**Contemplating A Finite, Multiply Connected Universe**
Now that you have become topology experts

I think it's about time to illuminate a bit the breathtaking possibilities arising from a possible

**finite, yet multiply connected spacial topology of our Universe**
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We proceed VERY slowly and intuitively!

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Let me start from a

**2d Universe first** since there we can still draw easily what we mean.

The simplest

**finite** and geometrically

**flat**,

**singly connected** 2d Universe is the blue square you can see below on the top left (Fig 1). In this flat universe the Pythagoras (triangle) theorem would clearly hold.

Look at the 2 galaxies depicted in Fig. 1), which I have adapted from a great lecture by Nick Bower/U. Chicago, that unfortunately has vanished from the net...

(

http://astro.uchicago.edu/home/web/olin ... nbower.htm)

An observer in the red galaxy can perceive the light emitted from the yellow galaxy only along ONE path, the yellow one in Fig. 1)! This corresponds to the fact that our assumed space topology so far is singly connected.

Now, let us make a multiply connected 2d space out of our finite-sized Universe "sheet"! Abstractly speaking this happens if we just

**identify** the opposite sides of our flat rectangle. Look e.g. at the white path of light emitted from the yellow galaxy! It can now reach our observer in the red galaxy: when the light hits the right-hand vertical border of our universe in Fig. 1) it reappears immediately at the opposite left-hand border as depicted (on account of the border identification). Eventually the white path hits the observer's eyes! Analogously, the red path and the top-bottom borders...

This identification of the opposite borders of our finite square Universe may sound somewhat artificial to you at first, but it is NOT at all.

Look at Figs. 2) and 3) what it actually implies. By identifying the opposite borders we have just made a nice

**doghnut shaped surface (space)** out of our original flat rectangle! Figs. 2) and 3) illustrate the steps intuitively. You immediately see how the white yellow and red paths of light from the yellow galaxy can be perceived by our observer in the red one. That 2d doghnut Universe is now multiply connected and correspondingly more than 1 image of the yellow galaxy can be seen by the observer. Since the distance s also differ, the different galaxy images correspond to largely different evolution stages of our yellow galaxy, given the long light travelling times involved.

Of course, the doghnut surface represents a

**curved 2d space** unlike the rectangle we have started with!

Correspondingly the Pythagoras theorem does not hold anymore for triangles inscribed on the doghnut surface.

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Next we increase the spacial dimension of our finite universe by 1 and realistically consider 3d. Now it's really becoming fun!

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Instead of our 2d sheet in Fig. 1) we now start from a

**finite** cubic 3d Universe as shown here

Analogously to the procedure above , let us now make a

**3d**-doughnut (torus) out of the cube by

**identifying the opposite walls of the cube**. This is indicated by the matching colors in the figure.

Unfortunately a 3d-doghnut cannot be drawn anymore but conceptionally everything is as in the 2d example above.

Still we may intuitively proceed: The identification may be simulated by

**replacing our identified walls by mirrors**. You, the observer now stands in the center of a finite-sized

**mirror hall**, which quite a few of you must have experienced in reality already.

To you the

**finite** multiply connected 3d doghnut Universe appears as if it was

**infinite**, since through the mirrors you perceive in all directions infinitely many copies of yourself and of the

**finite cubic** "base" Universe. By looking towards the right-hand mirror wall you may e.g. see the

**back of your own head**!

That seems to be a good point to let you contemplate a bit about the amazing implications. Please don't hesitate to ask if there is something unclear at this point.

Cheers,

Fridger