Editing Polyhedron (section)

===17th–19th centuries===
[[René Descartes]], in around 1630, wrote his book ''[[De solidorum elementis]]'' studying convex polyhedra as a general concept, not limited to the Platonic solids and their elaborations. The work was lost, and not rediscovered until the 19th century. One of its contributions was [[Descartes' theorem on total angular defect]], which is closely related to Euler's polyhedral formula.<ref>{{citation | last = Federico | first = Pasquale Joseph | isbn = 0-387-90760-2 | mr = 680214 | publisher = Springer-Verlag | series = Sources in the History of Mathematics and Physical Sciences | title = Descartes on Polyhedra: A Study of the "De solidorum elementis" | volume = 4 | year = 1982}}</ref> [[Leonhard Euler]], for whom the formula is named, introduced it in 1758 for convex polyhedra more generally, albeit with an incorrect proof.<ref>{{citation
 | last1 = Francese | first1 = Christopher
 | last2 = Richeson | first2 = David
 | doi = 10.1080/00029890.2007.11920417
 | issue = 4
 | journal = [[The American Mathematical Monthly]]
 | mr = 2281926
 | pages = 286–296
 | title = The flaw in Euler's proof of his polyhedral formula
 | volume = 114
 | year = 2007| s2cid = 10023787
 }}</ref> Euler's work (together with his earlier solution to the puzzle of the [[Seven Bridges of Königsberg]]) became the foundation of the new field of [[topology]].<ref>{{citation
 | last = Alexanderson | first = Gerald L.
 | doi = 10.1090/S0273-0979-06-01130-X
 | issue = 4
 | journal = American Mathematical Society
 | mr = 2247921
 | pages = 567–573
 | series = New Series
 | title = About the cover: Euler and Königsberg's bridges: a historical view
 | volume = 43
 | year = 2006| doi-access = free
 }}</ref> The core concepts of this field, including generalizations of the polyhedral formula, were developed in the late nineteenth century by [[Henri Poincaré]], [[Enrico Betti]], [[Bernhard Riemann]], and others.<ref>{{citation | last = Eckmann | first = Beno | contribution = The Euler characteristic – a few highlights in its long history | doi = 10.1007/978-3-540-33791-1_15 | isbn = 978-3-540-33791-1 | mr = 2269092 | pages = 177–188 | publisher = Springer | title = Mathematical Survey Lectures 1943–2004 | year = 2006}}</ref>

In the early 19th century, [[Louis Poinsot]] extended Kepler's work, and discovered the remaining two regular star polyhedra. Soon after, [[Augustin-Louis Cauchy]] proved Poinsot's list complete, subject to an unstated assumption that the sequence of vertices and edges of each polygonal side cannot admit repetitions (an assumption that had been considered but rejected in the earlier work of A. F. L. Meister).<ref>{{citation | last = Grünbaum | first = Branko | editor-last = Grattan-Guinness | editor-first = I. | contribution = Regular polyhedra | contribution-url = https://books.google.com/books?id=ZptYDwAAQBAJpg | isbn = 0-415-03785-9 | mr = 1469978 | pages = 866–876 | publisher = Routledge | title = Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences | volume = 2 | year = 1994}}</ref> They became known as the [[Kepler–Poinsot polyhedra]], and their usual names were given by [[Arthur Cayley]].<ref>{{citation|title=Regular-faced polyhedra: remembering Norman Johnson|work=AMS Feature column|first=Joseph|last=Malkevitch|year=2018|url=https://www.ams.org/publicoutreach/feature-column/fc-2018-01|publisher=American Mathematical Society|access-date=2023-05-27}}
</ref> Meanwhile, the discovery of higher dimensions in the early 19th century led [[Ludwig Schläfli]] by 1853 to the idea of higher-dimensional polytopes.{{sfnp|Coxeter|1947|pages=141–143}} Additionally, in the late 19th century, Russian crystallographer [[Evgraf Fedorov]] completed the classification of [[Parallelohedron|parallelohedra]], convex polyhedra that tile space by translations.<ref>{{citation|last=Austin|first=David|title=Fedorov's five parallelohedra|work=AMS Feature Column|publisher=American Mathematical Society|url=https://www.ams.org/samplings/feature-column/fc-2013-11|date=November 2013}}</ref>