Editing Kepler conjecture (section)

==Background==
[[Image:Closepacking.svg|thumb|Diagrams of cubic close packing (left) and hexagonal close packing (right).]]
Imagine filling a large container with small equal-sized spheres: Say a porcelain gallon jug with identical marbles. The "density" of the arrangement is equal to the total volume of all the marbles, divided by the volume of the jug. To maximize the number of marbles in the jug means to create an arrangement of marbles stacked between the sides and bottom of the jug, that has the highest possible density, so that the marbles are packed together as closely as possible.

Experiment shows that dropping the marbles in randomly, with no effort to arrange them tightly, will achieve a density of around 65%.<ref>{{cite journal |last1=Li |first1=Shuixiang |last2=Zhao |first2=Liang |last3=Liu |first3=Yuewu |date=April 2008 |title=Computer simulation of random sphere packing in an arbitrarily shaped container |journal=Computers, Materials and Continua |volume=7 |pages=109–118 |url=https://www.researchgate.net/publication/280882105}}</ref> However, a higher density can be achieved by carefully arranging the marbles as follows:
# For the first layer of marbles, arrange them in a hexagonal lattice ([[honeycomb#honeycomb pattern anchor|the honeycomb pattern]])
# Put the next layer of marbles in the lowest lying gaps you can find above and between the marbles in the first layer, regardless of pattern
# Continue with the same procedure of filling in the lowest gaps in the prior layer, for the third and remaining layers, until the marbles reach the top edge of the jug.

At each step there are at least two choices of how to place the next layer, so this otherwise unplanned method of stacking the spheres creates an uncountably infinite number of equally dense packings. The best known of these are called ''cubic close packing'' and ''hexagonal close packing''. Each of these arrangements has an average density of

:<math>\frac{\pi}{3\sqrt{2}} = 0.740480489\ldots</math>

The '''Kepler conjecture''' says that this is the best that can be done—no other arrangement of marbles has a higher average density: Despite there being astoundingly many different arrangements possible that follow the same procedure as steps&nbsp;1–3, no packing (according to the procedure or not) can possibly fit more marbles into the same jug.