Metrika

, Volume 77, Issue 2, pp 247–256 | Cite as

Cuboidal dice and Gibbs distributions

  • Wolfgang Riemer
  • Dietrich Stoyan
  • Danail Obreschkow
Article

Abstract

What are the face-probabilities of a cuboidal die, i.e. a die with different side-lengths? This paper introduces a model for these probabilities based on a Gibbs distribution. Experimental data produced in this work and drawn from the literature support the Gibbs model. The experiments also reveal that the physical conditions, such as the quality of the surface onto which the dice are dropped, can affect the face-probabilities. In the Gibbs model, those variations are condensed in a single parameter, adjustable to the physical conditions.

Keywords

Cuboidal dice Asymmetric dice Tossing Gibbs distribution Boltzmann distribution 

References

  1. Budden F (1980) Throwing non-cubical dice. Math Gazette 64(429):196–198CrossRefGoogle Scholar
  2. Gibbons JD, Chakraborti S (2003) Nonparametric statistical inference, 4th edn. Marcel Dekker, New YorkMATHGoogle Scholar
  3. Heilbronner E (1985) Crooked dice. J Recreat Math 17:177Google Scholar
  4. Hyšková M, Kalousová A, Saxl I (2012) Early history of geometric probability and stereology. Image Anal Stereol 31:1–16CrossRefMathSciNetGoogle Scholar
  5. Newton I, Whiteside DT (ed) (1967) The mathematical papers of Isaac Newton. Cambridge University Press, Cambridge, vol I, 1664–1666, pp 60–61Google Scholar
  6. Obreschkow D (2006) Broken symmetry and the magic of irregular dice. Online project. http://www.quantumholism.com/cuboid
  7. Riemer W (1991) Stochastische Probleme aus elementarer Sicht. Bibliographisches Institut, Mannheim, Wien, ZürichmMATHGoogle Scholar
  8. Simpson T (1740) The nature and laws of chance. Cave, LondonGoogle Scholar
  9. Singmaster D (1981) Theoretical probabilities for a cubical die. Math Gazette 65:208–210CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Wolfgang Riemer
    • 1
  • Dietrich Stoyan
    • 2
  • Danail Obreschkow
    • 3
  1. 1.Zentrum für Lehrerbildung KölnPulheimGermany
  2. 2.Institut für StochastikFreibergGermany
  3. 3.International Centre for Radio Astronomy Research, M468University of Western AustraliaCrawleyAustralia

Personalised recommendations