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.