304 
subject have been entertained and expressed 
even by scientific writers. 
Place one of the balls on the table and 
arrange four others around and touching it 
with equal intervals between them in the form 
of a right-angled cross. Then place one ball 
directly on top of the central one, and finally 
one directly beneath it. This forms a group 
of seven balls which suggests one of the “ jack- 
stones” (generally made of iron or lead) that 
children play with. The group has a perfectly 
symmetrical arrangement which admits of in- 
definite extension on the same system in all 
directions by the addition of balls. In such 
a system any one ball, except of course those 
on the outer boundaries of the assemblage, is 
symmetrically surrounded by six others all 
touching the one that is central for this indi- 
vidual group of seven, but no two of these 
surrounding balls touch each other. The 
planes mutually tangent to each pair of balls 
at their common point of contact will ob- 
viously form by their intersections a system of 
equal cubes with common interfaces, each cube 
circumscribed about a ball. It is plain that 
these cubes “stack” together so as to make a 
solid volume, or in other words there will be 
no voids between—no waste of space. It 
will be clear that exactly the same arrange- 
ment results from placing on the table a num- 
ber of balls in contact and in a single layer 
in “square” order, or with the balls in rows 
both ways at right angles like the squares on a 
checker board, and then placing another layer 
in the same formation with each ball directly 
over a-ball of the first layer, and so on. The 
balls will have to be stuck together or very 
carefully placed or they will not retain this 
formation but they will fall down or spread 
and the pile will collapse under the influence 
of the gravitation of the earth. 
It soon becomes apparent that this cubical 
arrangement is not the most compact possible 
or not the one which permits placing the 
greatest number of the balls in a given volume. 
For example, after placing the first layer in 
square formation greater concentration is at- 
tained by placing each ball of the second layer 
over an interval or space among the balls of 
SCIENCE 
[N. S. Vou. XLVIII. No. 1241 
each group of four in the first layer rather 
than directly over another ball, and so con- 
tinuing the succeeding layers. 
Now undertake to effect the most compact 
arrangement possible beginning with one ball, 
and place about a central ball on the table 
as many others of the same size as there is 
room for in one layer with all touching the 
central ball. There will of course be six side 
balls, all tangent to each other throughout as 
well as to the central ball, in hexagonal order. 
Then three more balls can be placed above 
touching the central one—and only three, 
though there are six intervals among the balls 
of the foundation layer—and likewise three 
others can be placed below, making twelve 
surrounding balls or a group of thirteen, all 
in mutual contact throughout, so that the 
position of each ball in the group is definitely 
fixed relative to its neighbors. This arrange- 
ment may be extended without limit and it 
is the most compact possible for an indefinite 
number of balls, but it is not perfectly sym- 
metrical throughout. The mutually tangent 
planes at the points of tangency between the 
balls make a system of rhombic dodecahedrons, 
each one surrounding a ball. Equal rhombic 
dodecahedron will stack together without voids 
when similarly oriented but they do not form 
a completely symmetrical division of space, 
since the rhombie dodehedron is not one of 
the regular polyhedrons, or not a solid with all 
equal regular polygons for faces. All of the 
diedral angles of this solid are 120 degrees, 
but its twelve faces are equilateral oblique 
angled parallelograms or rhombs and the plane 
angles meeting at the vertices or solid angles 
are not therefore all equal. 
It should be noted that the formation re- 
sulting from starting with a layer in square 
order and placing the balls of the next layer 
over the intervals in the first one and so on, is 
also this same rhombic dodecahedronal ar- 
rangement, only differently disposed with re- 
spect to the table or the horizontal plane. It 
is what we so often see in a pile of oranges 
in the groceries and on the fruit stands. In 
all horizontal layers of such a pile the balls 
are in square order, but there are other sys- 
