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The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra

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References

  1. A. D. Aleksandrov, Elementary deduction of the theorem about the center of a convex parallelohedron in 3 dimensions [In Russian]. Trudy fiz.-mat. Inst. Akad. Nauk im. Steklov 4 (1933), 89–99.

    Google Scholar 

  2. A. D. Alexandrov, Convex Polyhedra. Springer, Berlin 2005. Russian original: Moscow 1950; German translation: Berlin 1958.

  3. S. Bilinski, Über die Rhombenisoeder. Glasnik Mat. Fiz. Astr. 15 (1960), 251–263. The quite detailed review by J. J. Burckhardt in Zentralblatt v. 99, p. 155 #15506, does not mention that this contains a correction of Fedorov’s claim. Coxeter in MR 24#A1644 mentions that “…the ‘second’ has never before been noticed” – but does not mention Fedorov.

  4. M. Brückner, Vielecke und Vielflache. Teubner, Leipzig 1900.

  5. J. J. Burckhardt, Über konvexe Körper mit Mittelpunkt. Vierteljschr. Naturforsch. Ges. ZÜrich 85 (1940), Beiblatt. Festschrift R. Fueter, pp. 149–154.

  6. J. L. Cowley, Geometry Made Easy: A New and Methodical Explanation of the ELEMNENTS [sic] of GEOMETRY. Mechell, London 1752.

    Google Scholar 

  7. H. S. M. Coxeter, The classification of zonohedra by means of projective diagrams. J. de math. pures et appliq. 41 (1962), 137–156. Reprinted in: Twelve Geometric Essays, Southern Illinois Univ. Press, Carbondale, IL, 1968 = The Beauty of Geometry. Twelve Essays. Dover, Mineola, NY, 1999.

  8. B. N. Delone, Sur la partition régulière de l’espace à 4 dimensions. Izv. Akad. Nauk SSSR Otdel Fiz.-Mat. Nauk 7 (1929), 79–110, 147–164.

    Google Scholar 

  9. E. S. Fedorov, Nachala Ucheniya o Figurah [In Russian] (= Elements of the theory of figures) Notices Imper. Petersburg Mineralog. Soc., 2nd ser., 24 (1885), 1–279. Republished by the Acad. Sci. USSR, Moscow 1953.

  10. E. S. Fedorov, Elemente der Gestaltenlehre. Z. für Krystallographie und Mineralogie 21 (1893), 679–694.

    Google Scholar 

  11. E. v. Fedorow (E. S. Fedorov) Erwiderung auf die Bemerkungen zu E. v. Fedorow’s Elemente der Gestaltenlehre von Edmund Hess. Neues Jahrbuch für Mineralogie, Geologie und Paleontologie, 1894, part 2, pp. 86–88.

  12. E. S. Fedorov, Reguläre Plan- und Raumtheilung. Abh. K. Bayer. Akademie der Wiss. Vol. 20 (1900), pp. 465–588 + 11 plates. Russian translation with additional comments: “Pravilnoe Delenie Ploskosti i Prostranstva” (Regular Partition of Plane and Space). Nauka, Leningrad 1979.

  13. B. Grünbaum, An enduring error. Elemente der Math. 64 (2009), 89–101.

    Article  MATH  Google Scholar 

  14. B. Grünbaum, Tilings by some nonconvex parallelohedra. Geombinatorics, 19 (2010), 100–107.

  15. B. Grünbaum, Census of rhombic hexecontahedra (In preparation). Mentioned in [25].

  16. B. Grünbaum and G. C. Shephard, Tilings and Patterns. Freeman, New York 1987.

    MATH  Google Scholar 

  17. G. W. Hart, Dodecahedra. http://www.georgehart.com/virtual-polyhedra/dodecahedra.html (as of Oct. 15, 2009).

  18. G. W. Hart, A color-matching dissection of the rhombic enneacontahedron. http://www.georgehart.com/dissect-re/dissect-re.htm (as of Oct. 15, 2009).

  19. E. Hess, Ueber zwei Erweiterungen des Begriffs der regelmässigen Körper. Sitzungsberichte der Gesellschaft zur Beförderung der gesammten Naturwissenschaften zu Marburg, No. 1–2 (1875), pp. 1–20.

  20. E. Hess, Bemerkungen zu E. v. Fedorow’s Elementen der Gestaltenlehre. Neues Jahrbuch für Mineralogie, Geologie und Paleontologie, 1894, part 1, pp. 197–199.

  21. E. Hess, Weitere Bemerkungen zu E. v. Fedorow’s Elementen der Gestaltenlehre. Neues Jahrbuch für Mineralogie, Geologie und Paleontologie, 1894, part 2, pp. 88–90.

  22. J. Kappraff, Connections. 2nd ed. World Scientific, River Edge, NJ 2001.

    Book  MATH  Google Scholar 

  23. J. Kepler, Harmonice Mundi. Lincii 1619; English translation of Book 2: J. V. Field, Kepler’s Star Polyhedra, Vistas in Astronomy 23 (1979), 109–141.

    Article  MathSciNet  Google Scholar 

  24. E. A. Lord, A. L. Mackay, and S. Ranganathan, New Geometries for New Materials. Cambridge Univ. Press 2006.

    MATH  Google Scholar 

  25. J. McNeill, Polyhedra. http://www.orchidpalms.com/polyhedra/ In particular http://www.orchidpalms.com/polyhedra/rhombic/RTC/RTC.htm (as of Oct. 10, 2009).

  26. L. Michel, S. S. Ryshkov, and M. Senechal, An extension of Voronoï’s theorem on primitive parallelohedra. Europ. J. Combinatorics 16 (1995), 59–63.

    Article  MATH  MathSciNet  Google Scholar 

  27. H. Minkowski, Allgemeine Lehrsätze über die konvexen Polyeder. Nachr. Gesell. Wiss. Göttingen, math.-phys. Kl. 1897, pp. 198–219 = Gesamm. Abh. von Hermann Minkowski, vol. 2, Leipzig 1911. Reprinted by Chelsea, New York 1967, pp. 103–121.

  28. T. Ogawa, Three-dimensional Penrose transformation and the ideal quasicrystals. Science on Form: Proc. First Internat. Sympos. for Science on Form, S. Isihzaka et al., eds. KTK Publisher, Tokyo 1986, pp. 479–489.

  29. M. O’Keeffe, 4-connected nets of packings of non-convex parallelohedra and related simple polyhedra. Zeitschrift für Kristallographie 214 (1999), 438–442.

    Article  MathSciNet  Google Scholar 

  30. A. Schoenflies, Symmetrie und Struktur der Krystalle. Encykl. Math. Wissenschaften. Bd. 7. Krystallographie. Teil B, (1906), pp. 437–478.

  31. M. Senechal and R. V. Galiulin, An Introduction to the Theory of Figures: the geometry of E. S. Fedorov. Structural Topology 10 (1984), 5–22.

    MATH  MathSciNet  Google Scholar 

  32. S. K. Stein, Factoring by subsets. Pacif. J. Math. 22 (1967), 523–541.

    MATH  Google Scholar 

  33. S. K. Stein and S. Szabó, Algebra and Tiling. Math, Assoc. of America, Washington, DC 1994.

  34. E. Steinitz, Polyeder und Raumeinteilungen, Enzykl. Math.Wiss. (Geometrie) 3 (Part 3 AB 12) (1922) 1–139.

  35. S. Szabo, A star polyhedron that tiles but not as a fundamental domain. Intuitive Geometry (Siófok, 1985), Colloq. Math. Soc. János Bolyai, 48, North-Holland, Amsterdam 1987.

  36. J. E. Taylor, Zonohedra and generalized zonohedra. Amer. Math. Monthly 99 (1992), 108–111.

    Article  MATH  MathSciNet  Google Scholar 

  37. H. Unkelbach, Die kantensymmetrischen, gleichkantigen Polyeder. Deutsche Mathematik 5 (1940), 306–316. Reviewed by H. S. M. Coxeter in Math. Reviews 7 (1946), p.164.

  38. G. Voronoï, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. J. reine angew. Math. 134 (1908), 198–287; 135 (1909), 67–181.

  39. R. Williams, Natural Structure. Eudaemon Press, Mooepark, CA 1972. Corrected reprint: The Geometrical Foundation of Natural Structure. Dover, NY 1979.

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  1. Department of Mathematics, University of Washington 354350, Seattle, WA, 98195-4350, USA

    Branko Grünbaum

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Grünbaum, B. The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra. Math Intelligencer 32, 5–15 (2010). https://doi.org/10.1007/s00283-010-9138-7

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