Essay #1: A Hierarchy of Structures - Circle/Sphere/Hypersphere

Table I: "Volume" and "Surface" of the Circle/Sphere/Hypersphere
N	Structure	Formula	Vn ("Volume")	Sn-1 ("Surface")
2	Circle	x2 + y2 = r2	V2 = pr2	S1 = 2pr
3	Sphere	x2 + y2 + z2 = r2	V3 = 4pr3/3	S2 = 4pr2
4	Hypersphere	x2 + y2 + z2 + w2 = r2	V4 = p2r4/2	S3 = 2p2r3

A. "Volume" and "Surface" of the Circle/Sphere/Hypersphere

Although many topologists are familiar with the space described in Riemann's four-dimensional (4D) hypersphere, it's not known very well outside this select group. So the Table I introduces this space by listing Riemann's 4D hypersphere with its cousins - the circle and sphere.

[In the following table, I'm using Internet Explorer coding for the Greek letters. If you view this with the Netscape browser, you'll see "p" where you're supposed to see "pi" (3.14159). Check Appendix 11 if you want details of the differences between the two browsers with respect to the Greek letters.]

"N" is the number of terms on the left side of the equation and is also the true number of spatial dimensions in the structure. The next two columns are self explanatory. The Vn column is the expression for the sum of the points less than or equal to the radius r; it's the topological "volume" of the structure in all n dimensions, even though "volume" in our Euclidean world is accurate only when n = 3.

S^{n - 1}, the "surface", is intriguing because the expression here is always the derivative with respect to r of the formula in the same row of the Vn column. Again, "surface" is accurate in Euclidean terms only when n - 1 = 2 (or n = 3), although S refers topologically to this type of space no matter what value n assumes. The formula here refers to the sum of all points in n - 1 dimensions equal to the radius r.

The unusual topological characteristics of an S-type structure have been referred to for years in a variety of ways:

• "Finite but unbounded", because it has finite size but no formal boundary;

• "Multiply-connected", since there is more than one way to reach another position; and

• "Curves back on itself", because an apparently straight path from a position in S-type space is a circular route - called a "geodesic" - and returns to the original position from the opposite direction.

Thus an S-type structure has a positive global curvature in such a manner that, if r = constant, a route around the circle/sphere/hypersphere is always 2pr. On the other hand, if r >>> 0 - that is, r is very large - we have no way of knowing from local observations whether the structure we occupy is Vn, presumably orthogonal, or Sn, globally curved.

[It would not be an exaggeration to say that the object of these essays is to specify the assumption about a very large elastically expanding structure which allows us to differentiate between Vn and Sn which, when n = 3, is the topological difference between Friedmann space and Riemannian space.]

Knowing the lower-dimensional formula is the derivative in calculus helps understand why the lower-dimensional structure is frequently referred to as the "skin", "membrane", or "surface" of the higher-dimensional structure. For example, we could say that the area defined by 4pr2 is the "skin" or "surface" (S2) of the 3D sphere whose volume is given by 4pr3/3 (V3).

Another example: the physicists Lisa Randall and Raman Sundrum in this article refer to a large (uncompactified) fourth spatial dimension and mention a "3-brane" in which we presumably observe most, but not all, physical activity. Their "3-brane" is what a topologist would typically refer to as the S3 "surface" of a V4 (Riemannian) hypersphere.

[But if you will recall the distinction presented in the Introduction, you will see that Randall and Sundrum agree with the general notion of a "membrane" or "brane", but they probably don't agree that the fourth spatial dimension exists in the way it is used in this Riemannian hypersphere; they assume it's much smaller.]

So even though 2,300 years of Euclidean geometry probably make a curved "Riemannian surface" in S3 difficult to understand at first, the familiar Euclidean relationships that S1 has with V2 and S2 has with V3 are virtually duplicated in the Riemannian relationship that S3 has with V4.

[Although I'm not interested in anything beyond the four-space Riemannian hypersphere in these essays, properties of even higher-dimensional hyperspheres are presented at this site by Eric Weisstein. This particular file gives you more trigonometric information about the hypersphere, including the derivation of the "surface" formula at the end.

Be sure to take the link to "Ball" at the bottom. Once you get there, go to a table about midway down the scroll bar. It's understood that you mentally insert the radius rn in the V (volume) column and r^n-1 in the S (surface) column. Once you do that you'll realize that this table shows the derivative relationship between V and S and extends it to the higher-dimensional structures.]

B. Embeddedness

The key to unraveling the presumed "mystery" of Riemann's 4D hypersphere is to understand the critical significance of one very simple property: embeddedness, which means that under certain circumstances (when one or more of the terms equals zero) the algebraic equation with fewer terms as well as the geometric structure it represents is a legitimate solution of the equation with the greater number of terms.

[Of course, we know intuitively that a circle in (x,y) is "embedded" in a spherical aurface in (x,y,z). And the table above clearly indicates that a spherical surface is "embedded" in Riemann's 4D hypersphere in like manner. But whether this concept can be used for proper analysis of space in the universe depends on the nature of that space, and is the topic of Appendix 1. In the type of S3/V4 space I will derive in Essay #3, "embeddedness" is OK, but it is not OK - I repeat, not OK - in the usual (open, flat, closed) Friedmann models.]

For example, this formula describes a hypersphere whose radius changes with time (t):

Equation 1: x²(t) + y²(t) + z²(t) + w²(t) = r²(t).

There are any number of assumptions we might make about how the hypersphere changes with time. We might assume, for example, that its necessarily curved space is "infinitely elastic", which means that this space can expand forever and/or that it has expanded from virtually nothing. Perhaps we might suppose the hypersphere collapses to zero size when we look around it exactly one time. Or we could assume that the rate of expansion of this hypersphere was either higher or lower in earlier times than it is today.

But whatever assumptions we make, we're certain of one thing: the geometric behavior which is applicable to the higher-dimensional hypersphere is equally applicable to the lower-dimensional circle and spherical surface. We know this is true because the formulas defining these structures are legitimate solutions of the higher-dimensional hypersphere in certain circumstances, when one or more of the non-temporal parameters just happens to be zero.

As a trivial example, let's suppose r = 4,000 miles in the above equation and that r = constant (so we can ignore t). This would define a hypersphere with the same radius as the earth; I call it a "terrestrial hypersphere". If we traveled along what seems to be a "straight" line in any direction for about 25,000 miles through this curved space, we would wind up right back where we started after going all the way "around" the hypersphere.

How do we know this?

Because the circle and the spherical surface with the same radius of 4,000 miles are "embedded" in this curved space and their circumference is also about 25,000 miles.

And "embeddedness" works in both directions: we might make certain assumptions about the way a circle/spherical surface are behaving, and then assume this behavior is applicable to the higher-dimensional hypersphere in which they are "embedded".

For example, if a 4D hypersphere is expanding, the rate of expansion we observe will be deceptively high.

Why?

Because the circle and the spherical surface are embedded in this metric (when two or one of the terms equal zero) and we know that the circumference of a circle and a spherical surface increases faster than its radius for the simple reason that the circumference C = 2pr.

[OK, Appendix 11 explains what the problem is with Greek letters on the two browsers. Here is the above formula with Netscape coding: C = 2 &pi r, which should look OK in Netscape, but pretty silly on the IE browser.]

Precisely the same principle is applicable to the curved space of the hypersphere because we can only look out along its circumference. So if the hypersphere correctly describes the universe we see, in one fell swoop we can sweep away one of the most formidable problems in the big bang model because a substantial amount of matter has been inferred from a rate of expansion (aka Hubble parameter) that may be too high because we're measuring it along the surface or circumference of the hypersphere, rather than its radius.

C. Two Myths about the Hypersphere

Before we get much further into the realm of the 4D hypersphere, let's dispel two myths about this non-Euclidean foreigner right now:

Myth #1: It is IMPOSSIBLE for us to comprehend what happens in any structure with more than the three spatial dimensions we're accustomed to in our Euclidean world.

Wrong!

It's not all that difficult for us to think in at least one more spatial dimension (that is, S3 on the "surface" of V4) provided (1) we understand that the three-dimensional spherical surface is embedded in this space, and (2) we realize the "extra" dimension lies at right angles to the conventional three-dimensional spherical surface.

[Note: Why the fourth spatial dimension lies at right angles to a spherical surface is explained in Appendix 2.]

So if we assume that light is constrained to the spherical surface and analyze what happens to our perception of objects at various positions on this surface, we know that not only does this behavior actually occur on the spherical surface (because of "embeddedness"), it also occurs at right angles to this spherical surface, as shown in Appendix 2.

Myth #2: The 4D hypersphere can't be visualized easily by those of us confined to three-dimensional thinking.

Again, this myth is false!

To show why it's false, let's examine first the surface of a sphere. Although we understand that it's curved, we also know that its area can be defined in two-dimensional terms: 4pr2, as shown in the table above. And we could approximate this curved surface reasonably well by representing each hemisphere as a very ordinary circle. The center on one circle would be the "north pole", the center of the other circle would be the "south pole", and the edge of each circle would be the "equator" on the earth's surface.

But if we did it this way, we would encounter the same problem we always encounter when we try to represent a curved surface like the earth using one less spatial dimension than it really has: we have to show a "break" or "discontinuity" that isn't there. The best we can do here is to show the two circles contiguous at only one position (like a pair of coins touching each other), even though we recognize the two circles are contiguous at all positions.

Thus the intrinsically curved surface of the earth in S2 can be approximated by a flat two-dimensional map in V2, and this is something we do routinely in, say, the more conventional Mercator projections.

A very similar exercise allows us to get a good idea of the geometry of the hypersphere. Obviously its space is curved, even though we can express the volume on its "surface" in three-dimensional terms: 2p2r3 ( again, see the above table). And, like what we did on the spherical surface, we can divide this curved space in half and represent each "hemi-hypersphere" as a very ordinary three-dimensional sphere.

Again, we've got exactly the same type of problem we had before: these two spheres are actually contiguous at all surface positions. But since we're representing the hypersphere in one less dimension than it really has, the best we can do is to show the two spheres contiguous at only one position (like a pair of baseballs touching each other).

The two-sphere configuration illustrates the hypersphere just as well as the two circles approximate the spherical surface. Both representations give us a very good idea of the geometry of the respective higher-dimensional structures using one less spatial dimension than they really have.

[Note: It's no accident why both lower-dimensional visualizations of the respective higher-dimensional structures are so similar. Appendix 3 explains how you create a 4D hypersphere from a 3D spherical surface and it gives a simple proof showing why this representation of the hypersphere using V3 geometry is tolerably accurate.

If it helps to actually see the two spheres, they are shown on page 4 (of the 32 page pdf version) of the article on Topological Lensing in Spherical Space by Evelise Gaussman, Roland Lehoucq, Jean-Pierre Luminet, Jean-Philippe Uzan, and Jeffrey Weeks.]

The most important thing to understand about the w term - at least the way I'm using it here (in Riemann's hypersphere) - is that it projects the global curvature of a spherical surface into the comparably curved space of the hypersphere. This is why, for example, that if a two-circle configuration is a reasonably good approximation for the spherical surface, the two-sphere configuration is an equally good approximation for the four-space hypersphere.

Provided we assume light is confined to a spherical surface, we always look toward its opposite side as we look "around" the surface. And, because of "embeddedness", the same thing happens on the hypersphere: if we're at the center of one sphere of the two-sphere configuration, we'll always look toward the center of the other sphere. The center of this second sphere is the "opposite side" of the hypersphere; it's as far away as we can see before our line of sight starts coming back.

Just as the route around a sphere with a constant radius r is 2pr and halfway around (to the opposite pole) is pr, then a route all the way around the hypersphere is likewise 2pr and the length of the path to the opposite side is pr.

[Perhaps a more relevant question is how far away is the antipode in a hypersphere that collapses to zero size when we look around it precisely one time. As you will see shortly in a file linked to the end of Essay #3, there's a simple way to get a "ballpark" estimate.]

But what happens to our observations if an antipode in S3 is close enough to affect what we see?

This is the primary topic of Essay #2.