Light Eye
03-11-2005, 11:11 AM
Dear Friends,
Interesting article by Janna Levin.
http://plus.maths.org/issue10/features/topology
Be Well, Be Love.
David
In space, do all roads lead to home? by Janna Levin
Figure 1: Getting nowhere
Imagine you walk down a strange street and pass a gate in a wall. You keep
walking in a straight line only to pass another gate in a wall. You keep walking
and pass yet another gate, yet another wall. You might begin to suspect it is
the same gate, the same wall. Even though you walk in as straight a line as
possible, never turning back, you come upon the two landmarks again and again.
This, of course, could actually happen on the surface of the Earth, on a much
larger scale. If you started in London and walked in as straight a line as
possible for a very long time, you would eventually come back to London. This is
because the Earth is curved and finite, and has a surface with the overall
geometry of a sphere.
Over the past few hundred years we have made ourselves quite familiar with the
Earth's compact surface, charting oceans and flying around the globe. Here I'm
using the word "compact" to mean that the surface has no edge, but rather is
smoothly connected to itself. With this definition, the surface of a sphere is
compact, and so is the surface of a bubble or a doughnut. The surface of a
square would not be compact, because it has edges. However, a compact surface
can be made from a square by smoothly gluing the edges of the square together as
we do below.
A remarkable possibility is that the entire universe is compact and connected.
In other words, if we were to launch a spaceship from Earth and fly in as
straight a line as possible, we could find ourselves returning home. As we see
the Earth receding in the distance behind us, we might also see it growing
nearer in front of us.
A compact universe is far more dramatic than a compact planet. From Einstein we
have learned that the universe has three spatial dimensions and one time
dimension. According to his theory of relativity, we all move along the natural
curves in that space. Even light follows these curves.
If that strange street lived in a short, compact universe, the world would get
even stranger. Light from the street lamps would wrap around the compact space,
following the natural curves. If you were to stand there you would see ahead of
you the light reflected off the street which had traveled all the way around the
space.
Further in the distance, you could see the same scene again: yourself standing
in the middle of the more distant but otherwise identical copy of the street.
Like a hall of mirrors, the pattern would go on infinitely and in all
directions. Though we know from personal experience that the universe is not
this small, it could be finite and compact on a much huger scale: thousands of
times the size of a galaxy, but finite nonetheless.
To visualize how this is possible we can can utilize the geometry of surfaces.
The geometry of surfaces can be classified according to two properties, the
local curvature of the surface and the global topology of the surface.
[Non-text portions of this message have been removed]
Interesting article by Janna Levin.
http://plus.maths.org/issue10/features/topology
Be Well, Be Love.
David
In space, do all roads lead to home? by Janna Levin
Figure 1: Getting nowhere
Imagine you walk down a strange street and pass a gate in a wall. You keep
walking in a straight line only to pass another gate in a wall. You keep walking
and pass yet another gate, yet another wall. You might begin to suspect it is
the same gate, the same wall. Even though you walk in as straight a line as
possible, never turning back, you come upon the two landmarks again and again.
This, of course, could actually happen on the surface of the Earth, on a much
larger scale. If you started in London and walked in as straight a line as
possible for a very long time, you would eventually come back to London. This is
because the Earth is curved and finite, and has a surface with the overall
geometry of a sphere.
Over the past few hundred years we have made ourselves quite familiar with the
Earth's compact surface, charting oceans and flying around the globe. Here I'm
using the word "compact" to mean that the surface has no edge, but rather is
smoothly connected to itself. With this definition, the surface of a sphere is
compact, and so is the surface of a bubble or a doughnut. The surface of a
square would not be compact, because it has edges. However, a compact surface
can be made from a square by smoothly gluing the edges of the square together as
we do below.
A remarkable possibility is that the entire universe is compact and connected.
In other words, if we were to launch a spaceship from Earth and fly in as
straight a line as possible, we could find ourselves returning home. As we see
the Earth receding in the distance behind us, we might also see it growing
nearer in front of us.
A compact universe is far more dramatic than a compact planet. From Einstein we
have learned that the universe has three spatial dimensions and one time
dimension. According to his theory of relativity, we all move along the natural
curves in that space. Even light follows these curves.
If that strange street lived in a short, compact universe, the world would get
even stranger. Light from the street lamps would wrap around the compact space,
following the natural curves. If you were to stand there you would see ahead of
you the light reflected off the street which had traveled all the way around the
space.
Further in the distance, you could see the same scene again: yourself standing
in the middle of the more distant but otherwise identical copy of the street.
Like a hall of mirrors, the pattern would go on infinitely and in all
directions. Though we know from personal experience that the universe is not
this small, it could be finite and compact on a much huger scale: thousands of
times the size of a galaxy, but finite nonetheless.
To visualize how this is possible we can can utilize the geometry of surfaces.
The geometry of surfaces can be classified according to two properties, the
local curvature of the surface and the global topology of the surface.
[Non-text portions of this message have been removed]