Rhaniad arwynebedd lleiaf silindr yn dair rhan


Rhaniad arwynebedd lleiaf silindr yn dair rhan
Tudur Davies, Lee Garratt a Simon Cox

Yn yr erthygl hon, dadansoddir datrysiadau dichonadwy i’r broblem geometrig o rannu silindr yn dair rhan â’r un cyfaint. Darganfyddir y datrysiadau yng nghyd-destun cyflwr egnïol isaf ewyn hylifol sych. Defnyddir y meddalwedd efelychu rhifiadol Surface Evolver er mwyn enrhifo’r holl ddatrysiadau a chyfrifo’r arwynebedd ym mhob achos. Darganfyddir y datrysiad arwynebedd lleiaf ar gyfer holl werthoedd cymhareb agwedd y silindr, sef hyd ei radiws wedi’i rannu â’i uchder. Dangosir mai pedwar datrysiad optimaidd sydd i’r broblem ar gyfer holl werthoedd y gymhareb agwedd. Rhoddir cyfwng ar gyfer cymhareb agwedd y silindr ar gyfer pob un o’r datrysiadau optimaidd.


Cyfeiriad:

 
  	Tudur Davies, Lee Garratt a Simon Cox, 'Rhaniad arwynebedd lleiaf silindr yn dair rhan', Gwerddon, 20, Hydref 2015, 30-43.
   

Allweddeiriau

 
    Mathemateg, optimeiddio, arwynebau lleiaf, ewyn, geometreg, problem Kelvin, deddfau Plateau.
    

Llyfryddiaeth:

 
  	
  1. Alfaro, M., Brock, J., Foisy, J., et al. (1990), ‘Compound soap bubbles in the plane. SMALL Geometry Group’, traethawd PhD, Williams College, Williamstown, MA.
  2. Alfaro, M., Brock, J., Foisy, J., et al. (1993), ‘The standard double soap bubble in R2 uniquely minimizes perimeter’, Pacific Journal of Mathematics, 159, 47–59.
  3. Almgren, F. a Taylor, J. (1976), ‘The geometry of soap films and soap bubbles’, Scientific American, 82–93.
  4. Bleicher, M. N. (1987), ‘Isoperimetric divisions into several cells with natural boundary’, Intuitive Geometry, Colloquia Mathematica Societatis János Bolyai, 48, 63–84.
  5. Brakke, K. (1992), ‘The Surface Evolver’, Experimental Mathematics, 1, 141–52.
  6. Canete, A. a Ritore, M. (2004), ‘Least-perimeter partitions of the disk into three regions of given areas’, Indiana University Mathematics Journal, 53, 883–904.
  7. Cantat, I., Cohen-Addad, S., Elias, F., et al. (2013), Foams Structure and Dynamics (Oxford: Oxford University Press).
  8. Cox, S. J. (2006), ‘Calculations of the minimal perimeter for n deformable cells of equal area confined in a circle’, Philosophical Magazine Letters, 86, 569–78.
  9. Cox, S. J. a Flikkema, E. (2010), ‘The minimal perimeter for N confined deformable bubbles of equal area’, The Electronic Journal of Combinatorics, 17, R45.
  10. Cox, S. J. a Graner, F. (2003), ‘Large two-dimensional clusters of equal-area bubbles’, Philosophical Magazine, 83, 2573–84.
  11. Cox, S. J., Graner, F., Vaz, M. F., et al. (2003), ‘Minimal perimeter for n identical bubbles in two dimensions: calculations and simulations’, Philosophical Magazine, 83, 1393–1406.
  12. Engelstein, M. (2010), ‘The least-perimeter partition of a sphere into four equal areas’, Discrete & Computational Geometry, 44, 645–53.
  13. Goldberg, M. (1934), ‘The isoperimetric problem for polyhedra’, Tohoku Mathematical Journal, 40, 226–36.
  14. Hales, T. C. (2002), ‘The honeycomb conjecture on the sphere’, arXiv:math/0211234.
  15. Heppes, A. a Morgan, F. (2005), ‘Planar clusters and perimeter bounds’, Philosophical Magazine, 85, 1333–45.
  16. Morgan, F. (1994), ‘Mathematicians, including undergraduates, look at soap bubbles’, The American Mathematical Monthly, 101, 343–51.
  17. Morgan, F. (2000), Geometric Measure Theory (Williamstown, MA: Academic Press).
  18. Plateau, J. A. F. (1873), Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires (Paris: Gauthier-Villars).
  19. Ros, A. (2005), ‘The isoperimetric problem’, yn Hoffman, D. (gol.), Global Theory of Minimal Surfaces, Clay Mathematics Proceedings, Volume 2, tt.175–209 (Cambridge, MA: American Mathematical Society).
  20. Taylor, J. E. (1976), ‘The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces’, Annals of Mathematics, 103, 489–539.
  21. Tomonaga, Y. (1974), ‘Geometry of Length and Area’, traethawd PhD, Utsonomiy University, Tokyo.
  22. Thomson, W. (1887), ‘On the division of space with minimum partitional area’, Philosophical Magazine, 24, 503.
  23. Weaire, D. (1994), The Kelvin Problem (London: Taylor & Francis).
  24. Weaire, D., a Hutzler, S. (2000), The Physics of Foams (Oxford: Oxford University Press).
  25. Weaire, D., a Phelan, R. (1993), ‘A counter-example to Kelvin’s conjecture on minimal surfaces’, Philosophical Magazine Letters, 69, 107–10.
  26. Wichiramala, W. (2004), ‘Proof of the planar triple bubble conjecture’, Journal für die  reine und agnewandte Mathematik, 567, 1–49.