Chapter 2 : Dimension three

M. C. Escher tells the adventures of two-dimensional creatures who are trying to imagine three-dimensional objects. 

1. The narrator

M.C. Escher (1898-1972) 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 (1685-1750) is another artist who fascinates mathematicians (among others!).  He too presents us with astonishing symmetries.  

Kurt Gödel (1906-1978) 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?

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 three-dimensional 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"

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.

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.

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 3-dimensional image from the successive cross-sections.   

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 "passe-muraille" 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 three-dimensional objects, whose images are projected onto our retinas, which are only two-dimensional?    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 2-dimensional images to mentally reconstruct a 3-dimensional 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. 

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 4-dimensional 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 Earth-sphere turns, the dance of the faces will give him an even more precise idea.  

It is this method which is shown in the second part of this chapter.  First we see the entire setup  as seen by a three-dimensional 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 3-dimensional 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.