Editing Polyhedron (section)

===Renaissance===
{{multiple image
 | image1 = Pacioli.jpg
 | caption1 = ''[[Portrait of Luca Pacioli|Doppio ritratto]]'', attributed to [[Jacopo de' Barbari]], depicting [[Luca Pacioli]] and a student studying a glass [[rhombicuboctahedron]] half-filled with water.<ref>{{citation | last = Gamba | first = Enrico | title = Imagine Math | editor-last = Emmer | editor-first = Michele | contribution = The mathematical ideas of Luca Pacioli depicted by Iacopo de' Barbari in the ''Doppio ritratto'' | doi = 10.1007/978-88-470-2427-4_25 | isbn = 978-88-470-2427-4 | pages = 267–271 | publisher = Springer | year = 2012}}</ref>
 | image2 = Leonardo polyhedra.png
 | caption2 = A skeletal polyhedron (specifically, a [[rhombicuboctahedron]]) drawn by [[Leonardo da Vinci]] to illustrate a book by [[Luca Pacioli]]
 | total_width = 400
}}
As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Italian [[Renaissance]]. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations into [[Perspective (graphical)|perspective]].<ref name=polyhedrists>{{citation|title=The Polyhedrists: Art and Geometry in the Long Sixteenth Century|first=Noam|last=Andrews|publisher=MIT Press|year=2022|isbn=9780262046640}}</ref> [[Toroidal polyhedron|Toroidal polyhedra]], made of wood and used to support headgear, became a common exercise in perspective drawing, and were depicted in [[marquetry]] panels of the period as a symbol of geometry.<ref>{{citation | last1 = Calvo-López | first1 = José | last2 = Alonso-Rodríguez | first2 = Miguel Ángel | date = February 2010 | doi = 10.1007/s00004-010-0018-4 | issue = 1 | journal = Nexus Network Journal | pages = 75–111 | title = Perspective versus stereotomy: From Quattrocento polyhedral rings to sixteenth-century Spanish torus vaults | volume = 12| doi-access = free }}</ref> [[Piero della Francesca]] wrote about constructing perspective views of polyhedra, and rediscovered many of the Archimedean solids. [[Leonardo da Vinci]] illustrated skeletal models of several polyhedra for a book by [[Luca Pacioli]],<ref>{{citation | last = Field | first = J. V. | authorlink = Judith V. Field | doi = 10.1007/BF00374595 | issue = 3–4 | journal = [[Archive for History of Exact Sciences]] | jstor = 41134110 | mr = 1457069 | pages = 241–289 | s2cid = 118516740 | title = Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler | volume = 50 | year = 1997}}</ref> with text largely plagiarized from della Francesca.<ref>{{citation | last = Montebelli | first = Vico | doi = 10.1007/s40329-015-0090-4 | issue = 3 | journal = Lettera Matematica | mr = 3402538 | pages = 135–141 | title = Luca Pacioli and perspective (part I) | volume = 3 | year = 2015 | s2cid = 193533200}}</ref> [[Polyhedral net]]s make an appearance in the work of [[Albrecht Dürer]].<ref>{{citation | last = Ghomi | first = Mohammad | issue = 1 | journal = [[Notices of the American Mathematical Society]] | mr = 3726673 | pages = 25–27 | title = Dürer's unfolding problem for convex polyhedra | url = https://www.ams.org/publications/journals/notices/201801/rnoti-p25.pdf | volume = 65 | year = 2018| doi = 10.1090/noti1609 }}</ref>

Several works from this time investigate star polyhedra, and other elaborations of the basic Platonic forms. A marble tarsia in the floor of [[St. Mark's Basilica]], Venice, designed by [[Paolo Uccello]], depicts a stellated dodecahedron.<ref>{{citation | last = Saffaro | first = Lucio | editor1-last = Taliani | editor1-first = C. | editor2-last = Ruani | editor2-first = G. | editor3-last = Zamboni | editor3-first = R. | contribution = Cosmoids, fullerenes and continuous polygons | contribution-url = https://books.google.com/books?id=dOk7DwAAQBAJ&pg=PA55 | location = Singapore | pages = 55–64 | publisher = World Scientific | title = Fullerenes: Status and Perspectives, Proceedings of the 1st Italian Workshop, Bologna, Italy, 6–7 February | year = 1992}}</ref> As the Renaissance spread beyond Italy, later artists such as [[Wenzel Jamnitzer]], Dürer and others also depicted polyhedra of increasing complexity, many of them novel, in imaginative etchings.<ref name=polyhedrists/> [[Johannes Kepler]] (1571–1630) used [[star polygon]]s, typically [[pentagram]]s, to build star polyhedra. Some of these figures may have been discovered before Kepler's time, but he was the first to recognize that they could be considered "regular" if one removed the restriction that regular polyhedra must be convex.<ref>{{citation | last = Field | first = J. V. | author-link = Judith V. Field | doi = 10.1016/0083-6656(79)90001-1 | issue = 2 | journal = Vistas in Astronomy | mr = 546797 | pages = 109–141 | title = Kepler's star polyhedra | volume = 23 | year = 1979| bibcode = 1979VA.....23..109F }}</ref>

In the same period, [[Euler's polyhedral formula]], a [[linear equation]] relating the numbers of vertices, edges, and faces of a polyhedron, was stated for the Platonic solids in 1537 in an unpublished manuscript by [[Francesco Maurolico]].<ref>{{citation|first= Michael|last=Friedman|publisher=Birkhäuser|year=2018|title=A History of Folding in Mathematics: Mathematizing the Margins|title-link=A History of Folding in Mathematics|series=Science Networks. Historical Studies|volume=59|isbn=978-3-319-72486-7|doi=10.1007/978-3-319-72487-4|page=71}}</ref>