Evening star. [volume] (Washington, D.C.) 1854-1972, 1930-10-13
SCIENTIST PICTURES
FOURTH DIMENSION.
Carries Geometry Additional
Step and Builds Model to
Illustrate Theorem.
BY PAUL r. HEYL.

line. The point has no dimensions Of
it is sharp enough the line it leaves
has one dimension-length. Imagine
the line moved perpendicularly to itself
so as to sweep out a surface. This sur
face has two dimenSions--length and
breadth-one more than the line. Again,
| imagine a surface shaped, let us say,
! like a square to be moved perpendicu-
larly TO its plane so AS to generate a
cube. This solid has three dimensions
length, breadth and thickness-one
more than the surface
| Now notice a peculiar condition of
the motion in each case. The line. when
! it generated the surface, moved in a
| direction not to be found in the line
| itself. Had it moved in its own direc-
| ton it would have produced merely a
| longer line. The surface, when it swept
out the solid, moved also in a direction
| not TO be found in itself. Had it moved
along either of its own dimensions or
any intermediate direction it would have
generated merely larger surface.
Now If We move solid, say cube, in
any direction contained in itself we pro-
duce merely larger solid. To generate
a four -dimensional figure it would be
necessary to move the cube in some di-
rection not to be found in solid space.
Assumption Made.
Here the mathematician makes an
assumption. Though such direction
Is unknown to us and uncomfortable, he
assumes its existence, and proceeds TO
deduce its logical consequences. Mere
nconceivability never troubles mathe-
matician; the only thing that stops
him Is inconsistency. And, strange as
it may seem, If We once make the fun
damental assumption of the existence OF
a fourth dimension, it Is possible to
build up q geometry of four dimen-
sional space as consistent and as logical
as plane or solid geometry, though per-
haps little more difficult.
The writer has recently constructed
set of wire models illustrating one of
the theorems of this four dimensional
geometry. These models will be kept
on exhibition at the Bureau of Stand
ards. The theorem refers to the number
of regular figures possible in four di-
mensional space.
In plane or two dimensional geom-
etry there Is an endless succession OF
regular figures, the equilateral triangle,
the square, the pentagon and so forth,
approaching the circle as limit. In
solid or three dimensional geometry
there are possible five regular solids
the tetrahedron, cube octahedron, do
decahedron and icosahedron. In four
dimensional geometry there are six, reg
ular figures, named and described as
follows:
The 4-hedroid. bounded by 4 tetrahe-
dra; the 8-hedroid, bounded by cubes;
the 16-hedroid, bounded by l6 tetra-
hedra; the 24-hedroid, bounded by "a
octahedra the 120-hedroid, bounded
by 100 dodecahedra. and the 60o-he-
L., fAkr
Figures Represented.
How can we represent such figures in
three dimensions?
When we study plane geometry we
are greatly assisted by the fact that 1t
Is possible to draw on the blackboard
or print on the pages of the text book
figures which correspond perfectly 10
the problem under discussion. Solid
geometry offers a little more difficulty
In that it makes a continual demand
on the imagination. Solid models may
occasionally be used. but for the most
part our dependence must be on dla-
grams drawn in plane perspective. just
as we can represent by picture on
fiat sheet of paper mans mead ana
face and recognize 1t as representa-
ton of a solid, so We may represent
the regular three dimensional figures in
perspective diagrams. Such diagrams
are almost complete. A cube may thus
be shown with it's bounding squares,
its IL edges and 8 corners. All that Is
lacking is the solid, three dimenslonal
content This Is easily supplied by
the imagination. for we have all seen
a cube and know just how 1t should
look.
In an analogous manner we may con-
strllct solid pictures, three dimensional
nersDectiye models of four dimenSional
figures Such models are likewise al
most complete, showing the bounding
solids, with their surfaces, edges ana
corners Everything is there except the
four dimensional content, and this the
Imagination of no man can supply.
There are but few sets OF such mod
els in existence About SO years ago
a set was made in Germany by Schlegel,
and about the same time Stringham.
in the United States, made a set Wnlcn
is now at Yale University. In 1896,
complete set of six figures was made for
the University of Pennsylvania. and
set OF the four simpler figures for Har-
ward University, both by the present
writer.
Concept Not New.
Four dimensional geometry is by no
means a new concept 1t has been Ia
miliar to mathematicians for about 100
wears. and some Of the greatest mathe-
maticians have contributed to its devel-
opment Until recent years the subject
has had no standing in physical science,
but the advent Of Einstein's theory OF
RELATIVELY has forced the subject upon
the attention OF physicists.
II is surprising how peacefully this
revolution has come about. One needs
to be scarcely so years old 10 remember
when any physical scientist bold enough
TO take the subject seriously ran the risk
of being sent TO academic coventry, If
not to the lunatic asylum. But today
Jeans, Eddington and host of others
may speak freely that which the late
C. n. Hinton thought, but dared not ut
ter except In a strictly limited esoteric
circle, and safe in the twentieth CEN
tury. they may quote with the approval
of their peers these words of Min
kowski:
The notions of time and space which
shall develop for you have sprung from
the soil of experimental physics. There
in lies their strength. Their tendency Is
radical From this hour space in itself
and time in itself sink TO mere. shadows,
and only kind OF blend of the two re-
tains an independent existence"
Reprinted from United States Daily by Per-
mlSSion.)-
U. S. BLACKSHIRTS GAIN