The cosmos is big, the speed of light is small

Back to astronomy - the reason why we got interested in Poincaré Dodecahedral Space (PDS, for short) in the first place. Is it possible that our cosmos, this actual universe that we're in, is topologically a PDS? Well, we don't see images of our back-view (twisted by 36°) hanging out there in space - not anywhere at all close by, at least. Certainly not close enough for me to appear as a human being, seen from behind. If anything 'I' would appear as a galaxy, or a cluster, or a supercluster of galaxies...

It seems clear that there is no 'repetition' as we look out into the cosmos, on a scale that is less than say a hundred million light years (i.e. the diameter of our local supercluster of galaxies). But on scales larger than that it is difficult to be so sure. One has to think rather carefully about what our image would actually look like. Not only would it be 'us' as seen from an unfamiliar point-of-view, but it would also be 'us' as we were several hundred million years ago. There would be no shortcut for the light coming to us from the distant image.

Is there any possibility of seeing two images of the same object which have the same 'date-stamp', i.e. such that the light which reaches us left both images at the same epoch? Yes, if the two images are equidistant from us. Imagine an object that is just half-way between me and my image. Then the object will have an image that is halfway between me and the second image of myself that I see by looking behind me - and the two images of my object will be equidistant from me.

Now, the cosmological (constant-curvature) PDS is constructed by identifying opposite faces of a non-Euclidean dodecahedron. Where am I in the dodecahedron? The at first sight surprising answer is that I might be anywhere - and it really doesn't matter. The space of the PDS is absolutely homogeneous; the curvature is the same everywhere, there are no special points. Admittedly space is not quite so isotropic - or 'the same in all directions' - as it was when space was topologically trivial. At any point there are now special directions: the six pairs of directions in which I see images of myself (or could do so, if I could see far enough).

The point is, that the edges and vertices of the dodecahedron are artifices of the construction by which we conceived of the PDS. Now we have got there, we can discard the original dodecahedron: it is no more than a map of the actual space. Other maps are possible, and there is nothing to stop me using a map which is centred on me.

To get from the space in which I find myself to the map, I need to measure out a regular dodecahedron, centred on myself, of such a size (and orientation) that any two opposite pentagonal faces are images of one another, in other words, identical.

Then, any object which happens to be located on one of the faces of this mapping-dodecahedron, will have a second image on the opposite face, and both images will be equidistant from me. I will get two different views of the same object, at the same moment of its existence.

Given that the scale of this dodecahedron is likely to be of the order of hundreds of millions if not billions of light-years, there is a danger that it will be too big for me to see any repeated images at all. This will be the case if a sphere drawn with me as centre, and radius the distance travelled by light since the origin of the universe (approximately 14 billion light years), is small enough to fit inside the dodecahedron without touching any of its faces.

There is also the possibility that this sphere is just about the same size as the dodecahedron, so that it intersects each of its twelve faces in a circle (see the picture below). This is the possibility that Roukema and his colleagues investigated. It is only objects lying within these 12 circles in the sky that can have double images (in opposite directions), and they must be far enough away that both the objects and their images lie within the visibility sphere (also called the cosmological horizon) - the sphere centred on me whose radius is the age of the universe times the speed of light. (The point is that if the object is too close to me, its image will be too far away to be seen by me.)

The visiblity sphere just beginning to burst through
the cosmic dodecahedron

In this scenario, only objects that are nearly as old as the universe are going to have double images. But there's not much we can see out there - with the notable exception of the primaeval plasma which emits the microwave background radiation, and at which (with the help of the WMAP sattellite) we have been gazing so intently during the last few years.

In other words, if the visibility sphere is only just big enough to intersect the dodecahedron, the only thing of which we are likely to be able to see a double image is the primaeval plasma. The glowing plasma is very nearly as old as the universe itself, so to find the double images we need to look in the direction of the 12 circles, above-mentioned, in which the visibility sphere intersects the dodecahedron. If Roukema's hypothesis is correct, when we look at one of these circles and its opposite number (which is also its topological image), we will be looking at the same bits of the primaeval plasma, and they should look the same so far as their tiny temperature variations are concerned.

Testing the theory is a question of looking at all possible opposing circles and testing them for correlations (not forgetting the 36° twist!). The unkowns are the orientation of the dodecahedron, and the size of the circles of intersection (which depends on the exact relationship between the size of the dodecahedron and the age of the universe).

In the latest (January 2008) paper by Roukema et al. (link) the configuration of the dodecahedron which yields the best correlations turns out to be such that the face-centres are at the following positions (specified by galactic longitude l and latitude b; I have also converted to Right Ascension and Declination - using Nasa'sCoordinate Converter - so that you know where to look in the sky; and have added the nearest constellation in each case):

Face 1 l=117° b=020° RA=22h17  Decl=81°  Cepheus
Face 2 l=184° b=062° RA=10h47  Decl=37°  Leo Minor
Face 3 l=046° b=049° RA=15h57  Decl=28°  Corona B.
Face 4 l=060° b=-013° RA=20h30  Decl=17°  Dolphin
Face 5 l=125° b=-042° RA=01h00  Decl=21°  Andromeda
Face 6 l=176° b=-004° RA=05h20  Decl=30°  Taurus

Note that the five faces 2..6 are arranged symmetrically around face 1. The remaining six face-positions are antipodal to the six given ones. (You may find it easier to think in terms of vertices of an icosahedron instead of face-centres of a dodecahedron. The angular distance between adjacent vertices is 63.43°, or tan^-1(2).)

The circles that give the best correlation have an angular radius of 11°. Even if the data were entirely random, there would no doubt be a dodecahedron, and a circle radius that yielded a maximum for the correlation. The question is, whether the quality of the optimum correlation is such as to make one believe that there really is a 'signal' - indicating the PDS topology - in all those masses of data from WMAP, rather than just noise. Much of the latest paper from Roukema et al. is devoted to trying to convince us that there really is a signal. I haven't yet understood the argument. But to quote from the paper's conclusion:

Do we really live in a Poincaré Dodecahedral Space? Further constraints either for or against the model are certainly still needed, but the evidence in favour of a PDS-like signal in the WMAP data does seem to be cumulating.

'Bye for now!

What is needed is Non-Euclidean geometry

Suppose for a moment that regular dodecahedra did fit together to 'tessellate' 3-D space. Each vertex would be the common property of 4 dodecahedra, and four edges would converge at each vertex. Their configuration would be just like what is called a tetrahedral bond in chemistry - think for example of the carbon atom surrounded by 4 hydrogens in methane. Looking down one of the bond directions, the other three are equally spaced with apparent angles of 120 deg between them. If the carbon is at the origin, we can take the hydrogens to be at (1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1). The cosine of the angle between any two of the bonds will be -1/3, giving us the angle as 109.471° = α. Here we have another measure of the failure, in actual fact, of dodecahedra to fit snuggly together. For the angle at the vertex of a regular pentagon is only 108°. Another case of 'angle deficit'.

But hold on - so far our unspoken assumption has been that we are working in Euclidean geometry. In non-Euclidean geometry, the angles of a triangle add up to less than 180° (in hyperbolic, negative-curvature geometry), or more than 180° (elliptic, or positive-curvature geometry) - and the more so, in both cases, the larger the area of the triangle. The same will go for for the internal angles of a pentagon - so what we need to get a pentagon with angles of 109.471° is elliptic geometry. Altogether, the angles will exceed 540° by 7.355° or 0.128 of a radian, implying that the area of the pentagon must be 0.128 times the square of the radius of curvature - or just 0.128 if we take the radius of curvature as one unit.

We can obtain the length of the side of the pentagon, by doing a bit of spherical trigonometry (which applies in elliptic geometry). The second fundamental formula of spherical trigonometry goes as follows:

cos(A) = - cos(B) cos(C) + sin(B) sin(C) cos(a).

Here A, B, C are the three angles of a triangle, and a is the length of the side opposite A (again we assume the radius of curvature is one unit). The formula is usually thought of as determining the angle A when a, B and C are known. However, it can also be used to determine a when A, B and C are known. In the present case, A is 72°, B and C are each α/2. We obtain

cos(a) = ( cos(72) + cos²(α/2))/sin²(α/2) = 0.9635

hence a = 0.2709.

In flat space, the volume of a dodecahedron whose side is a, is (2 + 3.5 φ) times a³ (here φ stands for the golden number), giving us in the present case 0.1524. This is of course only an approximation (an underestimate, in fact), for the dodecahedron about which we are now talking exists in positively curved space.

It is interesting to compare this with the volume comprised in a 3-sphere of unit radius, which is just 2π²: enough to fit in approximately 2π²/0.1524 ≈ 129 of our dodecahedra.

Poincaré Dodecahedral Space - topology

Suppose that 3 adjacent windows of my dodecahedral cell (see previous posts), having a vertex in common, are coloured blue, red and yellow. Then the images I see through these windows will also have three coloured faces. Here is a picture of what I will see (click on the image for better resolution).

Now, consider any pair of the images - for example, the one that is seen through the yellow window (call it Y) and the one that is seen through the red window (call it R). Suppose a person standing in Y looks out through the window that faces towards R - i.e. through the pentagon which adjoins his red and blue windows but which isn't yellow. Furthermore, suppose that, looking out of this window, he sees an image of himself and his cell, rotated by 36°, in just the same way that I see images of myself outside each of my windows.

Remarkably, the image of his own cell which he sees will then be just as if he could see just the same image R which I see through my red window. In other words, R is related to Y by just the same sort of translation-and-twist by which R is related to my own cell.

It is a bit like this. My friend is sitting at his table looking at a plate of food in front of him. I catch sight of my friend, and his plate of food, in a mirror. What is the image of my friend looking at? He is of course looking directly at the image of his plate of food. If I weren't so used to this property of mirrors, I might be rather surprised!

However, the keen-witted will have noticed that the three images Y, R and B can't be in quite the correct geometrical relationship with one another. The whole configuration would only work exactly if the dihedral angle between adjacent faces of a regular dodecahedron were exactly 120°. But it isn't! It's more like 116.565° (the angle whose tangent is -2). So there's a bit of rattling around, or 'angle deficit' (meaning the angles aren't quite big enough for everything to fit together snuggly).

We won't let that get us down, however. For the moment let's pretend that we are topologists - who allow themselves to stretch their objects and pull them around as though they were made of rubber. So an angle deficit of 3.435° is easily dealt with by moulding the rubbery dodecahedron.

Now Poincaré Dodecahedral Space (PDS) was originally conceived by Poincaré as a purely topological object. In fact when first conceived by Poincaré it had nothing to do with the dodecahedron at all! That connection came quite a few years later. But we will stick with the dodecahedral way of getting at it.

Think of PDS as a three-dimensional analogue of the construction of a toroidal surface in two dimensions. Start with a rectangular piece of paper. It is decreed that a closed surface is to be constructed by identifying the top edge of the paper with its bottom edge, and its left-hand edge with its right-hand edge. In other words, the paper is to be conceived as a sort of map of the surface, which is quite faithful in most places, but falls down at the edges. For the paper comes to an end at any of its edges, but the actual surface has no edges. The top and bottom edges of the map correspond to a single line drawn on the surface. If you are wandering around on the surface and happen to cross this line, then your path on the map will go off the edge (the top edge, say) and reappear at the corresponding point on the bottom edge.

If you cross back over the line, your representative point will go back off the bottom edge and reappear at the top edge.

In this way the surface is perfectly well defined, in an 'intrinsic' sort of way. But most of us are never quite satisfied with the 'intrinsic' point-of-view, and want to know what the surface is really like. Fortunately, in this case, this desire can be fulfilled, by going into three dimensions. We can 'indentify' the top and bottom edges of the paper simply by bending it into a tube. Then the left and right-hand edges can be 'identified' by bending the tube round into a torus. A paper tube wouldn't bend like that, of course. But we're doing topology, remember? So this is rubbery paper.

So the surface we were told to construct turns out to be topologically identical ('homeomorphic' in topology-jargon) with the surface of a torus - which is topologically distinct from the surface of a sphere, by the way, because one can draw on it closed paths which cannot be continuously shrunk to a point (they are condemned to go round the torus, one way or another).

Next, suppose the instructions were like this. Take a hexagonal piece of paper; identify its opposite edges. What do we 'really get' this time? This is a bit more tricky. Roll up the hexagon into a tube, bend the tube round - and the ends of the tube don't fit together properly. What we have to do is to stretch the tube out lengthways, twist one end of it through 180°, and then join the ends. We're back at the torus - a bit twisted, but that makes no difference from the toplogical point-of-view (though it might from others).

Ready to move on to three dimensions? You could start by taking a cuboid, and identifying its opposite faces. This defines a new space, but can we visualize what it's 'really like'? Working in the usual three dimensions, we can only get part of the way. Stretch out the cuboid in one direction, bend it round and join one pair of opposite faces. What we've got is a torus with a rectangular cross-section. Stretch the solid torus out into a long cylinder, bend it round and join a second pair of opposite faces. Now we've got a hollow torus, made out of a certain thickness of material. This solid has an outer surface and an inner one. Unfortunately, these two surfaces (inner and outer) are the descendants of the third pair of faces that are to be joined - and there is clearly no way of joining them without going outside three-dimensional space. All we can say is that the constructed space is a 'three-dimensional torus' - like a two-dimensional one, only more so.

OK, we're ready for the PDS. Take a rubbery dodecahedron, stretch it out in one direction, bend round and join opposite faces, giving them a twist of 36° (in the anti-clockwise sense) just before joining them.

Then do the same with the remaining 5 pairs of opposite faces.

Well, that sounds easy enough! Not so easy to visualize, though...

Our original, solid dodecahedron is the 'map' of the PDS. At each pentagonal face, one goes off the boundary of the map - but the actual space (of which it is a map) has no boundaries. Instead it has six pentagonal surfaces drawn in it (each one corresponding to two opposite faces of the dodecahedron). As one wanders back and to across one of these surfaces, ones representative point jumps from one face of the dodecahedron to the opposite one.

To be sure that the constructed space is well-defined, we need to make sure of what happens when you wander about near one of the lines in the PDS which correspond to the edges of the dodecahedral map - for example E in the figure below (click to enlarge).

Now E is common to two of the pentagonal faces, each of which is supposed to be joined up with its opposite number (with a twist) in the construction of the PDS. This means that E will be identified not just with the edge e1 (when the top face is identified with the bottom face) but also with the edge e2 (when the front face is identified with the back face).

But e1 is also the boundary of another face, the one at bottom-front-left, which needs to be identified with the face at top-back-right. But that's no problem: in the identification-with-twist, e1 will be identified directly with e2. So far so good: which ever way you look at it, E, e1 and e2 will all correspond to the same line in the PDS.

In fact, this line will be the common boundary of three pentagonal surfaces (corresponding to the pairs top-bottom, front-back and bottom-front-left, top-back-right). In the figure, we also see how the wedge-shaped bits of dodecahedron just near to the three edges fit together when the edges are identified to make a nice bit of three-dimensional space. Note that the wedges each have to be given a relative twist (anti-clockwise) as they are fitted together.

Having satisfied ourselves about the edges, it only remains to check what happens at the vertices of the dodecahedron. A vertex - such as V, below - belongs to three faces, and so is party to three identifications, leading to V being identified with three partners - marked v1, v2, v3.

Note that given one of the four partners, the other three three may be determined as follows. The given vertex belongs to three pentagonal faces. Choose one of them. Go across it following the left-hand diagonal (as explained in the figure) and from there go down the edge that is not in the plane of the pentagon. You arrive at one of the partner vertices. Get the other two by starting across different pentagons.

The partners are mutually equidistant - in other words, they are the vertices of a regular tetrahedron. In fact, all 20 vertices of the dodecahedron are grouped into 5 sets of four, yielding 5 tetrahedra, all symmetrically placed with respect to one another, the so-called 'compound of five tetrahedra'. (Note that there is a second compound, the mirror-image of the first, that would be obtained by traversing the faces on the right-hand diagonal instead of the left.)

Our picture also shows how the solid angles at the vertices V, v1, v2 and v3 come together at one point in the PDS. Once again, the pyramidal pieces need to be twisted as they are brought together.

One of the compounds of five tetrahedra

Poincare Dodecahedral Space (contd.)

View through the front window of my dodecahedral cell. Note that the back window of my image's cell matches the front window of mine. And that my image's front window is twisted by 36 degrees relative to mine.

View through two adjacent windows of my dodecahedral cell (click on the image to get better resolution). The edges of one pentagonal face of my cell are marked. Convinve yourself that the image cells are marked correctly.

View from the right-hand image, looking out of the blue window towards the left-hand image. On the left, the blue front window of the right-hand cell, with the marked pentagon abutting on it. On the right, how the blue window of the left-hand image appears. Note the rotation of 36 degrees.

Are we in Poincare Dodecahedral Space after all?

For a few years, people have been floating the idea that our world exists inside a space that has 'exotic topology' - that is, a space that joins up with itself in peculiar ways.

I hadn't paid much attention, until my eye was caught by an article in New Scientist magazine (12th January 2008) entitled 'Our finite, wrap-around universe'. It reported on a recent paper by Boudewijn Roukema, of Nicholas Copernicus University (how appropriate!) in Poland. (Feeling strong? You can see the paper at www.arxiv.org/abs/0801.0006 . Click on 'PDF' under where it says 'Download' in the top right hand corner.)

Roukema identifies 12 spots in the starry sky, arranged symmetrically, so that the angle between any two of these spots matches the angle between two faces of a regular dodecahedron (another way of putting it - the points could be the vertices of a regular icosahedron).

These special points have the property, that, if you could look far enough in the direction of one of these spots, you would see the universe beginning to repeat itself. It would be a bit like looking at yourself between two parallel mirrors - except that, in that scenario, you see as many images of the back of your head as of the front (i.e. - your face). In Roukema's scenario, you see only the back of your head. The image of yourself that you see is facing the same way as you. You are not looking in a mirror; you are located in a convoluted space, such that light leaving the back of your head goes on a long journey round the universe and enters in at your eyes in the usual way.

Now for the catch: your image is facing the same way as you (forwards!) but doesn't seem to be standing upright. She or he is leaning over as if in some sort of strange kaleidoscope - at an angle of 36°!

Beyond the first image you can see a second - twisted through another 36°. And so on, unto the tenth image, which is once more as upright as you are.

Now turn your head and look in the direction of another of Roukema's spots. Again you see an image of youself, with turned head, from the back - and twisted through 36°.

Why 36°?

(What follows does not answer the question Why?, at least not directly. Think of it as fleshing out the question...) Suppose you were standing in a large dodecahedral cell, with glass walls. Your image would then stand in a similar dodecahedral cell. What the rotation of 36° allows, is that the front wall of your dodecahedron matches the back wall of your image's cell. If you imagine your cell expanding, eventually it would be so big that it could actually fit onto its image!

The same applies to the image that you saw by turning your head: there is another cell out in that direction, also capable of fitting together with your cell, here.

Now for a question that may occur to you (if you have any trace of the mathematical disease in your blood): we've seen that the two image cells, seen in different directions, fit onto adjacent faces of the original cell, here, that the real you (?) is standing inside. One wonders: do they fit onto each other?

Well, that opens a can of worms... I think we'd better leave it 'til tomorrow.