Three-Sphere Visualiser (and/or related stuff)
1. Overview....................................................................................................................1
2. Introduction................................................................................................................1
3. Description using words ............................................................................................1
4. Description using equations.......................................................................................1
5. Riemann's description – Two Ball models ................................................................2
6. Dante’s Inferno ..........................................................................................................3
7. Stereographic Projection............................................................................................3
1. Overview
This has to do with N-spheres and ideally the project would come up with, and
implement a visualisation approach that would give some impression of what it would
be like to live in a 4-D world (compare the book and movie “flat-land”
http://www.youtube.com/watch?v=C8oiwnNlyE4 ).
Part of the background is the fairly recent and very noteworthy proof of the “Poincare
conjecture” in Mathematics. There are many relevant web-sites to look at, e.g.
http://www.physicsforums.com/archive/index.php/t-119730.html).
2. Introduction
The contents of this section (from http://www.claymath.org/poincare/3-sphere.html and
http://theory.org/geotopo/3-sphere/html/node2.html) introduce the notion of spheres of
various dimensions.
We all know what a circle and a sphere look like. We say a circle (S1) is a one-dimensional
sphere and a (“normal”) sphere (S2) is two-dimensional. A circle can be drawn with a
compass [we mean the perimeter only], while S2 occurs, for example, as the surface of a
ping-pong ball. A circle is one-dimensional, since a point is determined by specifying one
value (an angle). A sphere (S2) is two-dimensional since a point is determined by two
numbers, latitude and longitude. One might ask: is there a three-dimensional analogue of the
sphere, what we might call a three-sphere (S3), or a hypersphere?i
3. Description using words
The short answer is "yes." A somewhat longer answer is
(1) S1 is the set of points in the plane at fixed distance (the radius) from a given point (the
center)
(2) S2 is the set of points in three-space at fixed distance from a given point
(3) thus by analogy S3 is the set of points in four-space at a fixed distance from a given point.
This way of thinking about the circle, sphere, and hypersphere makes evident the family
resemblance of these shapes.
4. Description using equations
The translation of (1) is that S1 is the set of points where x2 + y2 = r2.
This is the equation for a circle of radius r centered at (0, 0).
The translation of (2) is similar: S2 is the set of points where
x2 + y2 + z2= r2.
This is the equation for a sphere of radius r centered at (0, 0, 0).
The translation of (3) follows the same pattern: S3 is the set of points where
x2 + y2 + z2 + w2= r2.
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This is the equation for a hypersphere of radius r centered at (0, 0, 0, 0).
Again, the family resemblance of these shapes is made apparent by the family resemblance of
the equations. We could call the shapes defined the one-sphere, the two-sphere, and the threesphere, respectively.
5. Riemann's description – Two Ball models
Here is yet another way of thinking of the hypersphere, or three-sphere.
(1) Take two wires. Join the ends of one to the ends of the other. The result is a misshapen
circle. It can be reshaped, without cutting or tearing the wire, into a true circle (S1).
(2) The two balls model of S2 First, imagine that we were beings who inhabited a twodimensional universe. Visualizing three-dimensional space would (probably) not come
naturally to us, and thus neither would visualizing S2, which needs to be embedded in at least
three dimensions.
However, we could visualize S2 by thinking of it as two disks, plus a special rule. The special
rule goes like this: whenever you cross the boundary of one disk, you reappear at the
corresponding point on the boundary of the other disk. Each disk is a hemisphere, and the
boundary is the equator. Thought of this way, the rule becomes obvious: whenever you leave
one hemisphere, and cross the equator, you reappear at the corresponding point in the other
hemisphere.
Note that when our two dimensional being lays the two disks flat out in R2, there are two
ways that they can be laid down. First, both disks can simply be flattened onto R2. If this is
done, then a vector pointing out of the sphere on the lower hemisphere will be pointing down
from R2, and a vector pointing out of the sphere on the upper hemisphere will be pointing up
on R2. Alternately, the bottom hemisphere can be reflected or flipped, so that a vector
pointing out on the sphere will be pointing up from R2, on both hemispheres. Note that this
act of flipping means that we will have to be extra careful about how we identify the edges of
the hemisphere; specifically, points will be mapped to the corresponding point on the
hemisphere after the reflection. For practical purposes, we will consider the first model where
the lower hemisphere is not flipped.
(3) The two balls model of S3 With this little thought experiment complete, let us try
something more ambitious. Take two solid balls in three-space. Each one consists of a twosphere (S2) and the material inside it, just as a disk consists of a circle and the material inside
it. Now imagine that the two-sphere for one of the balls is glued (magically) to the two-sphere
of the other ball in such a way that latitudes and longitudes of one sphere correspond to
latitudes and longitudes of the other sphere. The result is the hypersphere, or three-sphere
(S3).
What this amounts to is to imagine that we are beings who inhabit a three dimensional
universe. Visualizing four-dimensional space would not come naturally to us, and neither
would visualizing S3, which needs to be embedded in at least four dimensions. But we can
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visualize S3 by thinking of it as two filled balls, or filled S2s, plus a special rule, just like in
the two dimensional case.
Finally, similarly to the two disks model of S2, we can flatten the hemispheres onto S3 without
flipping one, or with flipping one. Just as before, we will use the model without the flip.
(4) Using a two balls model to visualize higher dimensional spheres If we wanted to visualize
S4, we could first visualize S3 in this way, and then make two copies of S3, and consider each
to be a D4 or filled S3. We could then glue each point in the S3 to a corresponding point in the
other S3, and we would have S4. Unfortunately, this becomes a bit cumbersome and
complicated for dimensions higher than three.
6. Dante’s Inferno
A model of the three-sphere similar to the one just described appears in Dante (there is a
video game based called Dante’s Inferno!). The earth is a ball. We live on its surface, a twosphere. Below the surface of the earth are concentric but smaller spheres in lower rings of the
underworld. These spheres contract to a single point, the center. Perhaps the abode of Hades,
or the Devil.
As you pass upwards from the surface of the
earth into the sky, then the heavens, you enter
another realm, also composed of concentric
spheres. When you go high enough, the
spheres start to become smaller and smaller.
Eventually the spheres shrink to a single
point. This is the abode of the Deity.
The universe, in Dante's view, is made of two
three-dimensional balls. A point in each is
determined by latitude, longitude and depth
(or height). While we can visualize each ball
as a separate entity, we cannot visualize the
hypersphere in its entirety. Nonetheless, we
can understand it well enough to say what happens as we travel from part to part.
7. Stereographic Projection – Implementation Route?
Stereographic projection is a way of projecting Sn onto Rn. Intuitively, it can be thought of as
the process of puncturing the sphere and stretching it out flat, onto Euclidean space. It too can
be used as an aid to visualization.
Stereographic Projection of S1
To stereographically project S1 onto R1, first choose an equator, and a ``north pole.'' In the
case of S1, the equator simply is two antipodal points on the circle, and the north pole is the
point on the circle equally between them. For simplicity, let's assume the circle is the unit
circle, the north pole is the point (0,1), and the equator the set of points {(1, 0), (-1,0)}. We
are now going to map, or stereographically project, the circle onto the x axis.
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The map is as follows: Draw a line from the north pole, through a point P on the circle, and
extend it until it hits the x-axis at the point P’.. Stereographic projection is simply the map
which takes each point P on the circle and maps it to the point P’ in R1.
In the case with S1, the map sends P=(x, y)
to P’ = x/(1-y).
Notice that every point on S1 has an image
on R1, except for the north pole itself. The
north pole is considered to be mapped to a
point at infinity. So we can say that
stereographic projection S1 -> R1∪{∞} is a
bijection.
As mentioned above, stereographic
projection can be thought of as puncturing
the sphere, and stretching it out flat onto the
correct Euclidean space. Points in the
``southern hemisphere'' are mapped to points between 1 and -1, and points in the ``northern
hemisphere'' are mapped to points outside.
Stereographic Projection of S2
To stereographically project S2, imagine S2, centered at the origin. To project the sphere onto
the x-y plane, we choose the ``north pole'' to be the point (0,0,1). Then, to project a point P =
(x, y, z) in S2 onto the x-y plane, we draw a line from the north pole, through P to its image
point P’ on the x-y plane. Thus, the map sends P=(x, y, z) to P’ = (x/(z-1), y/(z-1)).
Again, the north pole is mapped to the point
at infinity. Now the hypothetical twodimensional beings have another way to
visualize S2: they can think of it as regular
R2, plus a point at infinity. If they were
moving, say, a little square around the sphere,
and watching how its image changed in the
plane as it moved, they would see it get larger
as it got further from the south pole, which is
mapped to the origin on the plane. If they
moved the square from the south pole, through the north pole, and back to the south pole, the
two dimensional beings would see the square get larger and larger as it approached the north
pole. When the north pole was inside the square, they would see the square all around them.
As the square moved back to the south pole, they would see the square approaching from the
other direction, getting smaller as it approached.
Stereographic Projection in Sn
Stereographic projection is possible in general, for Sn. The map is:
P = (x1, x2, …, xn+1) in Sn is mapped onto P’ = (x1/(xn+1-1), x2/(xn+1-1), …, xn/(xn+1-1)) in Rn
Now we have another way to visualize the three sphere: we can think of it as R3, plus a point
at infinity. If we were moving a cube around the three sphere, and looking at its image in
Euclidean three-space under stereographic projection, it would get bigger as it got further
from the south pole, which is the origin in R3. Like the image of S2 in R2, if we were moving
a cube along a great circle from the south pole, through the north pole, back to the south pole,
we would see it start out small at the origin, get larger as it went further from the origin.
When the north pole crossed inside the little cube, we would see the cube all around us. Then,
as the cube passed back towards the south pole, we would see it shrink again as it approached.
Stereographic projection offers another technique of visualizing or thinking about higher
dimensional spheres, and another point of view on S3.
i
For completeness, we consider that that a point is a zero-dimensional sphere (S0).
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