Editing Polyhedron (section)

===20th–21st centuries===
Mathematics in the 20th century dawned with [[Hilbert's problems]], one of which, [[Hilbert's third problem]], concerned polyhedra and their [[Dissection problem|dissections]]. It was quickly solved by Hilbert's student [[Max Dehn]], introducing the [[Dehn invariant]] of polyhedra.<ref>{{citation
 | last = Zeeman | first = E. C. | author-link = Christopher Zeeman
 | date = July 2002
 | doi = 10.2307/3621846
 | issue = 506
 | journal = [[The Mathematical Gazette]]
 | jstor = 3621846
 | pages = 241–247
 | title = On Hilbert's third problem
 | volume = 86| s2cid = 125593771 }}</ref> [[Steinitz's theorem]], published by [[Ernst Steinitz]] in 1992, characterized the graphs of convex polyhedra, bringing modern ideas from [[graph theory]] and [[combinatorics]] into the [[Polyhedral combinatorics|study of polyhedra]].<ref>{{citation
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | doi = 10.1016/j.disc.2005.09.037
 | hdl = 1773/2276 | hdl-access = free
 | issue = 3–5
 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
 | mr = 2287486
 | pages = 445–463
 | title = Graphs of polyhedra; polyhedra as graphs
 | volume = 307
 | year = 2007}}</ref>

The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called [[stellation]]. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by [[H.S.M. Coxeter]] and others in 1938, with the now famous paper ''[[The Fifty-Nine Icosahedra]]''.<ref name=fiftynine>{{citation
 | last1 = Coxeter | first1 = H.S.M. | author1-link = Harold Scott MacDonald Coxeter
 | last2 = Du Val | first2 = P.
 | last3 = Flather | first3 = H.T.
 | last4 = Petrie | first4 = J. F.
 | isbn = 978-1-899618-32-3
 | mr = 676126
 | orig-date = 1938
 | publisher = Tarquin Publications
 | title = The Fifty-Nine Icosahedra
 | title-link = The Fifty-Nine Icosahedra
 | year = 1999}}.</ref> Coxeter's analysis signaled a rebirth of interest in geometry. Coxeter himself went on to enumerate the star uniform polyhedra for the first time, to treat tilings of the plane as polyhedra, to discover the [[regular skew polyhedron|regular skew polyhedra]] and to develop the theory of [[complex polytope|complex polyhedra]] first discovered by Shephard in 1952, as well as making fundamental contributions to many other areas of geometry.<ref>{{citation|title=King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry|first=Siobhan|last=Roberts|author-link=Siobhan Roberts|publisher=Bloomsbury Publishing|year=2009|isbn=9780802718327}}</ref>

In the second part of the twentieth century, both [[Branko Grünbaum]] and [[Imre Lakatos]] pointed out the tendency among mathematicians to define a "polyhedron" in different and sometimes incompatible ways to suit the needs of the moment.<ref name="lakatos"/><ref name=sin/> In a series of papers, Grünbaum broadened the accepted definition of a polyhedron, discovering many new [[Regular polyhedron#History|regular polyhedra]]. At the close of the twentieth century, these latter ideas merged with other work on incidence complexes to create the modern idea of an abstract polyhedron (as an abstract 3-polytope), notably presented by McMullen and Schulte.<ref>{{citation|last1=McMullen|first1=Peter|author1-link=Peter McMullen|last2=Schulte|first2=Egon|author2-link=Egon Schulte|title=Abstract Regular Polytopes|series=Encyclopedia of Mathematics and its Applications|volume=92|publisher=Cambridge University Press|year=2002}}</ref>

{{multiple image
 | total_width = 350
 | align = right
 | image1 = Circogonia icosahedra.jpg
 | caption1 = The [[radiolarian]] ''Circogonia icosahedra''
 | image2 = Dymaxion projection.png
 | caption2 = [[Dymaxion map]], created by the net of a regular icosahedron
}}
Polyhedra have been discovered in many fields of science. The Platonic solids appeared in biological creatures, as in The ''[[Braarudosphaera bigelowii]]'' has a regular dodecahedral structure.<ref name=hagino-13>{{citation
 | last1 = Hagino | first1 = K.
 | last2 = Onuma | first2 = R.
 | last3 = Kawachi | first3 = M.
 | last4 = Horiguchi | first4 = T.
 | year = 2013
 | title = Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-A in ''Braarudosphaera bigelowii'' (Prymnesiophyceae)
 | journal = PLOS ONE
 | volume = 8
 | issue = 12
 | article-number = e81749
 | doi = 10.1371/journal.pone.0081749
| doi-access = free
 | pmid = 24324722
 | pmc = 3852252
 | bibcode = 2013PLoSO...881749H
 }}</ref> [[Ernst Haeckel]] described a number of species of [[radiolarians]], some of whose shells are shaped like various regular polyhedra.<ref name=haeckel-04>{{citation
 | last = Haeckel | first = E
 | year = 1904
 | title = [[Kunstformen der Natur]]
}}. Available as Haeckel, E. ''Art forms in nature'', Prestel USA (1998), {{isbn|3-7913-1990-6}}. Online version at [http://www.biolib.de/haeckel/kunstformen/index.html Kurt Stüber's Biolib] (in German)</ref> The outer protein shells of many [[virus]]es form regular polyhedra. For example, [[HIV]] is enclosed in a regular icosahedron, as is the head of a typical [[myovirus]].<ref>{{citation
 | title = Virus Taxonomy
 | chapter = Myoviridae
 | publisher = Elsevier
 | year = 2012
 | pages = 46–62
 | doi = 10.1016/b978-0-12-384684-6.00002-1
 | isbn = 9780123846846
 | ref={{sfnref|Elsevier|2012}}
}}</ref><ref name=strauss>{{citation
 | last1 = Strauss | first1 = James H.
 | last2 = Strauss | first2 = Ellen G.
 | title = Viruses and Human Disease
 | chapter = The Structure of Viruses
 | publisher = Elsevier
 | year = 2008
 | pages = 35–62
 | doi = 10.1016/b978-0-12-373741-0.50005-2
 | pmc = 7173534
 | isbn = 9780123737410
 | s2cid = 80803624
}}</ref> The regular icosahedron may also appeared in the applications of [[cartography]] when [[R. Buckminster Fuller]] used its net to his project known as [[Dymaxion map]], frustatedly realized that the [[Greenland]] size is smaller than the [[South America]].{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/7/mode/1up?view=theater 7]}}

Polyhedra make a frequent appearance in modern [[computational geometry]], [[computer graphics]], and [[geometric design]] with topics including the reconstruction of polyhedral surfaces or [[Polygon mesh|surface meshes]] from scattered data points,<ref>{{citation
 | last1 = Lim | first1 = Seng Poh
 | last2 = Haron | first2 = Habibollah
 | date = March 2012
 | doi = 10.1007/s10462-012-9329-z
 | issue = 1
 | journal = Artificial Intelligence Review
 | pages = 59–78
 | title = Surface reconstruction techniques: a review
 | volume = 42| s2cid = 254232891
 }}</ref> geodesics on polyhedral surfaces,<ref>{{citation
 | last1 = Mitchell | first1 = Joseph S. B. | author1-link = Joseph S. B. Mitchell
 | last2 = Mount | first2 = David M. | author2-link = David Mount
 | last3 = Papadimitriou | first3 = Christos H. | author3-link = Christos Papadimitriou
 | doi = 10.1137/0216045
 | issue = 4
 | journal = [[SIAM Journal on Computing]]
 | mr = 899694
 | pages = 647–668
 | title = The discrete geodesic problem
 | volume = 16
 | year = 1987}}</ref> [[Visibility (geometry)|visibility]] and illumination in polyhedral scenes,<ref>{{citation
 | last1 = Teller | first1 = Seth J. | author1-link = Seth J. Teller
 | last2 = Hanrahan | first2 = Pat | author2-link = Pat Hanrahan
 | editor-last = Whitton | editor-first = Mary C. | editor-link = Mary Whitton
 | contribution = Global visibility algorithms for illumination computations
 | doi = 10.1145/166117.166148
 | pages = 239–246
 | publisher = Association for Computing Machinery
 | title = Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1993, Anaheim, CA, USA, August 2–6, 1993
 | year = 1993| isbn = 0-89791-601-8 | s2cid = 7957200 }}</ref> [[polycube]]s and other non-convex polyhedra with axis-parallel sides,<ref>{{citation|title=Polycube Optimization and Applications: From the Digital World to Manufacturing|hdl=11584/261570|first=Gianmarco|last=Cherchi|date=February 2019|type=Doctoral dissertation|publisher=University of Cagliari}}</ref> algorithmic forms of Steinitz's theorem,<ref>{{citation
 | last = Rote | first = Günter
 | editor1-last = van Kreveld | editor1-first = Marc J.
 | editor2-last = Speckmann | editor2-first = Bettina
 | contribution = Realizing planar graphs as convex polytopes
 | doi = 10.1007/978-3-642-25878-7_23
 | pages = 238–241
 | publisher = Springer
 | series = Lecture Notes in Computer Science
 | title = Graph Drawing – 19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 21–23, 2011, Revised Selected Papers
 | volume = 7034
 | year = 2011| doi-access = free
 | isbn = 978-3-642-25877-0
 }}</ref> and the still-unsolved problem of the existence of polyhedral nets for convex polyhedra.<ref>{{citation|last1=Demaine|first1=Erik|author1-link=Erik Demaine|last2=O'Rourke|first2=Joseph|author2-link=Joseph O'Rourke (professor)|title=Geometric Folding Algorithms: Linkages, Origami, Polyhedra|title-link=Geometric Folding Algorithms|publisher=Cambridge University Press|year=2007}}</ref>