Editing Polyhedron (section)

==History==
===Before the Greeks===
[[File:Papyrus moscow 4676-problem 14 part 1.jpg|thumb|Problem 14 of the [[Moscow Mathematical Papyrus]], on calculating the volume of a [[frustum]]]]
Polyhedra appeared in early [[architecture|architectural forms]] such as cubes and cuboids, with the earliest four-sided [[Egyptian pyramids]] dating from the [[27th century BC]].<ref>{{citation | last = Kitchen | first = K. A. | date = October 1991 | doi = 10.1080/00438243.1991.9980172 | issue = 2 | journal = World Archaeology | pages = 201–208 | title = The chronology of ancient Egypt | volume = 23}}</ref> The [[Moscow Mathematical Papyrus]] from approximately 1800–1650 BC includes an early written study of polyhedra and their volumes (specifically, the volume of a [[frustum]]).<ref>{{citation | last1 = Gunn | first1 = Battiscombe | last2 = Peet | first2 = T. Eric | date = May 1929 | doi = 10.1177/030751332901500130 | issue = 1 | journal = The Journal of Egyptian Archaeology | pages = 167–185 | title = Four Geometrical Problems from the Moscow Mathematical Papyrus | volume = 15| s2cid = 192278129 }}</ref> The mathematics of the [[Old Babylonian Empire]], from roughly the same time period as the Moscow Papyrus, also included calculations of the volumes of [[cuboid]]s (and of non-polyhedral [[cylinder]]s), and calculations of the height of such a shape needed to attain a given volume.<ref>{{citation | last = Friberg | first = Jöran | issue = 2 | journal = Revue d'Assyriologie et d'archéologie orientale | jstor = 23281940 | pages = 97–188 | title = Mathematics at Ur in the Old Babylonian Period | volume = 94 | year = 2000}}</ref>

The [[Etruscan civilization|Etruscans]] preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery of an [[Etruscan civilization|Etruscan]]  [[dodecahedron]] made of [[soapstone]] on [[Monte Loffa]]. Its faces were marked with different designs, suggesting to some scholars that it may have been used as a gaming die.<ref>{{citation |title=An Etruscan dodecahedron|first=Amelia Carolina|last=Sparavigna|year=2012|arxiv=1205.0706}}</ref>

===Ancient Greece===
{{multiple image
 | align = right |total_width=500
 | image1 = Kepler Hexahedron Earth.jpg |width1=290|height1=304
 | image2 = Kepler Icosahedron Water.jpg |width2=306|height2=328
 | image3 = Kepler Octahedron Air.jpg |width3=328|height3=334
 | image4 = Kepler Tetrahedron Fire.jpg |width4=367|height4=328
 | image5 = Kepler Dodecahedron Universe.jpg |width5=330|height5=332
 | footer = five elements in each Platonic solids, but the assignment drawing was by Kepler's ''Harmonices Mundi''
}}
Ancient Greek mathematicians discovered and studied the [[Regular polyhedron#History|convex regular polyhedra]], which came to be known as the [[Platonic solid]]s. Their first written description is in the ''[[Timaeus (dialogue)|Timaeus]]'' of [[Plato]] (circa 360 BC), which associates four of them with the [[four elements]] and the fifth to the overall shape of the universe. A more mathematical treatment of these five polyhedra was written soon after in the ''[[Euclid's Elements|Elements]]'' of [[Euclid]]. An early commentator on Euclid (possibly [[Geminus]]) writes that the attribution of these shapes to Plato is incorrect: [[Pythagoras]] knew the [[tetrahedron]], [[cube]], and [[dodecahedron]], and [[Theaetetus (mathematician)|Theaetetus]] (circa 417 BC) discovered the other two, the [[octahedron]] and [[icosahedron]].<ref>{{citation | last = Eves | first = Howard | date = January 1969 | department = Historically Speaking | doi = 10.5951/mt.62.1.0042 | issue = 1 | journal = The Mathematics Teacher | jstor = 27958041 | pages = 42–44 | title = A geometry capsule concerning the five platonic solids | volume = 62}}</ref> Later, [[Archimedes]] expanded his study to the [[Uniform polyhedron|convex uniform polyhedra]] which now bear his name. His original work is lost and his solids come down to us through [[Pappus of Alexandria|Pappus]].<ref>{{citation | last = Field | first = J. V. | author-link = Judith V. Field | doi = 10.1007/BF00374595 | issue = 3–4 | journal = Archive for History of Exact Sciences | jstor = 41134110 | mr = 1457069 | pages = 241–289 | title = Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler | volume = 50 | year = 1997| s2cid = 118516740 }}</ref>

===Ancient China===
[[File:14-sided Chinese dice from warring states period.jpg|thumb|upright|Fourteen sided die from the [[Warring States period]]]]
Both cubical dice and 14-sided dice in the shape of a [[truncated octahedron]] in China have been dated back as early as the [[Warring States period]].<ref>{{citation
 | last1 = Bréard | first1 = Andrea | author1-link = Andrea Bréard
 | last2 = Cook | first2 = Constance A.
 | date = December 2019
 | doi = 10.1007/s00407-019-00245-9
 | issue = 4
 | journal = Archive for History of Exact Sciences
 | pages = 313–343
 | title = Cracking bones and numbers: solving the enigma of numerical sequences on ancient Chinese artifacts
 | volume = 74| s2cid = 253898304 }}</ref>

By 236 AD, [[Liu Hui]] was describing the dissection of the cube into its characteristic tetrahedron ([[orthoscheme]]) and related solids, using assemblages of these solids as the basis for calculating volumes of earth to be moved during engineering excavations.<ref>{{citation
 | last = van der Waerden | first = B. L.
 | contribution = Chapter 7: Liu Hui and Aryabhata
 | doi = 10.1007/978-3-642-61779-9_7
 | pages = 192–217
 | publisher = Springer
 | title = Geometry and Algebra in Ancient Civilizations
 | year = 1983}}</ref>

===Medieval Islam===
After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see [[Mathematics in medieval Islam]]).<ref>{{citation | last = Knorr | first = Wilbur | author-link = Wilbur Knorr | doi = 10.1016/0315-0860(83)90034-4 | issue = 1 | journal = Historia Mathematica | mr = 698139 | pages = 71–78 | title = On the transmission of geometry from Greek into Arabic | volume = 10 | year = 1983| doi-access = free }}</ref> The 9th century scholar [[Thabit ibn Qurra]] included the calculation of volumes in his studies,<ref>{{citation | last = Rashed | first = Roshdi | editor-last = Rashed | editor-first = Roshdi | contribution = Thābit ibn Qurra et l'art de la mesure | contribution-url = https://books.google.com/books?id=V5PTZi77YxwC&pg=PA173 | language = fr | isbn = 9783110220780 | pages = 173–175 | publisher = Walter de Gruyter | series = Scientia Graeco-Arabica | title = Thābit ibn Qurra: Science and Philosophy in Ninth-Century Baghdad | volume = 4 | year = 2009}}</ref> and wrote a work on the [[cuboctahedron]]. Then in the 10th century [[Abūl Wafā' Būzjānī|Abu'l Wafa]] described the convex regular and quasiregular spherical polyhedra.<ref>{{citation | last1 = Hisarligil | first1 = Hakan | last2 = Hisarligil | first2 = Beyhan Bolak | date = December 2017 | doi = 10.1007/s00004-017-0363-7 | issue = 1 | journal = Nexus Network Journal | pages = 125–152 | title = The geometry of cuboctahedra in medieval art in Anatolia | volume = 20| doi-access = free }}</ref>

===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>

===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>

===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>