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==References== {{Reflist|refs= <ref name=alexandrov>{{cite book | last = Alexandrov | first = A. D. | author-link = Aleksandr Danilovich Aleksandrov | contribution = 8.1 Parallelohedra | contribution-url = https://books.google.com/books?id=R9vPatr5aqYC&pg=PA349 | pages = 349–359 | publisher = Springer | title = Convex Polyhedra | title-link = Convex Polyhedra (book) | year = 2005 }}</ref> <ref name=berman>{{cite journal | last = Berman | first = Martin | year = 1971 | title = Regular-faced convex polyhedra | journal = Journal of the Franklin Institute | volume = 291 | issue = 5 | pages = 329–352 | doi = 10.1016/0016-0032(71)90071-8 | mr = 290245 }}</ref> <ref name=budden>{{citation | last = Budden | first = Frank | date = December 1985 | doi = 10.2307/3617571 | issue = 450 | journal = The Mathematical Gazette | jstor = 3617571 | pages = 271–278 | publisher = JSTOR | title = Cayley graphs for some well-known groups | volume = 69}}</ref> <ref name=crisman>{{cite journal | last = Crisman | first = Karl-Dieter | year = 2011 | title = The Symmetry Group of the Permutahedron | journal = The College Mathematics Journal | volume = 42 | issue = 2 | pages = 135–139 | doi = 10.4169/college.math.j.42.2.135 | jstor = college.math.j.42.2.135 }}</ref> <ref name=diudea>{{cite book | last = Diudea | first = M. V. | year = 2018 | title = Multi-shell Polyhedral Clusters | series = Carbon Materials: Chemistry and Physics | volume = 10 | publisher = [[Springer Science+Business Media|Springer]] | isbn = 978-3-319-64123-2 | doi = 10.1007/978-3-319-64123-2 | url = https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 | page = 39 }}</ref> <ref name=erdahl>{{cite journal | last = Erdahl | first = R. M. | doi = 10.1006/eujc.1999.0294 | issue = 6 | journal = European Journal of Combinatorics | mr = 1703597 | pages = 527–549 | title = Zonotopes, dicings, and Voronoi's conjecture on parallelohedra | volume = 20 | year = 1999| doi-access = free }}. Voronoi conjectured that all tilings of higher dimensional spaces by translates of a single [[convex polytope]] are combinatorially equivalent to Voronoi tilings, and Erdahl proves this in the special case of [[zonotope]]s. But as he writes (p. 429), Voronoi's conjecture for dimensions at most four was already proven by Delaunay. For the classification of three-dimensional parallelohedra into these five types, see {{cite journal | last1 = Grünbaum | first1 = Branko | author1-link = Branko Grünbaum | last2 = Shephard | first2 = G. C. | author2-link = Geoffrey Colin Shephard | doi = 10.1090/S0273-0979-1980-14827-2 | issue = 3 | journal = [[Bulletin of the American Mathematical Society]] | mr = 585178 | pages = 951–973 | series = New Series | title = Tilings with congruent tiles | volume = 3 | year = 1980| doi-access = free }}</ref> <ref name=jj>{{cite book | last1 = Johnson | first1 = Tom | last2 = Jedrzejewski | first2 = Franck | year = 2014 | title = Looking at Numbers | publisher = Springer | url = https://books.google.com/books?id=xtE-AgAAQBAJ&pg=PA15 | page = 15 | doi = 10.1007/978-3-0348-0554-4 | isbn = 978-3-0348-0554-4 }}</ref> <ref name=johnson>{{cite journal | last = Johnson | first = Norman W. | authorlink = Norman W. Johnson | year = 1966 | title = Convex polyhedra with regular faces | journal = [[Canadian Journal of Mathematics]] | volume = 18 | pages = 169–200 | doi = 10.4153/cjm-1966-021-8 | mr = 0185507 | s2cid = 122006114 | zbl = 0132.14603| doi-access = free }}</ref> <ref name=jtdd>{{cite book | last1 = Jensen | first1 = Patrick M. | last2 = Trinderup | first2 = Camilia H. | last3 = Dahl | first3 = Anders B. | last4 = Dahl | first4 = Vedrana A. | year = 2019 | editor-last1 = Felsberg | editor-first1 = Michael | editor-last2 = Forssén | editor-first2 = Per-Erik | editor-last3 = Sintorn | editor-first3 = Ida-Maria | editor-last4 = Unger | editor-first4 = Jonas | contribution = Zonohedral Approximation of Spherical Structuring Element for Volumetric Morphology | contribution-url = https://books.google.com/books?id=uLWZDwAAQBAJ&pg=PA131 | page = 131–132 | title = Image Analysis: 21st Scandinavian Conference, SCIA 2019, Norrköping, Sweden, June 11–13, 2019, Proceedings | publisher = Springer | doi = 10.1007/978-3-030-20205-7 | isbn = 978-3-030-20205-7 }}</ref> <ref name=kocakoca>{{cite book | last1 = Koca | first1 = M. | last2 = Koca | first2 = N. O. | year = 2013 | title = Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010 | contribution = Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes | contribution-url = https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48 | page = 48 | publisher = World Scientific }}</ref> <ref name=mizutani>{{cite book | last = Mizutani | first = Uichiro | year = 2001 | title = Introduction to the Electron Theory of Metals | url = https://books.google.com/books?id=zY5z_UGqAcwC&pg=PA112 | page = 112 | publisher = [[Cambridge University Press]] | isbn = 978-0-521-58709-9 }}</ref> <!-- <ref name=sg>{{cite journal | last1 = Schein | first1 = S. | last2 = Gayed | first2 = J. M. | year = 2014 | title = Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses | journal = Proceedings of the National Academy of Sciences | language = en | volume = 111 | issue = 8 | pages = 2920–2925 | doi = 10.1073/pnas.1310939111 | issn = 0027-8424 | pmc = 3939887 | pmid = 24516137 | bibcode = 2014PNAS..111.2920S | doi-access = free }}</ref> --> <ref name=tz>{{cite journal | last1 = Thuswaldner | first1 = Jörg | last2 = Zhang | first2 = Shu-qin | arxiv = 1811.06718 | doi = 10.1090/tran/7930 | issue = 1 | journal = Transactions of the American Mathematical Society | mr = 4042883 | pages = 491–527 | title = On self-affine tiles whose boundary is a sphere | volume = 373 | year = 2020 }}</ref> <ref name=williams>{{cite book | last = Williams | first = Robert | authorlink = Robert Williams (geometer) | year = 1979 | title = The Geometrical Foundation of Natural Structure: A Source Book of Design | publisher = Dover Publications, Inc. | url = https://archive.org/details/geometricalfound00will/page/78 | page = 78 | isbn = 978-0-486-23729-9 }}</ref> <ref name=yen>{{cite book | last = Yen | first = Teh F. | year = 2007 | title = Chemical Processes for Environmental Engineering | publisher = Imperial College Press | url = https://books.google.com/books?id=KHGelcDf1qQC&pg=PA388 | page = 338 | isbn = 978-1-86094-759-9 }}</ref> }}
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