Abstract
Let v 1,…,v n be unit vectors in ℝn such that v i ⋅v j =−w for i≠j, where \(-1<w<\frac{1}{n-1}\) . The points ∑ n i=1 λ i v i (1≥λ 1≥⋅⋅⋅≥λ n ≥0) form a “Hill-simplex of the first type,” denoted by \(\mathcal {Q}_{n}(w)\) . It was shown by Hadwiger in 1951 that \(\mathcal {Q}_{n}(w)\) is equidissectable with a cube. In 1985, Schöbi gave a three-piece dissection of \(\mathcal {Q}_{3}(w)\) into a triangular prism \(c\mathcal {Q}_{2}(\frac{1}{2})\times I\) , where I denotes an interval and \(c=\sqrt{2(w+1)/3}\) . In this paper, we generalize Schöbi’s dissection to an n-piece dissection of \(\mathcal {Q}_{n}(w)\) into a prism \(c\mathcal {Q}_{n-1}(\frac{1}{n-1})\times I\) , where \(c=\sqrt{(n-1)(w+1)/n}\) . Iterating this process leads to a dissection of \(\mathcal {Q}_{n}(w)\) into an n-dimensional rectangular parallelepiped (or “brick”) using at most n! pieces. The complexity of computing the map from \(\mathcal {Q}_{n}(w)\) to the brick is O(n 2). A second generalization of Schöbi’s dissection is given which applies specifically in ℝ4. The results have applications to source coding and to constant-weight binary codes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aguiló, F., Fiol, M.A., Fiol, M.L.: Periodic tilings as a dissection method. Am. Math. Mon. 107, 341–352 (2000)
Akiyama, J., Nakamura, G.: Dudeney dissection of polygons. In: Akiyama, J., Kano, M., Urabe, M. (eds.) Discrete and Computational Geometry, Tokyo, 1998. Lecture Notes in Comput. Sci., vol. 1763, pp. 14–29. Springer, Berlin (2000)
Barber, C.B., Dobkin, D.P., Huhdanpaa, H.T.: The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22, 469–483 (1996). Available from http://www.qhull.org
Boltianskii, V.G.: Hilbert’s Third Problem (transl. from the Russian by R.A. Silverman). Wiley, New York (1978)
Coffin, S.T.: The Puzzling World of Polyhedral Dissections. Oxford Univ. Press, London (1961)
Cohn, M.J.: Economical triangle–square dissection. Geom. Dedicata 3, 447–467 (1974–1975)
Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd. edn. Springer, New York (1998)
Conway, J.H., Torquato, S.: Packing, tiling, and covering with tetrahedra. Proc. Natl. Acad. Sci. USA 103, 10612–10617 (2006)
Coxeter, H.S.M.: Regular Polytopes, 3rd. edn. Dover, New York (1973)
Cromwell, P.R.: Polyhedra. Cambridge Univ. Press, Cambridge (1997)
Crowe, D.W., Schoenberg, I.J.: On the equidecomposability of a regular triangle and a square of equal areas. Mitt. Math. Semin. Giessen 164, 59–64 (1984)
Debrunner, H.E.: Tiling Euclidean d-space with congruent simplexes. In: Goodman, J.E., et al. (eds.) Discrete Geometry and Convexity, New York, 1982. Ann. New York Acad. Sci., vol. 140, pp. 230–261 (1985)
Dupont, J.L.: Scissors Congruences, Group Homology and Characteristic Classes. World Scientific, Singapore (2001)
Eves, H.: A Survey of Geometry, vol. 1. Allyn and Bacon, Boston (1966)
Frederickson, G.N.: Dissections: Plane and Fancy. Cambridge Univ. Press, Cambridge (1997)
Frederickson, G.N.: Hinged Dissections: Swinging & Twisting. Cambridge Univ. Press, Cambridge (2002)
Friedelmeyer, J.-P.: Dissections et puzzles. Pour la science, Dossier 59, April/June 2008
Gardner, M.: The Second Scientific American Book of Mathematical Puzzles and Diversions. Simon and Schuster, New York (1961)
Hadwiger, H.: Hillsche Hypertetraeder. Gazeta Mat. (Lisboa) 12(50), 47–48 (1951)
Hertel, E.: Verallgemeinerte Hadwiger-Hill Simplexe. Reports on Algebra and Geometry, Mathematisches Institut, Friedrich-Schiller-Universität Jena, Jena, Germany (2003)
Hill, M.J.M.: Determination of the volumes of certain species of tetrahedra without employment of the method of limits. Proc. Lond. Math. Soc. 2, 39–53 (1895–1896)
Lindgren, H.: Geometric Dissections. Van Nostrand, Princeton (1964); Revised edition with an appendix by G.N. Frederickson, Dover, New York (1972)
Macaulay, W.H.: The dissection of rectilinear figures. Math. Gaz. 7, 381–388 (1914); 8, 72–76 and 109–115 (1915)
Macaulay, W.H.: The dissection of rectilinear figures. Messenger Math. 48, 159–165 (1919); 49, 111–121 (1919); 52, 53–56 (1922)
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
MathWorks: MATLAB User’s Guide. MathWorks, Natick (2007)
McMullen, P.: Valuations and dissections. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. B, pp. 933–988. North-Holland, Amsterdam (1993)
McMullen, P., Schneider, R.: Valuations on convex bodies. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, Convexity and its Applications, pp. 170–247. Birkhäuser, Basel (1983)
Müller, C.: Equidecomposability of polyhedra with reference to crystallographic groups. Discrete Comput. Geom. 3, 383–389 (1988)
Paterson, D.A.: Geometric dissections in 4-D. J. Recreat. Math. 28, 22–37 (1996–1997)
Sah, C.-H.: Hilbert’s Third Problem: Scissors Congruence. Pitman, London (1979)
Schöbi, P.: Ein elementarer und konstruktiver Beweis für die Zerlegungsgleichheit der Hill’schen Tetraeder mit einer Quader. Elem. Math. 40, 85–97 (1985)
Schoenberg, I.J.: Mathematical Time Exposures. Math. Assoc. America (1982)
Steinhaus, H.: Mathematical Snapshots. Stechert, New York (1938)
Steinhaus, H.: Mathematical Snapshots, 3rd American edn. Oxford (1983)
Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and Its Applications. Wiley, Chichester (1987)
Sydler, J.-P.: Sur les tétraèdres équivalents à un cube. Elem. Math. 11, 78–81 (1956)
Tian, C., Vaishampayan, V.A., Sloane, N.J.A.: Constant weight codes: a geometric approach based on dissections. Preprint (2007)
Wells, D.: The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, London (1991)
Yandell, B.H.: The Honors Class: Hilbert’s Problems and Their Solvers. Peters, Natick (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sloane, N.J.A., Vaishampayan, V.A. Generalizations of Schöbi’s Tetrahedral Dissection. Discrete Comput Geom 41, 232–248 (2009). https://doi.org/10.1007/s00454-008-9086-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-008-9086-6