Notice: This material will be included in a forthcoming (summer 2000) book with the tentative title Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces. This new book will be an expanded and updated version of Experiencing Geometry on Plane and Sphere. This material is in draft form and may not be duplicated or quoted without the author's written permission, except for purposes of review or trying out the material with students. As always comments are welcome and will affect the final draft. Send comments to dwh2@cornell.edu.
Chapter 12
Let us, then, make a mental picture of our universe: ...as far as possible, a complete unity so that whatever comes into view, say the outer orb of the heavens, shall bring immediately with it the vision, on the one plane, of the sun and of all the stars with earth and sea and all living things as if exhibited upon a transparent globe.
Bring this vision actually before your sight, so that there shall be in your mind the gleaming representation of a sphere, a picture holding all the things of the universe... Keep this sphere before you, and from it imagine another, a sphere stripped of magnitude and of spatial differences; cast out your inborn sense of Matter, taking care not merely to attenuate it: call on God, maker of the sphere whose image you now hold, and pray Him to enter. And may He come bringing His own Universe...
— Plotinus, The Enneads, V.8.9 [A: Plotinus]
In this chapter you will explore hyperbolic 3-space and the 3-dimensional sphere which extrinsically sits in 4-space. But intrinsically, if we zoom in on a point in a 3-sphere or a hyperbolic 3-space, then locally the experience of the space will become indistinguishable from an intrinsic and local experience of Euclidean 3-space. This is also our human experience in our physical universe. We will study these 3-dimensional spaces because they are possible geometries for our physical universe and in order to see that these geometries are closely related to their 2-dimensional versions.
Try to imagine the possibility of our physical universe being a 3-sphere in 4-space. It is the same kind of imagination a 2-dimensional being would need in order to imagine that it was on a plane or 2-sphere (ordinary sphere) in 3-space.
Problem 12.1. Explain 3-Space to 2-D Person
How would you explain 3-space to a person living in two dimensions?
Think about the question in terms of this example: The person depicted in Figure 12.1 lives in a 2-dimensional plane. The person is wearing a mitten on the right hand. Notice that there is no front or back side to the mitten for the 2-D person. The mitten is just a thick line around the hand.
Figure 12.1. 2-dimensional person with mitten.
Suppose that you approach the plane, remove the mitten, and put it on the 2-D person's left hand. There's no way within 2-space to move the mitten to fit the other hand. So, you take the mitten off of the 2-D plane, flip it over in 3-space, and then put it back on the plane around the int:left hand. The 2-D person has no experience of three dimensions but can see the result — the mitten disappears from the right hand, the mitten is gone for a moment, and then it is on the left hand.
Figure 12.2. Where did the mitten go?
How would you explain to the 2-D person what happened to the mitten?
Suggestions
This person's 2-dimensional experience is very much like the experience of an insect called a water strider that we talked about in Chapter 2. A water strider walks on the surface of a pond and has a very 2-dimensional perception of the universe around it. To the water strider, there is no up or down; its whole universe consists of the surface of the water. Similarly, for the 2-D person there is no front or back; the entire universe is the 2-dimensional plane.
Living in a 2-D world, the 2-D person can easily understand any figures in 2-space, including planes. In order to explain a notion such as "perpendicular," we could ask the 2-D person to think about the thumb and fingers on one hand.
Figure 12.3. The 2-D person sees "perpendicular."
A person living in a 2-D world cannot directly experience three dimensions, just as we are unable to directly experience four dimensions. Yet, with some help from you, the 2-D person can begin to imagine three dimensions just as we can imagine four dimensions. One goal of this problem is to try to gain a better understanding of what our experience of 4-space might be. Think about what four dimensions might be like, and you may have ideas about the kinds of questions the 2-D person will have about three dimensions. You may know some answers, as well. The problem is finding a way to talk about them. Be creative!
One important thing to keep in mind is that it is possible to have images of things we cannot see. For example, when we look at a sphere, we can see only roughly half of it, but we can and do have an image of the entire sphere in our minds. We even have an image of the inside of the sphere, but it is impossible to actually see the entire inside or outside of the sphere all at once. Another similar example: sit in your room, close your eyes, and try to imagine the entire room. It is likely that you will have an image of the entire room, even though you can never see it all at once. Without such images of the whole room it would be difficult to maneuver around the room. The same goes for your image of the whole of the chair you are sitting on or this book you are reading.
Assume that the 2-D person also has images of things that cannot be seen in their entirety. For example, the 2-D person may have an image of a circle. Within a 2-dimensional world, the entire circle cannot be seen all at once; the 2-D person can only see approximately half of the outside of the circle at a time and can not see the inside at all unless the circle is broken.
Figure 12.4. The 2-D person sees a circle.
However, from our position in 3-space we can see the entire circle including its inside. Carrying the distinction between what we can see and what we can imagine one step further, the 2-D person cannot see the entire circle but can imagine in the mind the whole circle including inside and out. Thus, the 2-D person can only imagine what we, from three dimensions, can directly see. So, the 2-D person's image of the entire circle is as if it were being viewed from the third dimension. It makes sense, then, that the image of the entire sphere that we have in our minds is a 4-D view of it, as if we were viewing it from the fourth dimension.
When we talk about the fourth dimension here, we are not talking about time which is often considered the fourth dimension. Here, we are talking about a fourth spatial dimension. A fuller description of our universe would require the addition of a time dimension onto whatever spatial dimensions one is considering.
Try to come up with ways to help the 2-D person imagine what happens to the mitten when it is taken out of the plane into 3-space. It may help to think of intersecting planes rotating with respect to each other — How will a 2-D person in one of the planes experience it? Draw upon the person's experience living in two dimensions, as well as some of your own experiences and attempts to imagine four dimensions.
Problem 12.2. A 3-Sphere in 4-Space
We will now explore 3-dimensional spheres in 4-space which are possibly the shape of our physical universe. We include the following terminology to help clarify the terms and parameters of later problems in this chapter.
Definitions: Let R4 be the collection of 4-tuples of real numbers (x,y,z,w) with the distance function (metric)
d( (a,b,c,d), (e,f,g,h) ) = 
.
R4 can be considered as a 4-dimensional vector space or, with the metric, as 4-dimensional Euclidean space.
Note that every plane in R3 has exactly one line perpendicular to it at every point. A line is perpendicular to a plane if it intersects the plane and is perpendicular to every line in the plane which passes through the intersection point. In R4 we have similarly:
a. Show that every 2-dimensional subspace (a plane containing the origin O), P, in R4 has an orthogonal complement, P^, which is a 2-dimensional subspace (plane) which intersects P only at O such that every line through O in P is perpendicular to every line through O in P^.
[Hint: This was probably proved in your linear algebra course. The easiest (I think) proof is to change the orthonormal basis in R4 so that, in the new coordinates, P is the span of the first two basis vectors.]
Definitions: Let a 3-sphere, S3, be the collection of points in R4 that are at a fixed distance r from O, the center of R4. The number r is called the radius of the sphere.
We define a great circle on S3 to be the intersection of S3 with a plane in R4 through the center O.
We define a great 2-sphere on S3 to be the intersection of S3 with any 3-dimensional subspace of R4 (that passes through O.)
b. Show that every great 2-sphere in the 3-sphere has reflection-in-itself symmetry.
[Hint: Choose an orthonormal basis for R4 so that the great 2-sphere is in the 3-subspace spanned of the first three basis elements.]
c. Show that every great circle has the symmetries in S3 of rotation through any angle and reflection through any great 2-sphere perpendicular to the great circle. Since these are principle symmetries of a straight line in 3-space it makes sense to call these great circles geodesics in S3.
[Hint: Choose an orthonormal basis for R4 so that the great circle is in the plane spanned by the first two basis elements.]
d. If two great circles in S3 intersect, then they lie in the same great 2-sphere.
Suggestions for Problem 12.2
Thinking in four dimensions may be a foreign concept to you, but believe it or not, it is possible to visualize a 4-dimensional space. Remember, the fourth dimension here is not time, but a fourth spatial dimension. We know that any two intersecting lines that are linearly independent (that do not coincide) determine a 2-dimensional plane. If we then add another line that is not in this plane, the three lines span a 3-space. When lines such as these are used as coordinate axes for a coordinate system, then they are typically taken to be orthogonal — each line is perpendicular to the others. Now to get 4-space, imagine a fourth line that is perpendicular to each of these original three. This creates the fourth dimension that we are considering.
Although we cannot experience all four dimensions at once, we can easily imagine any three at a time, and we can easily draw a picture of any two. This is the secret to looking at four dimensions. These 3- or 2-dimensional subspaces look exactly the same as any other 3-space or plane that you have seen before. This holds true for any subspace of 4-space — since all four of the coordinate lines are orthogonal, any set of three of these will look the same and will determine a space geometrically identical to our familiar 3-space, and any set of two coordinate lines will look like any other and will determine a 2-dimensional plane.
Figure 12.5. Any three coordinate axes determine a 3-space.
For all of these problems, you should not be looking at projections of the 3-sphere into a plane or a 3-space, but rather looking at the part of the 3-sphere that lies in a subspace. For example, since the 3-sphere is defined as the set of points a distance r from the origin O in R4, if you take any 3-dimensional subspace (through O) of R4, then the part of the 3-sphere which lies in this 3-dimensional subspace is the set of points a distance r from its center O in the 3-space. So any 3-dimensional subspace of R4 intersects the 3-sphere in a 2-sphere, which you know all about by now, and you can easily visualize.
For all of the problems here, it is generally best to draw pictures of various planes (2-dimensional subspaces) through the 3-sphere because they are easy to draw on a piece of paper. Remember, only include in your picture those geometric objects that lie in the plane you are drawing. So, a great circle that lies in this plane would be drawn as a circle, while another great circle that passed through this plane would intersect this plane only in two points. See Figure 12.6.
For this particular problem, you are looking at the 3-sphere extrinsically. A good way to proceed is to draw several planes as outlined above, and try to get an idea of how the planes relate to one another when combined into a 4-dimensional space. Once you have an understanding of how the different planes interact in four dimensions, it is fairly easy to show how the great circles of a 3-sphere behave.
Figure 12.6. Intersecting great circles.
Problem 12.3. Hyperbolic 3-Space (Upper Half Space)
As previously mentioned, there is no smooth isometric embedding of a hyperbolic plane in 3-space and, thus, no analytic isometric description. In the same way there is no isometric analytic description of a hyperbolic 3-space in 4-space. Instead we will describe hyperbolic 3-space intrinsically in terms of the upper half space model that is analogous to the upper half plane model.
Definition: Let R3+ = {(x,y,z) in R3 | z > 0} and call it the upper half space.
In Chapter 5 we started with the annular hyperbolic plane and then defined a coordinate map z: R2+ ® H2. Now we do not have an isometric model H3 but, instead, we have to start with the upper half space and use z to define H3. Recall that z: R2+ ® H3 has distortion r/b at the point (a,b) in R2+, where r is the radius of the annuli. As we saw in Chapter 5 we can study the geometry of H2 by considering it to be the upper half plane with angles as they are in R2+ and distances distorted in R2+ by r/b at the point (a,b). So now we use this idea to define H3.
Definition: Define the upper half space model of hyperbolic space H3 to be the upper half space R3+ with angles as they are in R3+ and with distances distorted by r/c at the point (a,b,c). We call r the radius of H3.
We define a great semicircle in H3 to be the intersection of H3 with any circle that is in a plane perpendicular the boundary of R3+ and whose center is in the boundary of R3+ or the intersection of R3+ with any line perpendicular to the boundary of R3+. The boundary of R3+ are those points in R4 with w = 0.
We define a great hemisphere in R3+ to be the intersection of R3+ with a sphere whose center is on the boundary of R3+ in R4 or the intersection of R3+ with any plane which is perpendicular to the boundary of R3+ in R4.
a. Show that inversion through a great hemisphere in R3+ has distortion 1 in H3 and, thus, is an isometry in H3 and can be called a (hyperbolic) reflection through the great hemisphere.
[Hint: Look back at Problem 5.3. Note that any inversion in a sphere when restricted to a plane containing the center of the sphere is an inversion of the plane in the circle formed by the intersection of the plane and the sphere.]
b. Show that, given a great semicircle [or great hemisphere], there is a hyperbolic reflection (inversion through a great hemisphere) that takes the great semicircle [hemisphere] to a vertical half line [half plane] in the upper half space.
[Hint: Look back at Problem 4.2d. Note that any inversion in a sphere when restricted to a plane containing the center of the sphere is an inversion of the plane in the circle formed by the intersection of the plane and the sphere.]
c. Show that every great semicircle has the symmetries in H3 of rotation through any angle and reflection through any great hemisphere perpendicular to the great semicircle. Since these are principle symmetries of a straight line in 3-space it makes sense to call these great semicircles geodesics in H3.
[Hint: Look back at Problem 6.2b. Use Part c of this problem to show that any great hemisphere perpendicular to the great semicircle can be assumed to be the intersection with R3+ of a plane perpendicular to the boundary of R3+. Then (using Problem 6.2b) look at the composition of hyperbolic reflections (inversions) through great hemispheres which contain the great semicircle and whose centers are in the plane perpendicular to the great semicircle.]
c. If two great semicircles in H3 intersect, then they lie in the same great hemisphere.
[Hint: Use Part c to show that in the upper half space model there is a hyperbolic reflection that takes any great semicircle to a vertical half line.]
*Problem 12.4. Disjoint Equidistant Great Circles
a. Show that there are two great circles in S3 such that every point on one is a distance of one-fourth of a great circle away from every point on the other and vice versa.
Is there anything analogous to this in H3 or in ordinary 3-space? Why?
Suggestions
This problem is especially interesting because there is no equivalent theorem on the 2-sphere; we know that on the 2-sphere, all great circles intersect, so they can't be everywhere equidistant. The closest analogy on the 2-sphere is that a pole is everywhere equidistant from the equator. When we go up to the next dimension, this pole "expands" to a great circle such that every point on this great circle is everywhere equidistant from the equator. While this may seem mind-boggling, there are ways of seeing what is happening.
The main difference created by adding the fourth dimension lies in the orthogonal complement to a plane. In 3-space, the orthogonal complement of a plane is a line that passes through a given point. This means that for any given point on the plane, (the origin is always a convenient point), there is exactly one line that is perpendicular to the plane at that point. Now, what happens when you add the fourth dimension? In 4-space, the orthogonal complement to a plane is a plane. This means that every line in one plane is perpendicular to every line in the other plane. To understand how this is possible, think about how it works in 3-space and refer back to Figure 12.5. Now look at the x-y-plane and the z-w-plane. What do you notice? Why is every line through the center in one of these planes perpendicular to every line through the center in the other?
Figure 12.8. Orthogonal planes.
Knowing this, look at the two GREAT CIRCLES in terms of the planes in which they lie, and look at the relationships between these two planes, that is, where and how they intersect. Also, try to understand how GREAT CIRCLES can be everywhere equidistant.
If we rotate along a great circle on a 2-sphere, all points of the sphere will move except for the two opposite poles of the great circle. If you rotate along a great circle on a 3-sphere, then the whole 3-sphere will move except for those points that are a quarter great circle away from the rotating great circle. Therefore, if you rotate along one of the two great circles which you found above, the other great circle will be left fixed. But now rotate the 3-sphere simultaneously along both great circles at the same speed. Now every point is moved and is moved along a great circle!
b. Write an equation for this rotation and check that each point of the 3-sphere is moved at the same speed along some great circle. Show that all of the great circles obtained by this rotation are equidistant from each other (in the sense that the perpendicular distance from every point on one great circle to another of the great circles is a constant).
These great circles are traditionally called Clifford parallels. See [DG: Penrose] for a readable discussion of Clifford parallels in the article entitled, The Geometry of the Universe. These great circles are named after William Clifford (1845-1879, English).
*Problem 12.5. Hyperbolic and Spherical Symmetries
We are now ready to see that the symmetries of great circles and great 2-spheres in a 3-sphere [and great semicircles and great hemispheres in a hyperbolic 3-space] are the same as the symmetries of straight lines and (flat) planes in 3-space. If g is a great circle in the 3-sphere, then let g^ denote the great circle (from Problem 12.4) every point of which is p/2 from every point of g.
a. Check the entries in the following table (Figure 12.7.) which gives a summary of various symmetries of line, great circles, and great semicircles and of (flat) planes, great 2-spheres, and great hemispheres.
| symmetries of... | reflection through... | reflection through... | half-turn about... | rotation about... | translation along... |
| line l Ì R2 | l | line ^ l | point in l | NA | l |
| great circle g Ì S2 | g | gr. circle ^ g |
pt/pair in g | poles of g | g |
| gr. s-circle g Ì H2 |
g | gr. s-circle ^ g |
pt/pair in g | NA | g |
| line l Ì R3 | plane É l | plane ^ l | line ^ l intersecting l | l | l |
| great circle g Ì S3 | gr. sphere É g | gr. sphere ^ g |
gr. circle ^ g intersecting g |
g | g |
| gr. s-circle g Ì H3 |
gr. h-sphere É g | gr. h-sphere ^ g |
gr. s-cir. ^ g intersecting g |
g | g |
| plane P Ì R3 |
P | plane ^ P | line in P | line ^ P | line Ì P |
| great sphere G Ì S3 | G | gr. sphere ^ G | gr. circle in G | gr. circle ^ G | great circle Ì G |
| gr. h-sphere G Ì S3 | G | gr. h-sphere ^ G | gr. s-circle in G | gr. s-circle ^ G | gr. s-circle Ì G |
Figure 12.7. Symmetries in Euclidean, Spherical, and Hyperbolic Spaces
Definition: A surface in a 3-sphere or in a hyperbolic 3-space is called totally geodesic if, for any every pair of points on the surface, there is a geodesic (with respect to S3 or H3) which joins the two points and which also lies entirely in the surface.
b. Show that a great 2-sphere in S3 (with radius r) is a totally geodesic surface and is itself a sphere of the same radius r.
c. Show that a great hemisphere is a totally geodesic surface in H3 (with radius r) is isometric to a hyperbolic plane with the same radius r.
[Hint: In the upper half space model there is a hyperbolic reflection that takes every great hemisphere to a plane perpendicular to the boundary. (See Problem 12.3.)]
Problem 12.6. Triangles in 3-Dimensions
Show that if A, B, C are three points in S3 [or in H3] that do not all lie on the same geodesic, then there is a unique great 2-sphere [hemisphere], G2, containing A, B, C.
Thus, we can define DABC as the (small) triangle in G2 with vertices A, B, C. With this definition, triangles in S3 [or in H3] have all the properties which we have been studying of small triangles on a sphere [or triangles in a hyperbolic plane].
Suggestions
Think back to the suggestions in Problems 12.2 and 12.3 — they will help you here, as well. Take two of the points, A and B, and show that they lie on a unique plane through the center, O, of the 3-sphere [or a unique plane perpendicular to the boundary of R3+]. Then show that there is a unique (shortest) geodesic in this plane.
Figure 12.7. Great circle through A and B.
Think of A, B, and C as defining three intersecting great circles [or semicircles]. On a 3-sphere, look at the planes in which these great circles lie, and where the two planes lie in relation to one another. In hyperbolic 3-space, use a hyperbolic reflection to send one of the great semicircles to a vertical line.
Be sure to show that the great 2-sphere (hemisphere) containing A,B,C is unique.