{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:46:57Z","timestamp":1740109617072,"version":"3.37.3"},"reference-count":60,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2016,8,12]],"date-time":"2016-08-12T00:00:00Z","timestamp":1470960000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2016,10]]},"DOI":"10.1007\/s00454-016-9814-2","type":"journal-article","created":{"date-parts":[[2016,8,12]],"date-time":"2016-08-12T09:46:56Z","timestamp":1470995216000},"page":"657-692","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Wythoffian Skeletal Polyhedra in Ordinary Space, I"],"prefix":"10.1007","volume":"56","author":[{"given":"Egon","family":"Schulte","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9802-3646","authenticated-orcid":false,"given":"Abigail","family":"Williams","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,8,12]]},"reference":[{"key":"9814_CR1","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/PL00000128","volume":"59","author":"JL Arocha","year":"2000","unstructured":"Arocha, J.L., Bracho, J., Montejano, L.: Regular projective polyhedra with planar faces, part I. Aequat. Math. 59, 55\u201373 (2000)","journal-title":"Aequat. Math."},{"key":"9814_CR2","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1007\/PL00000122","volume":"59","author":"J Bracho","year":"2000","unstructured":"Bracho, J.: Regular projective polyhedra with planar faces, part II. Aequat. Math. 59, 160\u2013176 (2000)","journal-title":"Aequat. Math."},{"key":"9814_CR3","volume-title":"Sphere Packings, Lattices and Groups","author":"JH Conway","year":"1998","unstructured":"Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer, New York (1998)","edition":"3"},{"key":"9814_CR4","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1112\/plms\/s2-38.1.327","volume":"38","author":"HSM Coxeter","year":"1935","unstructured":"Coxeter, H.S.M.: Wythoff\u2019s construction for uniform polytopes. Proc. Lond. Math. Soc. (2) 38, 327\u2013339 (1935)","journal-title":"Proc. Lond. Math. Soc. (2)"},{"key":"9814_CR5","first-page":"33","volume":"43","author":"HSM Coxeter","year":"1937","unstructured":"Coxeter, H.S.M.: Regular skew polyhedra in 3 and 4 dimensions and their topological analogues. Proc. Lond. Math. Soc. (2) 43, 33\u201362 (1937)","journal-title":"Proc. Lond. Math. Soc. (2)"},{"key":"9814_CR6","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1007\/BF01181449","volume":"46","author":"HSM Coxeter","year":"1940","unstructured":"Coxeter, H.S.M.: Regular and semi-regular polytopes, I. Math. Z. 46, 380\u2013407 (1940)","journal-title":"Math. Z."},{"key":"9814_CR7","volume-title":"Regular Polytopes","author":"HSM Coxeter","year":"1973","unstructured":"Coxeter, H.S.M.: Regular Polytopes. Dover, New York (1973)"},{"key":"9814_CR8","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1007\/BF01161657","volume":"188","author":"HSM Coxeter","year":"1985","unstructured":"Coxeter, H.S.M.: Regular and semi-regular polytopes, II. Math. Z. 188, 559\u2013591 (1985)","journal-title":"Math. Z."},{"key":"9814_CR9","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/BF01161745","volume":"200","author":"HSM Coxeter","year":"1988","unstructured":"Coxeter, H.S.M.: Regular and semi-regular polytopes, III. Math. Z. 200, 3\u201345 (1988)","journal-title":"Math. Z."},{"key":"9814_CR10","volume-title":"Regular Complex Polytopes","author":"HSM Coxeter","year":"1991","unstructured":"Coxeter, H.S.M.: Regular Complex Polytopes, 2nd edn. Cambridge University Press, Cambridge (1991)","edition":"2"},{"key":"9814_CR11","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-21943-0","volume-title":"Generators and Relations for Discrete Groups","author":"HSM Coxeter","year":"1980","unstructured":"Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, New York (1980)","edition":"4"},{"key":"9814_CR12","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1098\/rsta.1954.0003","volume":"246","author":"HSM Coxeter","year":"1954","unstructured":"Coxeter, H.S.M., Longuet-Higgins, M.S., Miller, J.C.P.: Uniform polyhedra. Philos. Trans. R. Soc. Lond. Ser. A 246, 401\u2013450 (1954)","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"key":"9814_CR13","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1007\/s13366-011-0022-1","volume":"52","author":"AM Cutler","year":"2011","unstructured":"Cutler, A.M.: Regular polyhedra of index two, II. Contrib. Algebra Geom. 52, 357\u2013387 (2011)","journal-title":"Contrib. Algebra Geom."},{"key":"9814_CR14","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s13366-011-0015-0","volume":"52","author":"AM Cutler","year":"2011","unstructured":"Cutler, A.M., Schulte, E.: Regular polyhedra of index two, I. Contrib. Algebra Geom. 52, 133\u2013161 (2011)","journal-title":"Contrib. Algebra Geom."},{"key":"9814_CR15","first-page":"507","volume":"7","author":"M Ro Francos del","year":"2014","unstructured":"del Ro Francos, M.: Chamfering operation on k-orbit maps. ARS Math. Contemp. 7, 507\u2013524 (2014)","journal-title":"ARS Math. Contemp."},{"key":"9814_CR16","doi-asserted-by":"crossref","first-page":"2533","DOI":"10.1016\/j.jssc.2005.06.037","volume":"178","author":"O Delgado-Friedrichs","year":"2005","unstructured":"Delgado-Friedrichs, O., Foster, M.D., O\u2019Keefe, M., Proserpio, D.M., Treacy, M.M.J., Yaghi, O.M.: What do we know about three-periodic nets? J. Solid State Chem. 178, 2533\u20132554 (2005)","journal-title":"J. Solid State Chem."},{"key":"9814_CR17","doi-asserted-by":"crossref","first-page":"99","DOI":"10.26493\/1855-3974.26.6c7","volume":"1","author":"M Deza","year":"2008","unstructured":"Deza, M., Sikiric, M.D., Shpectorov, S.: Hypercube embeddings of Wythoffians. ARS Math. Contemp. 1, 99\u2013111 (2008)","journal-title":"ARS Math. Contemp."},{"key":"9814_CR18","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1007\/BF02188039","volume":"23","author":"AWM Dress","year":"1981","unstructured":"Dress, A.W.M.: A combinatorial theory of Gr\u00fcnbaum\u2019s new regular polyhedra, part I: Gr\u00fcnbaum\u2019s new regular polyhedra and their automorphism group. Aequat. Math. 23, 252\u2013265 (1981)","journal-title":"Aequat. Math."},{"key":"9814_CR19","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1007\/BF02189831","volume":"29","author":"AWM Dress","year":"1985","unstructured":"Dress, A.W.M.: A combinatorial theory of Gr\u00fcnbaum\u2019s new regular polyhedra, part II: complete enumeration. Aequat. Math. 29, 222\u2013243 (1985)","journal-title":"Aequat. Math."},{"key":"9814_CR20","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF00148018","volume":"26","author":"SL Farris","year":"1988","unstructured":"Farris, S.L.: Completely classifying all vertex-transitive and edge-transitive polyhedra, part I: necessary class conditions. Geom. Dedicata 26, 111\u2013124 (1988)","journal-title":"Geom. Dedicata"},{"key":"9814_CR21","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-1869-0","volume-title":"Finite Reflection Groups","author":"LC Grove","year":"1985","unstructured":"Grove, L.C., Benson, C.T.: Finite Reflection Groups, 2nd edn. Springer, New York (1985)","edition":"2"},{"key":"9814_CR22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01836414","volume":"16","author":"B Gr\u00fcnbaum","year":"1977","unstructured":"Gr\u00fcnbaum, B.: Regular polyhedra: old and new. Aequat. Math. 16, 1\u201320 (1977)","journal-title":"Aequat. Math."},{"key":"9814_CR23","doi-asserted-by":"crossref","unstructured":"Gr\u00fcnbaum, B.: Polyhedra with hollow faces. In: Polytopes: Abstract, Convex and Computational, pp. 43\u201370 (1994)","DOI":"10.1007\/978-94-011-0924-6_3"},{"key":"9814_CR24","doi-asserted-by":"crossref","unstructured":"Gr\u00fcnbaum, B.: Acoptic polyhedra. In: Advances in Discrete and Computational Geometry, pp. 163\u2013199 (1999)","DOI":"10.1090\/conm\/223\/03137"},{"key":"9814_CR25","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4613-0019-9","volume-title":"Convex Polytopes","author":"B Gr\u00fcnbaum","year":"2003","unstructured":"Gr\u00fcnbaum, B.: Convex Polytopes, 2nd edn. Springer, New York (2003)","edition":"2"},{"key":"9814_CR26","doi-asserted-by":"crossref","unstructured":"Gr\u00fcnbaum, B.: \u201cNew\u201d uniform polyhedra. In: Discrete Geometry: In Honor of W. Kuperberg\u2019s 60th Birthday, pp. 331\u2013350 (2003)","DOI":"10.1201\/9780203911211.ch23"},{"key":"9814_CR27","series-title":"Graduate Studies in Mathematics","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/103","volume-title":"Configurations of Points and Lines","author":"B Gr\u00fcnbaum","year":"2009","unstructured":"Gr\u00fcnbaum, B.: Configurations of Points and Lines. Graduate Studies in Mathematics, 103rd edn. American Mathematical Society, Providence, RI (2009)","edition":"103"},{"key":"9814_CR28","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1112\/jlms\/s1-40.1.577","volume":"40","author":"B Gr\u00fcnbaum","year":"1965","unstructured":"Gr\u00fcnbaum, B., Johnson, N.W.: The faces of a regular-faced polyhedron. J. Lond. Math. Soc. 40, 577\u2013586 (1965)","journal-title":"J. Lond. Math. Soc."},{"key":"9814_CR29","volume-title":"Tilings and Patterns","author":"B Gr\u00fcnbaum","year":"1987","unstructured":"Gr\u00fcnbaum, B., Shephard, G.C.: Tilings and Patterns. WH Freeman and Company, New York (1987)"},{"key":"9814_CR30","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/BF01263494","volume":"47","author":"Z Har\u2019El","year":"1993","unstructured":"Har\u2019El, Z.: Uniform solution for uniform polyhedra. Geom. Dedicata 47, 57\u2013110 (1993)","journal-title":"Geom. Dedicata"},{"key":"9814_CR31","doi-asserted-by":"crossref","first-page":"169","DOI":"10.4153\/CJM-1966-021-8","volume":"18","author":"NW Johnson","year":"1966","unstructured":"Johnson, N.W.: Convex solids with regular faces. Can. J. Math. 18, 169\u2013200 (1966)","journal-title":"Can. J. Math."},{"key":"9814_CR32","unstructured":"Johnson, N.W.: Uniform polytopes (In preparation)"},{"key":"9814_CR33","doi-asserted-by":"crossref","unstructured":"Martini, H.: Hierarchical classification of euclidean polytopes with symmetry properties. In: Polytopes: Abstract, Convex and Computational, pp. 71\u201396 (1994)","DOI":"10.1007\/978-94-011-0924-6_4"},{"key":"9814_CR34","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1007\/BF01837943","volume":"37","author":"P McMullen","year":"1989","unstructured":"McMullen, P.: Realizations of regular polytopes. Aequat. Math. 37, 38\u201356 (1989)","journal-title":"Aequat. Math."},{"issue":"1","key":"9814_CR35","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1007\/s000100300007","volume":"65","author":"P McMullen","year":"2003","unstructured":"McMullen, P., Monson, B.: Realizations of regular polytopes, II. Aequat. Math. 65(1), 102\u2013112 (2003)","journal-title":"Aequat. Math."},{"key":"9814_CR36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00454-004-0848-5","volume":"32","author":"P McMullen","year":"2004","unstructured":"McMullen, P.: Regular polytopes of full rank. Discrete Comput. Geom. 32, 1\u201335 (2004)","journal-title":"Discrete Comput. Geom."},{"key":"9814_CR37","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s00454-007-1342-7","volume":"38","author":"P McMullen","year":"2007","unstructured":"McMullen, P.: Four-dimensional regular polyhedra. Discrete Comput. Geom. 38, 355\u2013387 (2007)","journal-title":"Discrete Comput. Geom."},{"key":"9814_CR38","unstructured":"McMullen, P.: Geometric regular polytopes (In preparation)"},{"key":"9814_CR39","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/PL00009304","volume":"17","author":"P McMullen","year":"1997","unstructured":"McMullen, P., Schulte, E.: Regular polytopes in ordinary space. Discrete Comput. Geom. 17, 449\u2013478 (1997)","journal-title":"Discrete Comput. Geom."},{"key":"9814_CR40","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511546686","volume-title":"Abstract Regular Polytopes","author":"P McMullen","year":"2002","unstructured":"McMullen, P., Schulte, E.: Abstract Regular Polytopes. Cambridge University Press, Cambridge (2002)"},{"key":"9814_CR41","doi-asserted-by":"crossref","first-page":"2651","DOI":"10.1090\/S0002-9947-2013-05954-5","volume":"366","author":"B Monson","year":"2014","unstructured":"Monson, B., Pellicer, D., Williams, G.: Mixing and monodromy of abstract polytopes. Trans. AMS 366, 2651\u20132681 (2014)","journal-title":"Trans. AMS"},{"key":"9814_CR42","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1107\/S010876730800826X","volume":"64","author":"M O\u2019Keeffe","year":"2008","unstructured":"O\u2019Keeffe, M.: Three-periodic nets and tilings: regular and related infinite polyhedra. Acta Crystallogr. A 64, 425\u2013429 (2008)","journal-title":"Acta Crystallogr. A"},{"key":"9814_CR43","volume-title":"Crystal Structures, No. 1: Patterns and Symmetry","author":"M O\u2019Keeffe","year":"1996","unstructured":"O\u2019Keeffe, M., Hyde, B.G.: Crystal Structures, No. 1: Patterns and Symmetry. Mineralogical Society of America, Chantilly (1996)"},{"key":"9814_CR44","doi-asserted-by":"crossref","first-page":"6679","DOI":"10.1090\/S0002-9947-2010-05128-1","volume":"362","author":"D Pellicer","year":"2010","unstructured":"Pellicer, D., Schulte, E.: Regular polygonal complexes in space, I. Trans. Am. Math. Soc. 362, 6679\u20136714 (2010)","journal-title":"Trans. Am. Math. Soc."},{"key":"9814_CR45","doi-asserted-by":"crossref","first-page":"2031","DOI":"10.1090\/S0002-9947-2012-05684-4","volume":"365","author":"D Pellicer","year":"2013","unstructured":"Pellicer, D., Schulte, E.: Regular polygonal complexes in space, II. Trans. Am. Math. Soc. 365, 2031\u20132061 (2013)","journal-title":"Trans. Am. Math. Soc."},{"key":"9814_CR46","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s00454-010-9247-2","volume":"44","author":"D Pellicer","year":"2010","unstructured":"Pellicer, D., Weiss, A.I.: Combinatorial structure of Schulte\u2019s chiral polyhedra. Discrete Comput. Geom. 44, 167\u2013194 (2010)","journal-title":"Discrete Comput. Geom."},{"key":"9814_CR47","first-page":"174","volume":"53","author":"T Pisanski","year":"2000","unstructured":"Pisanski, T., Randic, M.: Bridges between geometry and graph theory. Geom. Work, MAA Notes 53, 174\u2013194 (2000)","journal-title":"Geom. Work, MAA Notes"},{"key":"9814_CR48","doi-asserted-by":"crossref","unstructured":"Pisanski, T., \u017ditnik, A.: Representing graphs and maps. In: Topics in Topological Graph Theory. Encyclopedia of Mathematics and Its Applications, vol. 128, pp. 151\u2013180 (2009)","DOI":"10.1017\/CBO9781139087223.011"},{"key":"9814_CR49","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s00454-004-0843-x","volume":"32","author":"E Schulte","year":"2004","unstructured":"Schulte, E.: Chiral polyhedra in ordinary space, I. Discrete Comput. Geom. 32, 55\u201399 (2004)","journal-title":"Discrete Comput. Geom."},{"key":"9814_CR50","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/s00454-005-1176-0","volume":"34","author":"E Schulte","year":"2005","unstructured":"Schulte, E.: Chiral polyhedra in ordinary space, II. Discrete Comput. Geom. 34, 181\u2013229 (2005)","journal-title":"Discrete Comput. Geom."},{"issue":"3","key":"9814_CR51","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1107\/S2053273314000217","volume":"70","author":"E Schulte","year":"2014","unstructured":"Schulte, E.: Polyhedra, complexes, nets and symmetry. Acta Crystallogr. Sect. A 70(3), 203\u2013216 (2014)","journal-title":"Acta Crystallogr. Sect. A"},{"key":"9814_CR52","volume-title":"A Textbook of Topology","author":"H Seifert","year":"1980","unstructured":"Seifert, H., Threlfall, W.: A Textbook of Topology. Academic Press, New York (1980)"},{"key":"9814_CR53","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-92714-5","volume-title":"Shaping Space: Exploring Polyhedra in Nature, Art and the Geometrical Imagination","author":"M Senechal","year":"2013","unstructured":"Senechal, M.: Shaping Space: Exploring Polyhedra in Nature, Art and the Geometrical Imagination. Springer, New York (2013)"},{"key":"9814_CR54","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1098\/rsta.1975.0022","volume":"278","author":"J Skilling","year":"1975","unstructured":"Skilling, J.: The complete set of uniform polyhedra. Philos. Trans. R. Soc. Lond. Ser. A 278, 111\u2013135 (1975)","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"key":"9814_CR55","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1017\/S0305004100052440","volume":"79","author":"J Skilling","year":"1976","unstructured":"Skilling, J.: Uniform compounds of uniform polyhedra. Math. Proc. Camb. Philos. Soc. 79, 447\u2013457 (1976)","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"9814_CR56","volume-title":"Three-Dimensional Nets and Polyhedra","author":"AF Wells","year":"1977","unstructured":"Wells, A.F.: Three-Dimensional Nets and Polyhedra. Wiley, New York (1977)"},{"key":"9814_CR57","unstructured":"Williams, A.: Wythoffian skeletal polyhedra. PhD Thesis, Northeastern University (2015)"},{"key":"9814_CR58","unstructured":"Williams, A.: Wythoffian skeletal polyhedra in ordinary space, II (In preparation)"},{"key":"9814_CR59","unstructured":"Wolfram Research Inc.: Mathematica 10.0. Wolfram Research, Inc., Champaign. http:\/\/www.wolfram.com (2010)"},{"key":"9814_CR60","unstructured":"Wythoff, W.A.: A Relation Between the Polytopes of the C600-family, vol. 20. Koninklijke Akademie van Wetenschappen te Amsterdam, Amsterdam (1918)"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-016-9814-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-016-9814-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-016-9814-2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,12]],"date-time":"2019-09-12T09:43:25Z","timestamp":1568281405000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-016-9814-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,8,12]]},"references-count":60,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,10]]}},"alternative-id":["9814"],"URL":"https:\/\/doi.org\/10.1007\/s00454-016-9814-2","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"type":"print","value":"0179-5376"},{"type":"electronic","value":"1432-0444"}],"subject":[],"published":{"date-parts":[[2016,8,12]]}}}