Polyhedra, which as an imaginative student of Maths, I used to imagine as a many-headed Hydra (hydra ~ hedra) covered in azure, emerald green and blood red feathers (poly ~ Polly, the parrot in Flaubert's Polly), polyhedra are nothing more (and nothing less) than 3-dimensional objects whose every face - every single one - is a regular polygon. The same polygon.
There are any number of polygons as you please. Not so with polyhedra. There are, as we will see, only five of them. We'll see how come only five. We will also take a quick leap outside, into 4- 5- and higher-dimension space. How many can we spot with mathematical eyes?
We will even look at what we mean by space, 3-d space, and the polyhedra within them. How do these relate to the space they are embedded in? Is it a case of distillation or representation? Art as representation is what a long train of thinkers that stretches all the way from Aristotle to the present have presented as the core essence of Art. Not for me. I prefer to think of Art as distillation.
Would that be compensation enough? All of the above?
Well, it will have to. This frustration that I feel has as it source something which is not my doing and which not I but someone else can possibly redress. There will be other occasions for me to burst out on the reader and there's precious little I can do about it except for these very modest amends. I ask your forgiveness in advance. Now, for the lesson.
PLATONIC SOLIDS:
Do they exist?
Construct one and see for yourself. We could easily construct one whose faces are square. You need a total of six squares arranged in a crucifix formation with 4 making the vertical stem and, at the second square from the top, put a square on either side as the arms of the crucifix. Then simply fold the squares along the lines they share with the neighbouring square so that they are at 90 degrees to each other. And there you have it:
They are known as Platonic because in the Timaeus, where Plato unfolds his carefully wrought and elaborate account of the formation of the universe, these solids make an appearance as the elementary building blocks. Four of them stand for four different elements of the material world - earth, water, air and fire. The fifth element, which in India, China and Tibet was associated with the void and in preSocratic Greece with aether, had all but disappeared from the Classical classification of the basic elements so Plato just associates the 5th Platonic solid with the arrangement of the constellations in heaven. The scraps, so to speak.
So, we have the cube - the one we constructed above - which stands for the Earth, the tetrahedron for Fire, the octahedron for Air and the icosahedron for Water. The 5th solid, with the obscure use of arranging the constellations in the heavens, is the dodecahedron. Here they are:
 |
| Tetrahedron (Fire) |
The association with fire is probably due to the fact that the shape of the solid has very sharp corners, evoking the prickliness of fire.
 | ||
| Cube/Hexahedron (Earth) |
 | ||
| Octahedron (Air) |
Why air? I have no idea, or rather, too many to decide on one.
 |
| Icosahedron (Water) |
Water. Maybe because the 20 faces make it slip through the fingers, just like water drops.
 |
| Dodecahedron (The Heavens) |
Notice that the faces of the solids are regular polygons. This is part of the definition of the polyhedron - a 3-dimensional geometric object whose faces are made up of the same regular polygon.
Now for the scheme, it goes like this:
SHAPE VERTICES EDGES FACES
Cube 8 12 6
Tetrahedron 4 6 4
Octahedron 6 12 8
Icosahedron 12 30 20
Dodecahedron 20 30 12
Where by vertix we mean the point where the lines meet on the surface i.e. the part you would use if you wanted to stab someone in the eye with a polyhedron, the edges are the lines that connect the vertices and the space enclosed by edges is what we mean by a face.
Now, just by studying the table you start getting the suspicion that there might be some pattern here. Compare the icosahedron with the dodecahedron and the octahedron with the cube - if you exchange the no of vertices with the number of faces, the number of edges stays the same. If we trust the Ancient Greek's intuition that there are only five of these solids, there might be a defining pattern which as a characteristic of polyhedra might even tell us why there can be only five of them.
This pattern was discovered by Leonard Euler and has since been known as the Euler characteristic:
V - E + F = 2,
with V for vertices, E for edges, F for faces
We now need to prove this. How? Graph theory:)
Graph theory is basically the study of 2-dimensional pathways - given a number of objects, which the theory models as nodes or points, we study the pairwise paths connecting the nodes. If you study graph theory well enough, you might even find a position as apprentice to a fishing-net mender.
Let's think of our polyhedra as graphs with the vertices as the nodes. The edges of the solid would then be the pathways. What about the faces? We will let the space enclosed by the edges correspond to the faces, but we will also count the space outside the graph as a face. This will allow us to transform polyhedra into graphs, enabling us to study them a bit more easily.
This is how we do it. Take a top view of the object. Consider the cube. Imagine we lit a light on top of a glass cube and look at the projection of the cube on the flat surface under the cube. It would look like this:
Count the number of vertices (the nodes of the graph), the edges (the paths of the graph) and the faces (the spaces enclosed by the paths of the graph) and see that the number is preserved, even for the faces where the top face of the cube corresponds to the space outside the graph.
So what we have here is what is known as a mapping - the graph is not a polyhedron. It inhabits a totally different space - 2nd dimensional and unmetered. But it bears a correspondence with the solids in terms of number of edges, vertices and faces.
OK. Let's prove that V - E + F = 2
We start with the simplest graph imaginable - just one node standing alone like the navel of a null universe. We have
1 - 0 + 1 = 2
Correct!
Now lets consider the graph of the cube:
We will start dismantling the graph by removing edges and checking that the Euler characteristic remains constant. Take away one edge.We have removed an edge but we have also removed a face, meaning that V - E + F remains the same. Why? Well,we have V - E + F. If we remove a face and an edge it becomes V - (E-1) +(F-1) = V - E + 1 + F - 1 = V - E + F -1 + 1 = V - E + F +0 = V - E + F.
So we can go ahead and remove all the edges around the perimeter of the graph and be sure V - E + F does not change. Next, we'll remove the edges left standing awkwardly alone on the outside. When we remove one edge we will also remove a vertex, so once more Euler's characteristic remains the same. As an exercise prove that ( V-1) - ( E - 1) + F = V - E + F. Then go ahead and remove all.
We are left with this:
Remove an edge. We have also removed a face (in fact, we have reduced the faces to the bare minimum of one), so the characteristic stays the same. Now remove a node and an edge at once. Once again, the characteristic stays the same. Now you prove it all the way! Continue till we get the single node/vertex/navel of a desert universe. In this manner all graphs of polyhedra can be reduced to this single node, preserving at all times the Euler characteristic.Therefore we can conclude that Euler's pattern is correct and a necessary characteristic of all polyhedra.
The question now is - what does V - E + F = 2 tell us about the possibility that there might be only 5 possible polyhedra? OK, time we said goodbye to graph theory and rolled up our sleeves for some algebra.
*The algebra we'll use is elementary but I understand that school education has a way of scaring people off doing Maths for the rest of their lives. If you are one of those victims, you can skip the following algebraic argument and accept the fact that V - E + F = 2 implies that there can only be 5 combinations possible. You do not have to feel bad about it. This argument has been conceived and reviewed by generations of mathematicians - it is logical and intelligent to assume that is correct without going through the mill yourself. In any case, when we discuss higher dimensions, I won't provide a proof of the claim I'll make.
First, we'll add two variables:
Nv as the number of edges meeting at each vertex
Nf as the number of edges that enclose a face
Let's now study the relationship between Nv and E, that is, the relationship between the number of edges meeting at every vertex and the total number of edges. Each edge connects two vertices, therefore if we multiply Nv by V we will have counted the edges twice. To get the right answer then we would have to divide (Nv x V) by two to get
E = (Nv x V)/2
and by a similar argument (keep in mind that each edge touches two faces)
E = (Nf x F)/2
Manipulating these equations we get
V = 2E/Nv (1) and F = 2E/Nf (2)
Substituting (1) and (2) into V - E + F = 2 we get
2E/Nv - E + 2E/Nf = 2
and simplifying
1/Nv + 1/Nf = 1/2 + 1/E,
which implies that
1/Nv + 1/Nf > 1/2 since E > 0
Now we have the following facts:
Nv >/= 3, since you need a minimum of 3 edges to enclose at each vertex in a 3-d object and
Nf >/= 3 since you need a minimum of 3 edges to enclose a face
1. Starting with Nv >/= 3, we have 1/Nv </= 1/3, meaning 1/Nf > 1/2 - 1/3 = 1/6, meaning Nf can be 3, 4, 5
2. By exactly the same argument, Nv can be 3, 4 or 5.
So, this is what we have discovered - that the number of edges connecting vertices or enclosing faces on polyhedra can only be 3, 4 or 5. Lets see what that means in terms of combinations possible.
a) When Nf = 3, 1/Nv > 1/6 implying that Nv < 6 i.e. Nv = 3, 4, 5
b) When Nf = 4, 1/Nv > 1/4 implying that Nv < 4 i.e. Nv = 3
c) When Nf = 5, 1/Nv > 3/10 implying that Nv < 10/3 i.e. Nv = 3
As you can see, there are only 5 possible combinations and these correspond to:
Nf Nv Solid
3 3 Tetrahedron
3 4 Octahedron
3 5 Icosahedron
4 3 Cube
5 3 Dodecahedron
Now, I should point out that the above proof is not rigorous but it is perfectly OK as a demonstration i.e.grounds enough for any intelligent person to accept the fact of there being only 5 Platonic solids. Though, truth be told, it is just as intelligent to assume all of the foregoing.
What about higher dimensions? We actually have a generalisation of Euler characteristic that holds for all polytopes (i.e. all regular 'solids' in any dimension - polygons, polyhedra, etc). What it tells us is that there are only six 4-polytopes (ie. only 6 regular shapes in 4-d) and only 3 in all the higher dimensions.
The proof is much more technical than the foregoing so we will have to give that a miss. The formula just says that the alternating sign sum of the characteristics will alternate between zero and two. So, with polygons it is V - E = 0, and you can check it's true. Which should also show you what there's an infinity of them. For the 4-th dimensional polytopes it's V - E + F - S = 0, where S stands for some quality that we cannot visualise (not anymore than an inhabitant of Flatland could visualise a face).
The main fact here is that from 3-d upwards, the set of possible polytopes for a given dimension becomes finite, which is what makes my mind draw analogies with distillation.. Mathematically, space is insubstantial - a set of point and points are not things. You can take Aristotelian exception to the fact that Euclid says the point is that which has no parts. Points do not occupy space. They are like phanthom arrows - they do not cover where they point. A set of points, whether 1-,2-,3-tuple points, is just another form assumed by the void. The protean Void in its many guises, one of which is the fabric of what we know as reality. In a process analogous to distillation, at least in my mind's eye, these points, these tuples enter into states of interdependence. In 3-d space, they form what we call solids by analogy with elements of (our) reality's miniscule and dispersed onthology. Space is actuated, made substantial rather than just represented.
In this view, the conception of polyhedra in the mind of a human being is a creative act. For art is not really about representation. Art is not even a thing. Art is an act, an act of distillation.
As an artist I hold this concept very close to heart as I have always disliked the classical formulation of art-as-representation. Art-as-distillation - distillation of experience, including of course the experience of space, this is how I have always thought about art. Never as representation.
In The Art of Looking Sideways, Alan Fletcher says
Space is substance. Cezanne painted and modelled space. Giacometti sculpted by "taking the fat off space". Mallarme conceived poems with absences as well as words. Ralph Richardson asserted that acting lay in pauses... Isaac Stern described music as "that little bit between each note - silences which give the form"... The Japanese have a word (ma) for this interval which gives shape to the whole. In the West we have neither word nor term. A serious omission.
Indeed, Ma is a key concept in traditional Japanese aesthetics. It is sometimes referred to as negative space, something that is often misinterpreted by Western art critics as blank space - it is anything but blank. It is full. Full space, goddamit!
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