Normal Piling and Theory of the Universe. on 
A pile can be built up in the form of a regular tetrahedron, 
and each of its 4 faces will then be composed of spheres in 
triangular formation. Or it can be built up in the form of a 
square pyramid; in which case the 4 sloping faces will be 
in triangular formation, while the base and all planes of 
spheres parallel to the base will be in square formation, The 
vertical planes of spheres parallel to the diagonals of the square 
base will also be in square formation. 
A normal pile can thus be split up into planes of spheres in 
7 different ways, there being 3 sets of parallel planes 
containing spheres in square formation, and 4 sets containing 
spheres in triangular formation. 
_ Perhaps the clearest idea of the relative position ot spheres 
in normal piling is obtained by imagining a solid chequer 
built up of equal cubes alternately black and white, so that 
every section taken parallel to the cube-taces is like a chess- 
board, Then, if we imagine all the cubes of one colour (say 
all the blacks) to be annihilated, and each white cube to be 
replaced by a sphere having the same centre, and of diameter 
equal to the diagonal of a cube-face, these spheres will just 
fit together as they stand, and will constitute a normal pile. 
A section through cube-centres, taken parallel to a cube-face, 
is in square formation ; and a section taken perpendicular to 
a body diagonal is in triangular formation. 
If we think of a normal pile as placed with a set of squarely 
formed layers horizontal, each sphere rests on 4 in the layer 
below, and in its turn supports 4 in the layer above, these 
last 4 being vertically over the first 4. Besides touching 
these 8 in adjoining layers, it also touches 4 in its own layer, 
making 12 in all. 
On the other hand, if we think of the pile as placed with 
one of its sets of triangularly-formed tiers horizontal, each 
sphere rests on3 in the tier below, supports 3 in the tier above, 
and also touches 6 in its own tier, again making 12 contacts 
in all. 
The following is another representation of normal piling :— 
Divide space into equal cubes by 3 sets of parallei planes. 
Then the centres of the spheres are to be at the cube-corners, 
and at the middle points of the cube-faces; the common 
sphere-diameter being made equal to the distance of the 
middle point of a face from a corner of the face. 
Another representation (the one adopted by Prof. O. Rey- 
nolds) consists in dividing space into cubic compartments as 
before, with the same sphere-diameter, but placing the sphere- 
centres at the cube-centres, and at the points of bisection of 
