Ocroser 11, 1918] 
tems or series of layers in the pile, inclined to 
each other and to the horizontal, in which the 
balls are all in the hexagonal order, which is 
the closest assemblage possible in any one 
layer or plane. 
We have thus developed one arrangement— 
the cubical—that gives universal symmetry 
with the balls in contact throughout, but not 
maximum concentration; and another one— 
the rhombic dodecahedral—that gives maxi- 
mum concentration and density, but not uni- 
versal symmetry. Now try for a formation 
that will give both. 
The sphere is itself a shape of the most 
perfect symmetry and it has the very maxi- 
mum quantity of contained volume or space 
for a given area of enclosing surface. It 
seems at first axiomatic and a foregone con- 
clusion that an assemblage of equal spheres 
must admit of an arrangement or grouping 
that will give to the aggregate collection char- 
acteristics exactly similar to those of the in- 
dividual sphere, with complete internal sym- 
metry and equilibrium. 
Recalling that in the second experiment the 
twelve side balls were placed about the central 
one all in mutual contact throughout so that 
the position of each ball in the group was 
definitely fixed and with no room for relative 
movement, it will perhaps be somewhat sur- 
prising to find that another arrangement for 
the twelve surounding balls is possible which 
gives a disposition perfectly symmetrical with 
respect to the central ball while the balls 
are nowhere in contact with each other at all, 
but each is equally and symmetrically spaced 
from all of its side neighbors. There is room 
to spare among the side balls but not enough 
for another ball. In this arangement the com- 
mon tangent planes between the central and 
the surrounding balls form a regular poly- 
hedron—the regular or pentagonal dodeca- 
hedron—about the central ball. This is a 
volume with twelve equal pentagons for faces, 
and of course having all its diedral angles as 
well as its vertices or polyhedral angles equal 
respectively. Each diedral angle of this solid, 
or the angle between any two adjacent faces, 
is 116° —34’ —54”; that is more than 90 de- 
SCIENCE 
355 
grees and somewhat less than 120 degrees, or 
between one quarter and one third of the com- 
plete angular space about one edge. Equal 
volumes or solids of this form may be as- 
sembled, face matching face, about a central 
one of the same size in a group of thirteen, 
but there must be a wedge shaped void, with 
a diedral angle at the edge of over ten 
degrees, between each two adjacent side mem- 
bers of the group where three edges of the 
solids coincide, and therefore the system can 
not be extended in the same formation by 
adding other equal solids of the same size and 
shape. From this it is apparent that a group- 
ing of spheres inscribed in the equal regular 
dodecahedrons does not admit of this sym- 
metrical arrangement beyond the group of 
thirteen. 
(A compact grouping of eight equal spheres 
which is symmetrical with respect to a central 
point, not within any one of them, may also 
be arranged as follows: Place three of the 
spheres in contact on the table, with a fourth 
over the interval, making a triangular pyr- 
amid group or a regular tetrahedral grouping. 
Then place a sphere over each of the four 
spaces or openings that will be found over the 
outer surfaces of the group, each opening sur- 
rounded by three tangent spheres. The limit 
in number for this grouping is eight spheres— 
there is no available space for any more placed 
symmetrically—and here is a suggestion of 
possibly some relation to the “ periodic law” 
of physical chemistry). 
To summarize: The only possible arrange- 
ment or grouping of equal spheres in contact 
that gives perfect symmetry as a fixed condi- 
tion throughout for a group of an indefinite 
number is the cubical system, and this does 
not give maximum density: while the only 
possible arrangement that gives maximum 
density as a fixed condition throughout such a 
group is the rhombic dodecahedral, but this 
does not give universal symmetry. There is 
no arrangement possible giving both maximum 
density and universal symmetry. 
It is scarcely necessary to add that these 
relations in no manner depend on absolute 
dimensions—they are true for spheres of the 
