Some Geometric Problems Related to Faustian Round Table Coin Game (1)
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Published:
May 31, 2019
Keywords:
Packing, covering, Newton number, Kissing number, Blocking number, Number of light-source, Hadwiger’s conjecture, Zong’s conjecture
Main Article Content
Abstract
In this article, we introduce a game, called Faustian round table coin game, and ask some questions related to this game. In addition, we give the definitions of related terms, e.g. packings, coverings, Newton numbers (or kissing numbers), blocking numbers and numbers of light-sources. Furthermore, we show the results related to these concepts. The goal of this article is to present some basic concepts and tools widely used in discrete geometry.
Article Details
How to Cite
Sriamorn, K. (2019). Some Geometric Problems Related to Faustian Round Table Coin Game (1). Mathematical Journal by The Mathematical Association of Thailand Under The Patronage of His Majesty The King, 63(696), 9–21. Retrieved from https://ph02.tci-thaijo.org/index.php/MJMATh/article/view/191929
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Academic Article
References
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[2] Boltjansky, V. G., Martini, H., and Soltan, P. (1997). Excursions into Combinatorial Geometry. Berlin: Springer.
[3] Böröczky, Jr. K. (2004). Finite Packing and Covering. Cambridge: Cambridge University Press.
[4] Dalla, L., Larman, D. G., Mani-Levitska, P. and Zong, C. M. (2000). The Blocking Numbers of Convex Bodies. Discrete Comput. Geom., 24: 267 - 277.
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[8] Grünbaum, B. (1963). Borsuk’s Problem and Related Questions. Proc. Symp. Pure Math., 7: 271 - 283.
[9] Levenshtein, V. I. (1979). On Bounds for Packings in n-dimensional Euclidean Space. Dokl. Akad. Nauk SSSR 245, 6: 1299 - 1303 (in Russian).
[10] Levi, F. W. (1955). Überdeckung eines Eibereiches durch Parallelverschiebungen seines offenen Kerns. Archiv der Mathematik, 6 (5): 369 - 370.
[11] Musin, O. R. (2008). The Kissing Number in Four Dimensions. Ann. Math. (2) 168: 1 - 32.
[12] Shannon, C. E. (1959). Probability of Error for Optimal Codes in a Gaussian Channel. Bell System Tech. J., 38: 611 - 656.
[13] Odlyzko, A. M. and Sloane, N. J. A. (1979). New Bounds on the Number of Unit Spheres that can touch a Unit Sphere in n dimensions. J. Comb. Theory Ser. A, 26: 210 - 214.
[14] Wyner, J. M. (1965). Capabilities of Bounded Discrepancy Decoding. Bell System Tech. J., 44: 1061 - 1122.
[15] Yu, L. and Zong, C. M. (2009). On the Blocking Number and the Covering Number of a Convex Body. Adv. Geom., 9: 13 - 29.
[16] Zong, C. M. (1996). Strange Phenomena in Convex and Discrete Geometry. New York: Springer.
[17] Zong, C. M. (1999). Sphere Packings. New York: Springer.
[18] Zong, C. M. (2006). What is Known about Unit Cubes. Bull. Amer. Math. Soc., 39: 533 - 555.
[19] Zong, C. M. (2006). The Cube: A Window to Convex and Discrete Geometry. Cambridge: Cambridge University Press.
[20] Zong, C. M. (2008). The Kissing Numbers, Blocking Numbers and Covering Numbers of a Convex Body. Contemp. Math., 453: 529 - 548.
[2] Boltjansky, V. G., Martini, H., and Soltan, P. (1997). Excursions into Combinatorial Geometry. Berlin: Springer.
[3] Böröczky, Jr. K. (2004). Finite Packing and Covering. Cambridge: Cambridge University Press.
[4] Dalla, L., Larman, D. G., Mani-Levitska, P. and Zong, C. M. (2000). The Blocking Numbers of Convex Bodies. Discrete Comput. Geom., 24: 267 - 277.
[5] Eckard, S. (2011). The Best-Known Packings of Equal Circles in a Square. Retrieved November 20, 2018, from http://hydra.nat.uni-magdeburg.de/ packing/csq/csq.html.
[6] Eckard, S. (2014). The Best-Known Packings of Equal Circles in a Circle. Retrieved November 20, 2018, from http://hydra.nat.uni-magdeburg.de/packing/
ccinew/cci.html
[7] Gohberg, I. Ts. and Markus, A. S. (1960). A certain problem about the covering of convex sets with homothetic ones. Izvestiya Moldavskogo Filiala Akademii Nauk SSSR (In Russian).
[8] Grünbaum, B. (1963). Borsuk’s Problem and Related Questions. Proc. Symp. Pure Math., 7: 271 - 283.
[9] Levenshtein, V. I. (1979). On Bounds for Packings in n-dimensional Euclidean Space. Dokl. Akad. Nauk SSSR 245, 6: 1299 - 1303 (in Russian).
[10] Levi, F. W. (1955). Überdeckung eines Eibereiches durch Parallelverschiebungen seines offenen Kerns. Archiv der Mathematik, 6 (5): 369 - 370.
[11] Musin, O. R. (2008). The Kissing Number in Four Dimensions. Ann. Math. (2) 168: 1 - 32.
[12] Shannon, C. E. (1959). Probability of Error for Optimal Codes in a Gaussian Channel. Bell System Tech. J., 38: 611 - 656.
[13] Odlyzko, A. M. and Sloane, N. J. A. (1979). New Bounds on the Number of Unit Spheres that can touch a Unit Sphere in n dimensions. J. Comb. Theory Ser. A, 26: 210 - 214.
[14] Wyner, J. M. (1965). Capabilities of Bounded Discrepancy Decoding. Bell System Tech. J., 44: 1061 - 1122.
[15] Yu, L. and Zong, C. M. (2009). On the Blocking Number and the Covering Number of a Convex Body. Adv. Geom., 9: 13 - 29.
[16] Zong, C. M. (1996). Strange Phenomena in Convex and Discrete Geometry. New York: Springer.
[17] Zong, C. M. (1999). Sphere Packings. New York: Springer.
[18] Zong, C. M. (2006). What is Known about Unit Cubes. Bull. Amer. Math. Soc., 39: 533 - 555.
[19] Zong, C. M. (2006). The Cube: A Window to Convex and Discrete Geometry. Cambridge: Cambridge University Press.
[20] Zong, C. M. (2008). The Kissing Numbers, Blocking Numbers and Covering Numbers of a Convex Body. Contemp. Math., 453: 529 - 548.