Abstract
Multi-shell clusters represent complex structures, the study of which needs rigorous definitions in graph theory, geometry, set theory, etc. Within this chapter, main definitions for polyhedra, regular (Platonic) polyhedra, semi-regular and uniform (Archimedean, Catalan, Johnson’s) polyhedra are given. Then, higher dimensional polytopes are introduced, basically the regular polytopes. Euler formula for polyhedra, and then the alternating sum of higher ranked facets are used to confirm an assumed structure. Abstract polytopes, posets (replacing the dimension concept with that of rank), Hässe diagrams are also discussed. Polytope realization is exemplified by P-centered clusters and “cell-in-cell” clusters, as the simplest 4-dimensional/ranked structures.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Balinski ML (1961) On the graph structure of convex polyhedra in n-space. Pac J Math 11(2):431–434
Buekenhout F, Parker M (1998) The number of nets of the regular convex polytopes in dimension ≤4. Discret Math 186:69–94
Catalan E (1865) Mémoire sur la Théorie des Polyèdres. J de l’École Polytechnique (Paris) 41:1–71
Coxeter HSM (1934) Discrete groups generated by reflections. Ann Math 35(3):588–621
Coxeter HSM (1940) Regular and semi-regular polytopes. J Math Zeit 46:380–407
Coxeter HSM (1973) Regular polytopes, 3rd edn. Dover, New York, NY
Coxeter HSM (1974) Regular complex polytopes. Cambridge University Press, Cambridge
Coxeter HSM (1982) Ten toroids and fifty-seven hemi-dodecahedra. Geom Dedicata 13:87–99
Coxeter HSM (1984) A symmetrical arrangement of eleven hemi-icosahedra. Ann Discrete Math 20:103–114
Coxeter HSM, Longuet-Higgins MS, Miller JCP (1954) Uniform Polyhedra. Phil Trans A, 403–439.
Cromwell PR (1997) Polyhedra. Cambridge University Press, Cambridge
Davis MW (2007) The geometry and topology of coxeter groups, London Mathematical Society monographs series, vol 32. Princeton University Press, Princeton, NJ
de la Vaissière B, Fowler PW, Deza M (2001) Codes in Platonic, Archimedean, Catalan, and related polyhedra: a model for maximum addition patterns in chemical cages. J Chem Inf Comput Sci 41:376–386
Euler L (1736) Solutio problematisad geometriam situs pertinentis. Comment Acad Sci I Petropolitanae 8:128–140
Euler L (1752–1753) Elementa doctrinae solidorum. Novi Comment Acad Sci I Petropolitanae 4:109–160
Grünbaum B (2003) Convex polytopes. In: Kaibel V, Klee V, Ziegler GM (eds) Graduate texts in mathematics, vol 221, 2nd edn. Springer, New York, NY
Grünbaum B (2009) Elemente der Mathematik 64 (3):89–101; Reprinted in Pitici M (ed) (2011) The best writing on mathematics 2010. Princeton University Press, pp 18–31
Grünbaum B, Shephard GC (1988) Duality of polyhedra. In: Senechal M, Fleck GM (eds) Shaping space – a polyhedral approach. Birkhäuser, Boston, MA, pp 205–211
Heath TL (1981) A history of Greek mathematics. Dover, New York, NY
Johnson NW (1966) Convex solids with regular faces. Can J Math 18:169–200 (The theory of uniform polytopes and honeycombs. Ph.D. Dissertation, University of Toronto)
Leinster T (2008) The Euler characteristic of a category. Doc Math 13:21–49
Lyusternik LA (1963) Convex figures and polyhedra. Dover, New York, NY
McMullen P, Schulte P (2002) Abstract regular polytopes, 1st edn. Cambridge University Press, Cambridge
Parvan-Moldovan A, Diudea MV (2015) Cell@cell higher dimensional structures. Stud Univ Babes-Bolyai Chem 60(2):379–388
Schläfli L (1901) Theorie der vielfachen Kontinuität Zürcher und Furrer, Zürich (Reprinted in: Ludwig Schläfli, 1814–1895, Gesammelte Mathematische Abhandlungen, Band 1, 167–387, Verlag Birkhäuser, Basel, 1950)
Schulte E (1980) Regular incidence complexes. PhD Dissertation, Dortmund University
Schulte E (1983a) Regulire Inzidenzkomplexe, II. Geom Dedicata 14:33–56
Schulte E (1983b) Regulire Inzidenzkomplexe, III. Geom Dedicata 14:57–79
Schulte E (1983c) On arranging regular incidence-complexes as faces of higher-dimensional ones. Eur J Comb 4:375–384
Schulte E (1985) Regular incidence-polytopes with Euclidean or toroidal faces and vertex-figures. J Comb Theory Ser A 40(2):305–330
Schulte E (2004) Symmetry of polytopes and polyhedra. In: Goodman JE, O’Rourke J (eds) Handbook of discrete and computational geometry, 2nd edn. Chapman & Hall, New York, NY
Schulte E (2014) Polyhedra, complexes, nets and symmetry. Acta Crystallogr A 70:203–216
Skilling J (1975) The complete set of uniform polyhedra. Philos Trans R Soc Lond Ser A Math Phys Sci 278(1278):111–135
Sopov SP (1970) Proof of the completeness of the enumeration of uniform polyhedra. Ukrain Geom Sbornik 8:139–156
Steinitz E (1922) Polyeder und Raumeinteilungen, Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometries) (IIIAB12), pp 1–139, Abgeschlossen am 31 August 1916
Tits J (1964) Algebraic and abstract simple groups. Ann Math 2nd Ser 80(2):313–329
Tits J, Weiss RM (2002) Moufang polygons. Springer monographs in mathematics. Springer, Berlin
Wenninger M (1974) Polyhedron models. Cambridge University Press, Cambridge
Wenninger M (1983) Dual models. Cambridge University Press, Cambridge
Ziegler GM (1995) Lectures on polytopes, graduate texts in mathematics 152. Chap. 4 “Steinitz’ theorem for 3-polytopes”. Springer, Berlin, p 103
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Diudea, M.V. (2018). Definitions in Polytopes. In: Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-64123-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-64123-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-64121-8
Online ISBN: 978-3-319-64123-2
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)