


Chapters 3 and 4 : The fourth
dimension
Mathematician
Ludwig Schläfli talks to us about objects in the
fourth dimension and shows us a procession of regular polyhedra in
dimension 4, strange objects with 24, 120 and even 600 faces!



1. Ludwig Schläfli and the others
We hesitated for quite some time
before choosing the presenter of this chapter. The idea of the
fourth dimension does not come from just one person and many
creative spirits were needed before it was definitively
established and assimilated into mathematics. Among the precursors, one
can cite the great Riemann who will present the final chapter and
who had, without doubt, a very clear idea of the fourth dimension
as of the middle of the nineteenth century.


But we called on Ludwig
Schläfli
(18141895), mainly because this original mind is almost
forgotten today, even amongst mathematicians. He
is one of the first to have grasped the idea that even if our
physical
space seems to be of dimension 3, there is nothing to stop
us imagining a space of dimension 4, or even proving geometrical
theorems about 4 dimensional mathematical objects. For him, the
fourth dimension was a pure abstraction, but after years of work, he
must have felt more at ease
in four
dimensions than in three ! His major work is the "Theorie der
vielfachen
Kontinuität", published in 1852. It must be said that few at
the
time realised the importance of this treatise. It was not
until the beginning of the twentieth century
that mathematicians
understood the point of this monumental work. For more information
on Schläfli, see here or here.
Even within the mathematical community, the fourth
dimension maintained an aspect of mystery and impossibility
for
many years. To the general public, the fourth dimension often suggests
science fiction full of paranormal phenomena, or sometimes,
Einstein's theory of relativity: "the fourth dimension is
time,
isn't it?" However, this is confusing mathematical questions
with
those of physics. We will return to this briefly later. Let
us
first try to grasp the fourth dimension as Schläfli
did, as a pure creation of the mind!



2. The idea of dimension
Schläfli uses the blackboard to remind us
of some of the things we have seen in the preceding chapters.
A
line is of dimension 1
because to locate a point on it, one needs only one number. This is
the abscissa, or xcoordinate, of the point, negative on the
left
of the origin, and positive on the right.



The plane of the blackboard is
of dimension 2
because
to locate a point in this plane, one can plot two perpendicular
straight lines on the blackboard and describe the position of points
with respect to these two axes: they are the abscissa
and the ordinate
(the x
and y
coordinates). For the space in which we live, one can
supplement the two axes of the blackboard by tracing a third axis,
perpendicular to the blackboard. Of course, it is rather
rare to
have chalk that plots straight lines which leave the blackboard, but
as we are ready to go off into the fourth dimension, we shall need
magic
chalk!
Any point in space can then be described by three
numbers noted
traditionally x, y and z, and that's
why we say that space has three
dimensions. One would of course like to be able to continue, but it is
not possible to trace a fourth axis perpendicular to the previous
three; this is not a surprise, because the physical space in
which we
live is of dimension 3 and we shouldn't look for the fourth
dimension here, but rather in our
imagination…


Schläfli suggests several ways in which
we can get
an idea of the fourth dimension. There is not just one single method,
just as
there are several ways to explain the third dimension to flat
lizards. It's the combination of different methods which allows us to
get a
look into the fourth dimension.
The first
method
method is the most pragmatic. We can simply decree that a point
in 4 dimensional space is nothing more than the set
of data
consisting of four numbers: x,
y, z, t.
The disadvantage of this approach it is that it is hard to visualise
anything. But it is completely logical and the majority of
mathematicians are happy with it. One can then try to copy the usual
definitions in dimension 2 and 3 and try to define objects in the
fourth dimension. For example, one can call (hyper)plane the
collection of points (x, y, z, t) satisfying a
linear equation of the form ax+by+cz+dt
= e, copying
the similar definition of a plane in space. With this kind of
definition, one can develop a consistent geometry, prove theorems and
so on. In fact, this is the only way in which to treat spaces of higher
dimension seriously. But the aim of this film is
not to be
"too serious" but rather to "show" the fourth dimension
and to explain the intuition that certain mathematicians have
of
it.
Schläfli then gives us an explanation
"by
analogy". The idea is to carefully observe dimensions 1, 2 and
3,
to notice certain phenomena, and then to suppose that these phenomena
continue to exist in the fourth dimension. It is a difficult game and
does not work all of the time. A lizard which leaves its world and
enters the third dimension must expect some surprises and needs time to
adapt. The same is true for the mathematician who climbs
into the fourth dimension "by analogy "…
Schläfli
takes the example of the sequence "line segment, equilateral
triangle,
regular tetrahedron". One senses that there is an analogy between these
objects, and
there is no doubt that the tetrahedron in some way
generalizes the equilateral triangle to dimension 3.


What then is the object that generalizes
the tetrahedron to the fourth dimension?
The segment has two vertices and it lies in
dimension 1.
The triangle has three vertices and it lies in dimension 2. The
tetrahedron has four vertices and it lies in dimension 3. It is
tempting to think that the sequence continues and that there is an
object in 4 dimensional space that has five
vertices and that continues the sequence. We can see that in
the
triangle and in the tetrahedron, there is an edge joining each pair of
vertices. If one tries to join five vertices to each other in
pairs, without thinking too hard about the space in
which
one makes the drawing, one sees that ten edges are needed. Then, it is
very natural to try to place triangular faces on each triplet of
vertices. Again, one finds ten of them. And then, one continues by
placing a tetrahedron on each quadruplet of edges. The object which we
have just built does not yet have a very clear
status… we know the vertices, the edges, the faces
and the
3 dimensional faces, but we do not yet see it very clearly. The
mathematician speaks about combinatorics
to describe what we know: we know which edges connect which vertices,
but we still don't have a geometrical view of the
object.
This object, whose existence we have just guessed, and that
continues
the sequence segment, triangle, tetrahedron, is called a simplex!


Click
on the image for a film. 

3. Schläfli's polyhedra
Polygons are drawn in the plane and polyhedra in
ordinary 3 dimensional space. Similar objects in dimension 4
(or more!) are generally called polytopes
though they are often simply called polyhedra.
Whereas Plato discussed the regular polyhedra in
ordinary 3 dimensional space, Schäfli described the regular
polyhedrons in dimension 4. Some are amazingly rich and the film
proposes to show them to 3 dimensional spectators
(you and me!) in the same way that the film showed the polyhedra of
Plato to the lizards, rather than a pot of flowers or a book
(admittedly, it would be very hard for the authors of the film to show
you flowers in dimension 4, which is a pity!).
Here we have one of Schläfli's most beautiful
contributions: the
complete and precise description of the six regular polyhedra in
dimension 4. As they lie in dimension 4, they
have vertices, edges, faces of dimension 2 and faces of dimension 3.
Here is a
table with the names of these polyhedra, the numbers of edges, of faces
etc


Simple name 
Name 
Vertices 
Edges 
2D Faces 
3D Faces 
Simplex 
Pentachoron 
5 
10 
10 triangles 
5
tetrahedra 
Hypercube 
Tesseract 
16 
32 
24 squares 
8 cubes 
16 
Hexadecachoron 
8 
24 
32 triangles 
16 tetrahedra 
24 
Icositetrachoron 
24 
96 
96 triangles 
24
octahedra 
120 
Hecatonicosachoron 
600 
1200 
720 pentagons 
120
dodecahedra 
600 
Hexacosichoron 
120 
720 
1200 triangles 
600 tetrahedra 

This will be useful to help appreciate
their
visualisations. For more information about polyhedra in dimension 4,
see here or here,
or also here.
4. "Seeing" in 4 dimensions
How does one "see" in 4 dimensions? Unfortunately
we can't give you 4D glasses, but there are other ways.


The method of sections
:
We begin as we did with the lizards. We are in our 3
dimensional
space and we imagine that an obect moves through 4 dimensional space
and progressively cuts our 3 dimension space.
The section is now in our space and instead of
being a
polygon which is deformed, it is now a polyhedron which is deformed. We
can get an intuition of the shape of the 4 dimensional polyhedron by
observing the sections as they slowly deform, ending up by
disappearing. Recognising the object in this way is not easy, and even
harder than for the lizards.
In the film, we get to know three of
these polyhedra : the hypercube and those called the
120 and the 600.
You see them cut through our space and show their sections which are 3
dimensional
polyhedra which deform. Impressive ! But not easy to understand.
The image on the right shows the 600 going through
our 3 dimensional space.


Click on the image for a film. 

As the fourth dimension is not easy to understand,
it is not a bad idea to use several complementary methods.



The method of shadows
:
The other method we give in this chapter is almost
more
evident than the sections method. We could also have used it with the
lizards. It is the technique of a painter who wishes to represent a
landscape containing 3 dimensional objects on a 2 dimensional canvas.
He projects the image onto the canvas. For instance, he can place a
light source behind the object and observe the shadow of the object on
the canvas. The shadow of the object only gives partial information,
but if one turns the object in front of the light and if one observes
the way in which the shadow changes, one can often get a very precise
idea of the object. All this is the art of perspective.
Here it's the same thing : think of the 4
dimensional
object that we want to represent as lying in 4 dimensional
space
and that a lamp projects its shadow onto a canvas which is now our 3
dimensional space. If the object turns in 4 dimensional space, the
shadow is modified, and we get an idea of the object even if we don't
see it!
First we see the hypercube, much more clearly than we did with
the sections.

Click on the image to start a film 

Then the 24, the object of
which we think
Schläfli was the proudest ! The reason is that this new
arrival is
really new ; it does not generalise any 3 dimensional polyhedron, as is
the case for the other polyhedra. Moreover it has the wonderful
property of being selfdual :
for example, it has as many 2 dimensional faces as 1 dimensional faces
(edges), and as many 3 dimensional faces as 0 dimensional faces
(vertices).
Finally we see the polyhedra 120 and 600 whose
sections
we have already examined. This new point of view shows us other aspects
of these 4 dimensional polyhedra, which are indeed complicated. These
two methods, sections and shadows, have advantages, but it has to be
admitted that they do not do justice to all of the symmetries of these
magnificent objects.
In the next chapter, we shall use another method,
called
stereographic projection ! Maybe that will help us to see more clearly
?
5. "Seeing" in 4 dimensions :
stereographic projection
(cfr. the
film,
Chapter 4 : the fourth dimension, continued)
Schläfli gives us one last
method to represent polyhedra in 4 dimensions.
It consists of using quite simply stereographic projection. But of
course this is not the same projection that Hipparchus showed us in
chapter 1 !
Consider a sphere in 4 dimensional
space. To
define such a sphere, we use the usual definition : it consists of the
set of points in this space which are all at the same distance from a
point called the centre. We have seen that the sphere in 3 dimensional
space is 2 dimensional, as its points are described by a longitude and
a latitude. In some sense, the sphere in 3 dimensional space is only 2
dimensional because "it is missing a dimension" : the height above the
sphere. In the same way, the sphere in 4 dimensional space is 3
dimensional, and it also is "missing" a dimension which is again the
height above the sphere.
What is the sphere in the plane, i.e. in 2
dimensional
space ? It's the set of all points at the same distance from a centre,
otherwise known as a circle. A circle is thus a sphere in 2 dimensional
space ! And it is clearly 1 dimensional as one number suffices
to
describe a position on a circle.
More surprising : what is a sphere in 1
dimensional
space, that is, on a line ? The set of points at the same distance from
a given point on a line. There are only two, one on the left and the
other on the right ... The sphere in 1 dimension then
contains only two points... Hardly surprising that we say that
it
is of dimension 0.
To sum up : in n dimensional
space, the sphere is of dimension n1 and it's for
this reason that mathematicians use the symbol S^{n1}.



The beginning of the chapter explains what the
sphere S^{3 }is,
but of course not even Schläfli can show it to us.
The best he can do is to show you a sphere S^{2}
and encourage you to proceed as if you were in 4 dimensions and imagine
the sphere S^{3}
... Stereographic projection as presented by Hipparchus projects the
sphere S^{2}
onto its tangent plane at the south pole. One can proceed in exactly
the same way with S^{3}.
One takes the tangent plane at the south pole of the sphere S^{3}
which is a 3 dimensional space and one can then project any point of S^{3}
(except the north pole) onto this space. It suffices to
extend
the straight line from the north pole passing throught the point until
it meets the tangent space to the south pole... Even if this takes
place in 4 dimensions, the figure is analogous to what we have already
seen.




Suppose then that Schläfli wants to show
us one of
these 4 dimensional polyhedra. He does what we have already done with
the reptiles. He inflates the polyhedron until it is traced out on the
sphere S^{3}.
Then he can project
stereographically onto the tangent plane at the south pole, which is
our 3 dimensional space, and we can observe this projection.
We can also roll the sphere S^{3}
on its tangent plane and then project so that we can watch the
polyhedron dance. Notice that when the rotation of the sphere takes a
face of the polyhedron through the projection pole, the corresponding
face has an infinite projection and we get the impression that the face
explodes on the screen. We had the same impression in chapter 1 when
polyhedra were projected on the plane.
This is the display proposed in chapter 4 :
projecting Schläfli polyhedra stereographically
and turning them at the same time.

Click on the image for a film. 

The geometry of 4 dimensional spaces is just the
beginning, as there are also spaces of dimension 5, 6... and inifinite
dimensional spaces ! Initially thought of as pure abstractions, modern
physics makes great use of them. Einstein's theory of relativity
uses a 4 dimensional spacetime. A
point of this spacetime is described by three numbers describing a
position and by a fourth describing an instant in time.
But the force of the theory of relativity lies
precisely
in mixing these four coordinates in some way without trying to
privilege
time or space, and these thus lose their individual specifics.
We
shall not explain this theory here
because Schläfli didn't know it ! Einstein's theory
dates from 1905, well after the birth of the mathematical idea of 4
dimensions. This is not the first, nor the last, time that physics and
mathematics interact fruitfully, each contributing its methods, with
differing goals and motivations, yet so close...
Besides, doesn't today's physics postulate the
existence
of spaces of dimension 10 or more, and doesn't quantum physics work
with infinite dimensional spaces ? There will be a bit of a wait before
we produce a film on spaces of dimension 10....


