- The different shapes for space:
- Euclidean – infinite in all directions
- 'Aristotelian' – a ball (a 3D sphere plus its insides)
- Spherical – in any dimensions
- Cylindrical
- Toroidal – spaces with edgeless holes
- 'Twisted' spaces – Klein Bottle
- Moving about on a torus and Klein bottle: geometry www.geometrygames.org.
Here's a case where points of space are identified with each other – like the edges of a sqare to make a 2D torus or the faces of a cube to make a 3D torus.
- Note that finite spaces need not have edges.
- Recent evidence suggests that our universe may have faces that
join up, like the torus!
Spherical pentagons cover the sphere in two dimensions (left) and spherical dodecahedrons in three dimensions (right): in the three dimensional case opposite faces can be identified to give 'Poincaré Dodecahedral Space' – perhaps the topology of the universe according to recent data! (From "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background" Luminet et al, Nature 9/10/2004 p593-5)
2. Is Aristotle's Space Possible?
- Archytas' argument – what would happen at the edge?
- Reply: infinite barrier.
- infinite wall
- black hole
- infinite force at edge: e.g, let 1/n be the fraction of the way
to
the edge, and F = -1/(1 - 1/n2).
- Reply: Poincaré's sphere (not his spherical dodecahedral space).
| 1/n |
1-1/n2 |
My
height |
My
stride |
| 0 |
1 |
1.8m |
1m |
| 1/10 |
99/100 |
1.78m |
99cm |
| 5/10 |
3/4 |
1.35m | 75cm |
| 9/10 |
19/100 |
34cm |
19cm |
| 99/100 |
199/10,000 |
3.6cm |
1.99mm |
- Infinite number of paces to edge – it appears infinite.
- Cannot reach the edge at any fixed rate of paces: e.g., one per
second.
- Can reach or pass edge if I can accelerate as fast as I like – allows infinitely many paces per second at the edge.
- But this would also allow escape from an infinite space – so
does
it support Archytas?
A plot of distance traveled (horizontal) vs time taken (vertical) of a space 'escapee'. Here x = 1/(1-t) - 1, a trajectory for which the speed is always finite for t < 1 second, but for which the escapee is nowhere in space from t = 1 second on! The escapee escapes space in a finite time without ever traveling infinitely fast! Similarly, if we take this a plot of strides taken against time in Poincare's universe, such a trajectory would get us to the edge of space – you'd have to accelerate even faster to escape altogether.
- Reply: logical impossibility – as a matter of logical necessity, everything that exists exists in space.
- logic foils the apparent physical possibility of crossing the edge.
- things passing the edge cease to exist.