


Chapter
2 : Dimension three
M. C. Escher tells the
adventures of twodimensional creatures who are trying to imagine
threedimensional objects.




M.C.
Escher (18981972) was an extraordinary artist
whose works attract the interest of many mathematicians. His
engravings show us paradoxical worlds, tilings with amazing symmetries
and infinite perspectives:
the sorts of things that really
fascinate mathematicians!
You can find a biography and a large
collection of reproductions of his engravings on the
official website .
J.
S. Bach
(16851750) is
another artist who fascinates mathematicians (among others!). He
too presents us with astonishing symmetries.
Kurt
Gödel
(19061978) was
a mathematician who revolutionized logic, also
exploiting symmetries between a whole entity and one of its parts.
A
remarkable book "Gôdel,
Escher, Bach" explores the
profound relationships
between the works of these three exceptional figures.



One of the most famous of
Escher's engravings is entitled Reptiles. Let's
take some time to admire it here, since unfortunately it goes by very
quickly in the film.
One one page of a sketchbook we see a
tiling in which flat lizards fit together perfectly.



This is the image of a
flat world: the
lizards who live in this page know only the page; they are ignorant of
the space surrounding them.
We see them, and we know that their flat
world is only a page of a notebook lying in our space, but the flat
lizards do not know this.
One of these lizards finds
a means of escaping the plane and visiting our world: we
see him at the bottom gradually gaining thickness, climbing up on a
book, using a drafting tool as a bridge leading to a promontory in the
form of a dodecahedron, before going back down and resuming a position
in his flat world, enriched by his new experience like an explorer who
has just discovered a new continent.
The engraving invites
philosophical reflection:
if these lizards are unaware of the
existence of an exterior world around them, could we be in the same
situation? Couldn't
there be a world "exterior" to ours, to which our senses do not give us
access? In
fact, there are many philosophical allusions in this engraving. We
see the four elements that according to Plato made up the world: water
in the glass, air blowing out of the nostrils of the lizard, earth in
the pot and fire suggested by the matchbox, and we even see the
dodecahedron, which represented Plato's fifth element, the
"quintessence"...Might the "Job" brand cigarette paper be a biblical
allusion?

All
M.C. Escher
Works © 2008 The M.C. Escher Company,the Netherlands.
All rights reserved. www.mcescher.com
Used with permission. 

The purpose of this chapter is
to prepare us for the fourth dimension. In
order to envision a fourth dimension which envelops us, we will begin
by imagining the strategies we might use to explain the existence of a
third dimension to the flat lizards. We
will imagine that we are this lizard chosen by heaven (the philosopher? the
mathematician?)
who had the privilege of being authorized to
leave the page and climb onto a dodecahedron. We
are in threedimensional space, we see objects such as a pot, a book, a
dodecahedron, and our mission is to "show" these objects to the other
lizards who can't see them because they are confined to a plane that
they cannot leave.
2. "Flatland"


This chapter could
have equally well been narrated by Edwin
Abbott,
an English pastor in the nineteenth century who wrote a book
wonderfully entitled FlatLand. This
book tells the story of a flat society, in which the characters are
triangles, squares, circles and line segments. In
this society the rules of life are very complex, and the charm of this
book is that the author uses them to caricature the Victorian society
of the nineteenth century in which he lived, which was full of its own
complexities.
The hero of the
book, a square, is a sort of analog of our lizard, who leaves his
plane and little by little gains consciousness of other dimensions. The
subtitle of the book is "A romance of many dimensions." This
little book is a real jewel and it is also one of the very first
popularizations of science.
The entire
text is available online.




3. The
Platonic Solids
What objects in our space shall
we "show" to the flat lizards?
We could show them a flowerpot or a book, but
instead we will remain in a philosophical mode and show them the five
Platonic solids.







Tetrahedron 
Octahedron 
Cube 
Dodecahedron 
Icosahedron 



Some
of these objects are very familiar to us, such as the cube. We
occasionally encounter others such as the tetrahedron. Still
others are quite rare, and you need to be very observant to find them
in nature.
For example, let's take
the icosahedron, with its 12 vertices, and slice
off each of the vertices as in the figure at left. We
obtain an object consisting of 20 hexagons and 12 pentagons. The
pentagons come from the 12 vertices which we sliced off, and correspond
to the faces of a dodecahedron.
You may recognize the final result as
the pattern on a soccer ball...


These objects are called polyhedra,
which in Greek means literally that they have
many faces! We do not intend here to go into a complicated theory of
polyhedra. We
simply want to choose five pretty objects in space and try to show them
to the lizards, or perhaps somehow explain to a lizard what a soccer
ball is.
There are many polyhedra (an
infinite number, in fact) but only five of them are regular.
Here again, we don't want to enter into the details of the definition
of this word, but just observe that for each of these five regular
polyhedra all of the faces are of the same type (for example, all of
the faces of the dodecahedron are regular pentagons, all of whose edges
have the same length), and that all of the vertices are of the same
type (for example, there are exactly three edges going out of each
vertex of a cube).
These properties are enough (almost) to
characterize the five objects that we want to show to the lizards.


Image 
Name 
Faces 
Vertices 
Edges
(length L) 
Surface 
Volume 

Tetrahedron 
4 
4 
6 



Octahedron 
8 
6 
12 



Cube 
6 
8 
12 



Dodecahedron 
12 
20 
30 



Icosahedron 
20 
12 
30 



To learn more about polyhedra,
you can consult this page, and to learn much more about the
five regular polyhedra, their history and their symmetries, you can
consult this page.
These objects are among the objects most revered by mathematicians
because they symbolize the concept of symmetry, which is unfortunately
not described in the film.
4. Sections


A first idea of how to
explain to the lizards what a tetrahedron is, is to cut it in slices. This
idea is very old, and Edwin Abbott uses it often in his book. This
is pretty much the idea used in tomography,
a technique of medical imagery which consists of examining the human
body slice by slice and then reconstructing a 3dimensional image from
the successive crosssections.
When a polyhedron moves in
space and encounters the lizards' plane, the intersection with this
plane is a polygon.
When the polyhedron moves, the polygon
deforms, and eventually disappears when the polyhedron has finished
passing through the plane (assuming polyhedrons can pass through walls
like the "passemuraille" of Marcel
Aymé!).
The lizards see only the polygons, but they see them in a dynamic way: they
can watch how they deform.
With a little experience they can
(perhaps) eventually gain an intuitive idea of what a polyhedron really
is, even though they cannot see it in space.




All of this raises many
questions. For
example, if the lizards are in a plane, how can they see a polygon? Difficult
question! And
hard to ask them.
But, if you think a bit, you will
understand that we face the same problem ourselves. How
do we see threedimensional objects, whose images are projected onto
our retinas, which are only twodimensional? There
are many possible answers to this.
First of all, we have two eyes which
don't see exactly the same thing, and our brain uses these two
2dimensional images to mentally reconstruct a 3dimensional image.
In addition, the effects
of shadows, light, etc. give partial information about the distance
which separates us from objects.
Finally, and perhaps most
important of all, we have experience with the world we live in: when
we see a photograph of a soccer ball, we recognize it even though the
image is in a plane,
because we have already seen and touched
other soccer balls.
So let us not hesitate to
assume our lizards have two eyes and a great deal of experience with
their world.
If a hexagon appears before them, they
are entirely capable of recognizing it as such. In
Abbott's book, all of these questions are discussed very amusingly.

Click
on the image for a film. 

In the film we see the five
regular polyhedra pass through the plane, and we can watch the
sections/polygons
as they deform. This deformation is not easy
to predict, since the sections depend on the way in which the polyhedra
pass through.
For example, if a cube approaches so that one
of its faces is parallel to the plane, we are not surprised to see that
the sections are squares.
But if we cut a cube by a plane which goes
through its center and is perpendicular to a diagonal, the intersection
is a regular hexagon...which is perhaps less obvious!
After having watched all of the
polyhedra pass through the plane, Escher proposes some exercises for
you. He
shows you a sequence of polygonal sections in the plane and you must
discover the polyhedron which is passing through, as though you were a
flat lizard.
Good luck with this exercise, which is not
easy (as you will see).
The method of sections has its limits, so we
look for other methods...
5. Stereographic
projection
Here is a second idea, which
might seem bizarre but which will be extremely useful in what follows
(when it is our turn to be "flat," confined to three dimensions while
someone is trying to show us objects in his 4dimensional world...). We
learned how to project the sphere onto a plane by stereographic
projection, and we have seen that even if this projection changes
lengths it still gives a fairly exact picture of the geography of the
Earth, especially if watch while the earth rolls on the plane.
We could try do the same thing
and roll the five polyhedra on a plane, projecting them
stereographically.
The
problem is that we can't roll a cube because
it is not round!
So, we inflate the polyhedra like balls so
that they become round.
For example, we start by inscribing a cube
inside a sphere.


The surface of the cube
consists of six square faces.
We project these six faces radially onto
the sphere, from the center.
You might say we are inflating the cube
until it becomes spherical.
The sphere is now covered by six
regions, which are no longer squares, of course, since their edges are
circle arcs.
But we obtain a good image of a cube
with the advantage that we can roll it like a ball.
Now we can imagine the
Earth with six continents, which are the six faces of the inflated cube. We
can do with this inflated cube exactly what we did with the Earth: project
it stereographically onto a plane and roll it around. The
dance of the continents becomes the dance of the six faces of a cube! Of
course, since the faces of the inflated cube are circle arcs, and we
have seen that stereographic projection sends circles on the sphere to
circles and lines on the plane, the projection of the inflated cube
onto the plane has "square" faces whose sides are circle arcs or line
segments. The
flat lizard sees the projection:
he must imagine that he is in a plane
tangent to the south pole of a sphere which he cannot see and find the
six faces of the inflated cube which are projected onto the plane. What
he can see in the plane gives him all of the information he needs to
understand the cube:
he can count the vertices, the edges,
and the faces, and he can understand their relative positions. And
if the Earthsphere turns, the dance of the faces will give him an even
more precise idea.


Click
on the image for a film. 

It is this method which is shown
in the second part of this chapter. First
we see the entire setup
as seen by a threedimensional being who sees
everything: the
polyhedron, the inflated polyhedron, the sphere, the projection onto
the lizards' plane.
Then we take the point of view of the flat
lizards who can see only the projection. Escher
finally appeals to our imagination to figure out which polyhedron we
are seeing.
The exercise is still not easy, but it does
seem easier
than using the method of slices.
These exercises will be useful
for what follows.
Remember:
in a little while you will be in the position
of a poor 3dimensional human incapable of seeing in the fourth
dimension! Someone
having the gift of being able to see in four dimensions will
try to show you what he sees, and he, too, will use slices and
projections.


