Contact number

In reading this text or in down­loading a film you may have used the solu­tion of the balls contact number problem in 8-dimen­sional space.

How many equiv­a­lent spheres can be placed around a given sphere?

Let us consider a plane case. We have a touch an equal coin?

It is possible to dispose six coins in this manner with their centeres in the vertices of a regular polygon.

Is it possible to dispose more coins?

Each coin takes up an angle of 60 degrees. Complete angle can be divided into six such angles. This means that six is the greatest number.

This may be shown in a slightly another way. On the circum­fer­ence of the central coin we consider the arc that is filled with one coin that touches the central coin. If you divide the length of the circum­fer­ence by the length of this arc you will find out that there is room just for six such arcs.

Thus we have found the dispo­si­tion of six coins and we have proved that this number is the greatest number.

The majority of extremal prob­lems, that is prob­lems of finding maximum and minimum, are solved just in this way. We give a certain construc­tion and then we prove that it is the best one under condi­tions of a problem.

In our ordi­nary three-dimen­sional space the problem proved to be much more diffi­cult.

How many iden­tical billiard-balls is it possible to arrange around a given billiard-ball of the same radius.

Twelve balls could be placed at the possible to roll the balls on the central ball.

Whether more than 12 balls could be placed? This ques­tion was the subject of a cele­brated discus­sion between the Scot­tish scien­tist David Gregory and Isaac Newton in 1694. It was Newton who studying astro­nomic ques­tions noticed that 12 balls could be placed at the vertices of an icosa­he­dron.

David Gregory gener­al­ized the upper esti­mate of the amount of coins placed around a given one. He calcu­lated the square of a spher­ical cap occu­pied by one ball and divided the square of the central sphere ball by the square of this cap. Conduct calcu­la­tion and you will be surprised that the answer is almost 15. As this number turned out to be less than fifteen it goes to prove that 15 balls could not be placed. However, Gregory could not guess that it is impos­sible to dispose even 13 balls.

It took 200 years to prove for the first time, that contact number in three-dimen­sional space is 12.

The contact number problem is also solved in 4-dimen­sional, 8-dimen­sional and 24-dimen­sional spaces. The contact numbers are 24, 240, and 196560 respec­tively. Balls are placed at the points of the chess lattice, Korkin-Zolotarev lattice and Leech lattice. The last advance­ment in this problem (solu­tion in the 4-dimen­sional case) has been obtained by Russian math­e­mati­cian Oleg Musin.

This beau­tiful and it would seem purely math­e­mat­ical problem is a special case of a spher­ical code problem. It has many impor­tant appli­ca­tions in tech­nique concerned with infor­ma­tion trans­mis­sion. In partic­ular, error-correcting code, used in modems, is closely related with contact number problem in 8-dimen­sional Euclidean space.

Some power restric­tions arise during trans­mis­sion of infor­ma­tion over a distance, for example, from the Earth to a satel­lite. To put it math­e­mat­i­cally, these restric­tions mean that trans­mitted signals are points of a sphere in Euclidean space. There is a recov­ered too.

If we know that the distor­tions during trans­mis­sion are small we can consider caps of a smaller size.

If we want to transmit more infor­ma­tion that is to have as many various words as possible we come to the problem: how to arrange as many spher­ical caps of a given size on a sphere as possible.

This creates, in balls language, the following problem: how many equiv­a­lent balls can simul­ta­ne­ously touch a given ball of another radius?

Open problem

In spite of a great applied impor­tance of the spher­ical code problem, it is solved only in a small number of special cases, both in 3-dimen­sional space and in high dimen­sional spaces. The exact solu­tion in the general case or even in some infi­nite series of cases has not yet been discov­ered.

Other etudes in “Points' best position”