Definitions in Polytopes

  • Mircea Vasile Diudea
Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 10)


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.


  1. Balinski ML (1961) On the graph structure of convex polyhedra in n-space. Pac J Math 11(2):431–434CrossRefGoogle Scholar
  2. Buekenhout F, Parker M (1998) The number of nets of the regular convex polytopes in dimension ≤4. Discret Math 186:69–94CrossRefGoogle Scholar
  3. Catalan E (1865) Mémoire sur la Théorie des Polyèdres. J de l’École Polytechnique (Paris) 41:1–71Google Scholar
  4. Coxeter HSM (1934) Discrete groups generated by reflections. Ann Math 35(3):588–621CrossRefGoogle Scholar
  5. Coxeter HSM (1940) Regular and semi-regular polytopes. J Math Zeit 46:380–407CrossRefGoogle Scholar
  6. Coxeter HSM (1973) Regular polytopes, 3rd edn. Dover, New York, NYGoogle Scholar
  7. Coxeter HSM (1974) Regular complex polytopes. Cambridge University Press, CambridgeGoogle Scholar
  8. Coxeter HSM (1982) Ten toroids and fifty-seven hemi-dodecahedra. Geom Dedicata 13:87–99CrossRefGoogle Scholar
  9. Coxeter HSM (1984) A symmetrical arrangement of eleven hemi-icosahedra. Ann Discrete Math 20:103–114Google Scholar
  10. Coxeter HSM, Longuet-Higgins MS, Miller JCP (1954) Uniform Polyhedra. Phil Trans A, 403–439.Google Scholar
  11. Cromwell PR (1997) Polyhedra. Cambridge University Press, CambridgeGoogle Scholar
  12. Davis MW (2007) The geometry and topology of coxeter groups, London Mathematical Society monographs series, vol 32. Princeton University Press, Princeton, NJGoogle Scholar
  13. 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 41:376–386CrossRefGoogle Scholar
  14. Euler L (1736) Solutio problematisad geometriam situs pertinentis. Comment Acad Sci I Petropolitanae 8:128–140Google Scholar
  15. Euler L (1752–1753) Elementa doctrinae solidorum. Novi Comment Acad Sci I Petropolitanae 4:109–160Google Scholar
  16. Grünbaum B (2003) Convex polytopes. In: Kaibel V, Klee V, Ziegler GM (eds) Graduate texts in mathematics, vol 221, 2nd edn. Springer, New York, NYGoogle Scholar
  17. 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 18–31Google Scholar
  18. Grünbaum B, Shephard GC (1988) Duality of polyhedra. In: Senechal M, Fleck GM (eds) Shaping space – a polyhedral approach. Birkhäuser, Boston, MA, pp 205–211Google Scholar
  19. Heath TL (1981) A history of Greek mathematics. Dover, New York, NYGoogle Scholar
  20. 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)Google Scholar
  21. Leinster T (2008) The Euler characteristic of a category. Doc Math 13:21–49Google Scholar
  22. Lyusternik LA (1963) Convex figures and polyhedra. Dover, New York, NYGoogle Scholar
  23. McMullen P, Schulte P (2002) Abstract regular polytopes, 1st edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  24. Parvan-Moldovan A, Diudea MV (2015) Cell@cell higher dimensional structures. Stud Univ Babes-Bolyai Chem 60(2):379–388Google Scholar
  25. 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)Google Scholar
  26. Schulte E (1980) Regular incidence complexes. PhD Dissertation, Dortmund UniversityGoogle Scholar
  27. Schulte E (1983a) Regulire Inzidenzkomplexe, II. Geom Dedicata 14:33–56Google Scholar
  28. Schulte E (1983b) Regulire Inzidenzkomplexe, III. Geom Dedicata 14:57–79Google Scholar
  29. Schulte E (1983c) On arranging regular incidence-complexes as faces of higher-dimensional ones. Eur J Comb 4:375–384CrossRefGoogle Scholar
  30. Schulte E (1985) Regular incidence-polytopes with Euclidean or toroidal faces and vertex-figures. J Comb Theory Ser A 40(2):305–330CrossRefGoogle Scholar
  31. 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, NYGoogle Scholar
  32. Schulte E (2014) Polyhedra, complexes, nets and symmetry. Acta Crystallogr A 70:203–216CrossRefGoogle Scholar
  33. Skilling J (1975) The complete set of uniform polyhedra. Philos Trans R Soc Lond Ser A Math Phys Sci 278(1278):111–135CrossRefGoogle Scholar
  34. Sopov SP (1970) Proof of the completeness of the enumeration of uniform polyhedra. Ukrain Geom Sbornik 8:139–156Google Scholar
  35. Steinitz E (1922) Polyeder und Raumeinteilungen, Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometries) (IIIAB12), pp 1–139, Abgeschlossen am 31 August 1916Google Scholar
  36. Tits J (1964) Algebraic and abstract simple groups. Ann Math 2nd Ser 80(2):313–329CrossRefGoogle Scholar
  37. Tits J, Weiss RM (2002) Moufang polygons. Springer monographs in mathematics. Springer, BerlinCrossRefGoogle Scholar
  38. Wenninger M (1974) Polyhedron models. Cambridge University Press, CambridgeGoogle Scholar
  39. Wenninger M (1983) Dual models. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  40. Ziegler GM (1995) Lectures on polytopes, graduate texts in mathematics 152. Chap. 4 “Steinitz’ theorem for 3-polytopes”. Springer, Berlin, p 103Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Mircea Vasile Diudea
    • 1
  1. 1.Department of Chemistry, Faculty of Chemistry and Chemical EngineeringBabes-Bolyai UniversityCluj-NapocaRomania

Personalised recommendations