A Recurrece Formula for Packig Hyper-Spheres DokeyFt. Itroductio We cosider packig of -D hyper-spheres of uit diameter aroud a similar sphere. The kissig spheres ad the kerel sphere form cells of equilateral hyper-triagles. We deote the maimum umber of kissig spheres by beig the dimesio of the Euclidia space E. may ot be the largest kissig umbers across all arragemets. For this particular packig patter ca be obtaied from by buildig a ()-D equilateral hyper-triagle from a -D equilateral hyper-triagle. We employ this approach to determie ad for illustratio. We also preset a recurrece formula that fits to 8 quite well.. Step-by-Step Determiatio of ad I this approach we assume the cetre of the kerel sphere is at the origi of the -D Cartesia coordiate system. The step-by-step approach is based o the followig cosideratios: Cetres of the kerel sphere ad the kissig spheres lie o the vertices of equilateral -D hyper-triagles. The origi is the commo verte shared by all hyper-triagles. The -D equilateral hyper-triagle is a base of a ()-D equilateral hyper-triagle. The out-of-plae ape of the ()-D hyper-triagle is at uit-distace from the origi. The orthogoal projectio of the ape o the base coicides with the cetroid of the base. Costructio of a ()-D equilateral hyper-triagle from a -D equilateral hyper-triagle ad determiatio of from cosist of the followig steps: Augmet the coordiates of the origi ad the eistig vertices from -D to ()-D. Determie the cetroid of each base. Locate the ew apees accordig to their distace from the origi beig uity. A verte is admissible as a ceter if it is ot less tha uity from ay other verte. is give by where is the umber of the ew admissible vertices.
(.A) Determiatio of from Coordiates of the si vertices ad the ceter of the regular heago together with the cetroids of the si equilateral triagles are show below after augmeted from -D to -D. ( ) () Let us cosider the base bouded by first. The ape of the equilateral tetrahedro which is directly above the cetroid of the base is at: ± ± () Likewise the ape of the tetrahedro over ( ) is situated at: ± ± () The si bases itroduce twelve ew apees. However oly the alteratig oes are acceptable. The cosecutive oes are ot because they are less tha uity apart. Hece we have. (.B) Determiatio of from The si tetrahedros have two patters. We ca choose oe to cosider say the oe below.
( ) ( ) () Each quartet i Eq. () is a base of a -D equilateral hyper-triagle. Let us cosider the first base as a eample. The coordiates of the vertices of the tetrahedro after augmeted to -D are: (5) The coordiates of the out-of-plae ape are foud to be: 8 5 ) 8 ( ) ( ) ( ± ± () Each of the si bases adds two out-of-plae apees oe o the ais ad oe o the ais. All of them are admissible as cetres. Therefore we obtai.. Recurrece Formula for 8 The above step-by-step approach becomes tedious as the dimesio of E icreases. We suggest a recurrece relatioship for estimatig 8 amely
k (k ) k k k k k (7) k k k k With Eq. (7) yields the followig estimates of to 8 : 5 7 8 8 Eq. (7) is related to the umber of -D bases. Let us umber the vertices. The table below shows how vertices are grouped to form bases. Each base yields two apees due to the quadratic orm. From -D To ()-D Vertices i -D Vertices i -D Bases No. of Bases { to } { to } 5 { to } 5 { to } 7 { to 8} 7 8 { to } {k k} k to {k k k} k to 5 {k k k k} k to 7 {k k k k 5k} k to 9 {k k k k 5k k} k to 9 {k k k k 5k k 7k} k to 59 8 8 The icremets i the umberig a k 7 follow a patter which is odd a (8) eve
5 Eq. (7) ad Eq. (8) are equivalet. Both yield the same. The epressios i Eq. (7) ca also be simplified ito: k k (k ) k (k ) k k k k k (9) A compariso of the estimated by usig Eq. (7) with the kow results is tabulated below. Lattice No-Lattice Lower Boud Upper Boud Eq. (7) 5 5 7 7 78 8 7 5 8 9 7 The recurrece formula implies M () The followig observatios about 5 7 ad 9 ca be made: M With ad 5 oly M will yield 5 5 M. M With 5 ad o iteger M ca be foud so that 7 78 M.
M With 8 ad 7 o iteger M eists such that 7 5 M. The estimated 9 is probably ot bad beig equal to the average of the bouds.. Summary Whe the packig patter is i the form of cells of -D equilateral hyper-triagles aroud the kerel sphere the kissig umber ca be obtaied from oe step at a time. This method however becomes tedious whe is higher tha. A empirical recurrece relatioship is foud to fit to 8 rather well. The formula is related to the umber of -D bases formed by groupig the eistig vertices.