Izbrane teme sodobne fizike in matematike
Članek obravnava razmerje med volumnoma centralne krogle pri enostavnem kubičnem pakiranju in hiperkocko. V članku je naprej predstavljen problem pakiranja krogel na splošno, nekaj zgodovine, najboljše pakiranje v tridimenzionalnem prostoru in sorodni problem Newtonovega števila oziroma “kissing number”. Kasneje je govora o enostavnem kubičnem pakiranju krogel, tj. posebnem primeru pakiranja krogel v kocko, najprej v 2−, 3− in 4−dimenzionalnem prostoru, nato pa še v poljubnem n−dimenzionalnem prostoru. Govora bo predvsem o centralni krogli, njenem radiju in volumnu. Predstavljen bo dokaz za izrek, ki pravi, da gre razmerje volumnom centralne krogle in kvadra preko vsake meje, ko gre dimenzija preko vsake meje. Na koncu pa se članek dotakne še uporabe pakiranja krogel v komunikacijskih sistemih.
The article discusses the ratio between volume of the central sphere of simple cubic packing and volume of the hypercube. In the paper, one can find something about sphere packing in general, some historical background, the best structure for 3−dimensional space and the relatable problem of kissing number. Later on, the paper provides more information about simple cubic packing, that is one of the simplest packings of spheres in the cube, presented in 2−, 3−, 4− and n−dimensional space. The central sphere, its radius and volume will be mostly discussed. In the article one can also find a proof of the theorem that tells us that the ratio between volume of the central sphere and volume of the hypercube rises to infinity while dimension rises to infinity. In the end, one can find usage of sphere packing in communication systems.