1. Is There a Centre? The Geometry of the Universe

2. Can We Be in a Finite Universe that has No Center or Edges?
Is this possible? It sounds bizarre…
Yes; but you have to wrap your mind once again around the concept of the curvature of space.
The universe may be ‘folded on itself’ with no boundaries and no ‘outside’.
A two-dimensional analogy may help!

3. A Starting Point
Consider the everyday experience of living on the curved surface of the Earth.
Locally, on small scales, it is flat.
On large scales, however, the curvature matters.

4. Is Greenland Really That Big?
It looks nearly as large as the whole continent of Africa!

5. An ‘Equal-Areas’ Projection

6. Havana to Moscow https://www.flightradar24.com

7. The Shortest Path! (a Geodesic)

8. Two Kinds of Curved Surfaces 1. Positive Curvature
Imagine standing on
the top of a sphere.
The surface ‘falls
away’ in front and
behind, and also to
the left and right.

9. Two Kinds of Curved Surfaces 2. Negative Curvature
Imagine standing ‘in
the saddle.’ The
surface rises in front
and behind, but falls
to the left and right.

10. Measuring the Curvature
Can we measure the curvature of the Earth with local measurements?
(say, as opposed to walking all the way around it)

11. Curved Surfaces Quantified

12. To Quadruple the Crop, Double the Radius of the Garden

13. It’s Different on a Positively-Curved Surface!

14. Curved Surfaces – Your Experience

15. Hence The ‘Torn’ Maps

16. …and the Imperfect Fit to a Pie Pan

17. Consider Only the Surface
Forget the ‘third dimension’ - the local ‘up and down’
A person on the surface could walk an infinite distance in any direction, never encountering an edge. It is unbounded. There is no edge.
Yet the total area to be explored is finite.

18. Moreover…
…no particular point on the surface is any more important than any other. The surface has no ‘special point’ -- no ‘center.’
[On Earth, we think of the poles as special. But if the Earth simply a ball of rock in empty space, they would have no particular significance.]

19. The Reason
The positively-curved
surface closes back on
itself.

20. A Digression: Fun With Curved/Folded Surfaces
A Moebius strip.

21. The Moebius Strip It has only one surface!

22. What If Light Itself Follows the Surface?
We would have no notion of ‘up and down,’ a direction perpendicular to the surface! We would live in a curved ‘flatland’

23. You Might See The Back of Your Own Head!

24. But Does Light ‘Follow the Surface?’
Clearly not along the surface of the Earth! The curvature (‘puckering’) of space is too small. But light does follows the curvature of space! Given the totality of matter (including dark matter) in the universe, it is possible that space ‘folds back’ on itself. If so, light will eventually return to its place of origin (unless it runs into something en route) like a nomad wandering over the surface of the Earth.

25. In Short
As we have seen, light can be deflected slightly by a star like the Sun, and significantly ‘lensed’ by the existence of large concentrations of matter (such as a cluster of galaxies) -- but also governed overall by the total distribution of matter in the entire [finite] universe itself. If the density of matter is high enough, the universe can be ‘closed’ – that is, finite, and folded back on itself – with no ‘outside’, no centre, and no edges.

26. Now Add Expansion (first cover the balloon uniformly with dots)

27. The Dots (Galaxies!) Do Not Expand - they are merely carried along by the expanding space

28. Result: A Finite Universe…
Observers located on any individual dot see all other dots recede from them as the ‘universe’ expands. No dot is the center of the expansion
There are a finite, countable number of dots
The entire observable universe is the surface of the balloon (and the ‘dots’ on it).

29. Now the Real Challenge!
Convert this understanding into one higher dimension!
That is, imagine a fully 3-dimensional space that curves back on itself so there is no outside, no edge, no boundary, no center.
The finite universe is not ‘inside’ some greater volume, surrounded by some vast void. It is all there is. And we can’t travel to ‘the edge’ and look beyond.