.</li> <li>Litchenberg, D. R. (1988), "Pyramids, Prisms, Antiprisms, and Deltahedra", <i>The Mathematics Teacher</i>, 81 (4): 261–265, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.5951%2FMT.81.4.0261">10.5951/MT.81.4.0261</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/27965792">27965792</a></li> <li>McNeill, J., <i>Isohedral Deltahedra</i></li> <li>Olshevsky, George, <a href="https://www.interocitors.com/polyhedra/papers/Polytopics%2028%20-%20Breaking%20Cundy's%20Deltahedra%20Record.pdf"><i>Breaking Cundy's Deltahedra Record</i></a> (PDF)</li> <li>Pedersen, M. C.; Hyde, S. T. (2018), "Polyhedra and packings from hyperbolic honeycombs", <i><a href="/facts/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America/w75CThX0">Proceedings of the National Academy of Sciences</a></i>, 115 (27): 6905–6910, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2018PNAS..115.6905P">2018PNAS..115.6905P</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1073%2Fpnas.1720307115">10.1073/pnas.1720307115</a>, <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6142264">6142264</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/29925600">29925600</a></li> <li>Petrucci, R. H.; Harwood, W. S.; Herring, F. G. (2002), <i>General Chemistry: Principles and Modern Applications</i> (8th ed.), Prentice-Hall, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-014329-7</li> <li>Pugh, Anthony (1976), <i>Polyhedra: A visual approach</i>, California: University of California Press Berkeley, pp. 35–36, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-520-03056-7.</li> <li>Rausenberger, O. (1915), "Konvexe pseudoreguläre Polyeder", <i>Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht</i>, 46: 135–142.</li> <li>Remhov, Arndt; Černý, Radovan (2021), "Hydroborate as novel solid-state electrolytes", in Schorr, Susan; Weidenthaler, Claudia (eds.), <i>Crystallography in Materials Science: From Structure-Property Relationships to Engineering</i>, <a href="/facts/De_Gruyter/2wJSEv3m">de Gruyter</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-11-067485-9.</li> <li><a href="/facts/G._C._Shephard/5AtJ6sMW">Shephard, G. C.</a> (2000), "Isohedral Deltahedra", <i>Periodica Mathematica Hungarica</i>, 39: 86–100, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FA%3A1004838806529">10.1023/A:1004838806529</a>.</li> <li>Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", <i>Mathematics Magazine</i>, 51 (1): 55–57, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2689647">10.2307/2689647</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2689647">2689647</a>.</li> <li>Tsuruta, Naoya; Mitani, Jun; Kanamori, Yoshihiro; Fukui, Yukio (2015), <a href="https://www.heldermann-verlag.de/jgg/jgg19/j19h2tsur.pdf">"Random Realization of Polyhedral Graphs as Deltahedra"</a> (PDF), <i>Journal for Geometry and Graphics</i>, 19 (2): 227–236.</li> <li>Weils, David (1991), <i>The Penguin Dictionary of Curious and Interesting Geometry</i>, Penguin Books, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780140118131.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.interocitors.com/polyhedra/Deltahedra/Cundy/index.html"><i>The Cundy Deltahedra or Biform Deltahedra</i></a></li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a>, <a href="https://mathworld.wolfram.com/Deltahedron.html">"Deltahedron"</a>, <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Cundy (1952)Cromwell (1997), p. 75Trigg (1978) - Cundy, H. Martyn (1952), "Deltahedra", Mathematical Gazette, 36 (318): 263–266, doi:10.2307/3608204, JSTOR 3608204 <a href="https://doi.org/10.2307%2F3608204" target="_blank">https://doi.org/10.2307%2F3608204</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Litchenberg (1988), p. 262Boissonnat & Yvinec (1989) - Litchenberg, D. R. (1988), "Pyramids, Prisms, Antiprisms, and Deltahedra", The Mathematics Teacher, 81 (4): 261–265, doi:10.5951/MT.81.4.0261, JSTOR 27965792 <a href="https://doi.org/10.5951%2FMT.81.4.0261" target="_blank">https://doi.org/10.5951%2FMT.81.4.0261</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Trigg (1978)Litchenberg (1988), p. 263Freudenthal & van der Waerden (1947) - Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine, 51 (1): 55–57, doi:10.2307/2689647, JSTOR 2689647 <a href="https://doi.org/10.2307%2F2689647" target="_blank">https://doi.org/10.2307%2F2689647</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Rausenberger (1915)Litchenberg (1988), p. 263 - Rausenberger, O. (1915), "Konvexe pseudoreguläre Polyeder", Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, 46: 135–142 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Cundy (1952)Cromwell (1997), p. 75Trigg (1978) - Cundy, H. Martyn (1952), "Deltahedra", Mathematical Gazette, 36 (318): 263–266, doi:10.2307/3608204, JSTOR 3608204 <a href="https://doi.org/10.2307%2F3608204" target="_blank">https://doi.org/10.2307%2F3608204</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Bernal (1964). - Bernal, J. D. (1964), "The Bakerian Lecture, 1962. The Structure of Liquids", Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 280 (1382): 299–322, Bibcode:1964RSPSA.280..299B, doi:10.1098/rspa.1964.0147, JSTOR 2415872, S2CID 178710030 <a href="https://ui.adsabs.harvard.edu/abs/1964RSPSA.280..299B" target="_blank">https://ui.adsabs.harvard.edu/abs/1964RSPSA.280..299B</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Kharas & Dahl (1988), p. 8. - Kharas, K. C. C.; Dahl, L. F. (1988), "Ligand-Stabilized Metal Clusters: Structure, Bonding, Fluxionarity, and the Metallic State", in Prigogine, I.; Rice, S. A. (eds.), Evolution of Size Effects in Chemical Dynamics Part 2: Advances in Chemical Physics Volume LXX, John Wiley & Sons, p. 8, ISBN 978-0-470-14180-9 <a href="https://books.google.com/books?id=ur7Nqe4ueBYC" target="_blank">https://books.google.com/books?id=ur7Nqe4ueBYC</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Burdett, Hoffmann & Fay (1978)Gillespie & Hargittai (2013), p. 152Kepert (1982), p. 7–21Petrucci, Harwood & Herring (2002), p. 413–414, See table 11.1.Remhov & Černý (2021), p. 270 - Burdett, Jeremy K.; Hoffmann, Roald; Fay, Robert C. (1978), "Eight-Coordination", Inorganic Chemistry, 17 (9): 2553–2568, doi:10.1021/ic50187a041 <a href="https://doi.org/10.1021%2Fic50187a041" target="_blank">https://doi.org/10.1021%2Fic50187a041</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Trigg (1978)Eppstein (2021) - Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine, 51 (1): 55–57, doi:10.2307/2689647, JSTOR 2689647 <a href="https://doi.org/10.2307%2F2689647" target="_blank">https://doi.org/10.2307%2F2689647</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Pedersen & Hyde (2018)Weils (1991), p. 78 - Pedersen, M. C.; Hyde, S. T. (2018), "Polyhedra and packings from hyperbolic honeycombs", Proceedings of the National Academy of Sciences, 115 (27): 6905–6910, Bibcode:2018PNAS..115.6905P, doi:10.1073/pnas.1720307115, PMC 6142264, PMID 29925600 <a href="https://ui.adsabs.harvard.edu/abs/2018PNAS..115.6905P" target="_blank">https://ui.adsabs.harvard.edu/abs/2018PNAS..115.6905P</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Cundy (1952)OlshevskyTsuruta et al. (2015) - Cundy, H. Martyn (1952), "Deltahedra", Mathematical Gazette, 36 (318): 263–266, doi:10.2307/3608204, JSTOR 3608204 <a href="https://doi.org/10.2307%2F3608204" target="_blank">https://doi.org/10.2307%2F3608204</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>McNeillShephard (2000)Tsuruta et al. (2015) - McNeill, J., Isohedral Deltahedra <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Trigg (1978)Tsuruta et al. (2015) - Trigg, Charles W. (1978), "An Infinite Class of Deltahedra", Mathematics Magazine, 51 (1): 55–57, doi:10.2307/2689647, JSTOR 2689647 <a href="https://doi.org/10.2307%2F2689647" target="_blank">https://doi.org/10.2307%2F2689647</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> </ol>