OcroseEr 11, 1918] 
istics in natural philosophy,” as recently re- 
ferred to by Professor Bumstead.t 
Going back to D’Arcy W. Thompson’s book 
on “Growth and Form,” there are found some 
exceedingly interesting discussions and refer- 
ences pertaining to the various possible divi- 
sions of space by plane surfaces that have a 
direct bearing on this subject. We can as- 
sume our equal spheres to be soap bubbles. 
The shape of a single bubble by itself is 
determined by the tendency of the enclosing 
film to contract due to its “tension,” or the 
mutual attraction among its own particles, 
and the opposition to this contraction tendency 
presented by the enclosed air. By the same 
principles that have been explained above it 
can be shown that in a group of such bubbles 
the tendency is to assume an arrangement 
that will give complete symmetry and a min- 
imum total partitioning area, that these con- 
ditions can not both obtain as a fixed and 
simultaneous status for the whole group, and 
that there can not be a condition of equili- 
brium and stability throughout such a group. 
The same will be true of any similar group of 
compartments or cells enclosing a fluid and 
with walls or partitions composed of substance 
that is of a fluid nature. Thompson seems to 
have fallen into some errors in his discussion, 
as where he calls the rhombic dodecahedron a 
“regular solid”? and where he understands 
that by means of an assemblage of equal and 
similar “tetrakiadecahedrons” space may be 
homogeneously partitioned into similar and 
similarly situated cells “ with an economy of 
surface in relation to area (volume ?) even 
greater than in an assemblage of rhombic 
dodecahedra” (p. 338). The “regular” tet- 
rakiadecahedron is a semi-regular polyhedron, 
a fourteen-sided volume with six equal square 
faces and eight that are regular and equal 
hexagons, the sides of these squares and hexa- 
gons all being equal. Such a solid may be 
forméd by cutting off the corners of a cube, 
also by cutting off the corners of a regular 
octahedron. Space can not be divided into 
equal volumes of this shape without surplus, 
1 See Scrence for January 18, 1918. 
2‘*Growth and Form,’’ p. 328. 
SCIENCE 
307 
or in other words these volumes can not be 
stacked together without leaving voids. This 
may readily be determined by a study of the 
diedral angles, or practically by constructing 
a number of the tetrakiadecahedrons and try- 
ing it. However, these errors are not material 
to the present purpose, and it was Thompson’s 
book that first suggested to the writer’s mind 
a still broader generalization of the principles 
herein referred to. 
It may be stated that all processes and 
phenomena of life are associated directly with 
some form of fluid substance. This includes 
not only gaseous matter and liquids but many 
forms of matter that are not solid in the 
ordinary sense nor yet liquid, but which have 
a certain degree of mobility among the con- 
stituent particles. The essential primary ele- 
ment of all organisms is the cell. All mate- 
rial substances whatever, whether affected by 
influences of life or whether only dead matter, 
are alike governed by the physical and mathe- 
matical laws here outlined. Is ‘there not 
therefore a remarkable and intimate relation 
between the “simple geometrical principle” 
above explained and all organic existence and 
processes ;—All ik, growth, repair, decay, and 
dissolution :—Even all mind, intelligence, emo- 
tion, and all reasoning and thought. The 
speculative philosopher might indeed go so far 
as to add all health, satisfaction, and pleasure; 
all sickness, distress and pain; all relations and 
struggles among humans, all endeavors of 
man, all events of history, everything :—And 
the psychologist may here note an analogy to 
the unending strife between good and evil 
which figures in so many of man’s supersti- 
tions and religious beliefs, primitive and 
otherwise. 
A further conception of the profound signifi- 
cance of these elementary geometrical rela- 
tions in connection with all activities and phe- 
nomena of the material universe may be 
formed by imaging a region or space, apart 
from any known real one, where it is possible 
for equal spheres to be so grouped that the 
arrangement will have at the same time max- 
imum concentration and universal symmetry— 
an imaginary space where the regular dodeca- 
