The easiest way to explain and visualize the hypersphere is to show how it is related to its lower-dimensional Euclidean cousins: the circle and the spherical surface. The following table is an appropriate entrée to the discussion:
Loci "inside" Loci "on surface" Equation Description Formula (< or = r) (only = r) 1 Circle x2 + y2 = r2 pi r2 2 pi r 2 Sphere x2 + y2 + z2 = r2 4 pi r3/3 4 pi r2 3 Hypersphere x2 + y2 + z2 + w2 = r2 pi2 r4/2 2 pi2 r3
1. The radius is "r".
2. "Pi" is the equivalent of the Greek letter substituted for 3.14159.
3. "Loci inside" versus "Loci on surface": if you differentiate the "inside" expression with respect to the radius,
4. The formula for the hypersphere in topology is equivalent to the formula defining the "Robertson-Walker (RW)
The key to unraveling the presumed "mystery" of the hypersphere is to understand the critical significance of one very simple property: embeddedness. Each lower-dimensional structure is always algebraically and geometrically "embedded" in the next higher-dimensional structure.
As an example, we know that under a certain circumstance Equation 1 is a legitimate solution of Equation 2 (when z = 0). And our knowledge of Euclidean geometry confirms what the algebra is telling us: that the circle in (x,y) dimensions is geometrically "embedded" in the higher-dimensional spherical surface defined in (x,y,z) dimensions.
Although you may not be all that familiar with hyperspheres, there should be no doubt that the following four spherical surfaces are all simultaneously "embedded" both algebraically and geometrically in the necessarily curved space of the hypersphere defined by Equation 3:
(Equation 3: x2 + y2 + z2 + w2 = r2)
Equation 3a (when w = 0): x2 + y2 + z2 = r2,
Equation 3b (when z = 0): x2 + y2 + w2 = r2,
Equation 3c (when y = 0): x2 + z2 + w2 = r2,
Equation 3d (when x = 0): y2 + z2 + w2 = r2.
Table 2 tells us one thing very clearly: among all possible solutions of the hypersphere is the very ordinary spherical surface existing in (x,y,z) dimensions of Equation 3a or Equation 2.
And we can repeat the process: not only can we assume one of the terms in Equation 3 is zero, there's no particular reason why we can't assume any two terms are zero. The two terms remaining, of course, define circles all simultaneously "embedded" on both the spherical surface and the hypersphere. There are six such circles defined along the (x,y), (x,z), (y,z), (x,w), (y,w), and (z,w) axes.
The next formula describes a hypersphere whose radius changes with time (t):
Equation 4: 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 higher in earlier times than it is today.
But whatever assumptions we make, we're certain of one thing: the geometric behavior that 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 Equation 3. 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 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".
But if a spherical surface has three spatial dimensions and the hypersphere is "higher-dimensional" relative to the spherical surface, how many spatial dimensions does it have?
And before you get too xenophobic about this non-Euclidean "foreigner", let's dispel two myths about it right now:
It's not all that difficult for us to think in at least one more spatial dimension 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.
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 hypersphere (because of "embeddedness"), it also occurs at right angles to this spherical surface.
[Note: The behavior of that fourth spatial dimension is a rather easy derivation of the way the embedded circles relate to each other. How you do it is to look at the way the three circles in (x,y), (y,z), and (x,z) are behaving at their respective points of intersection on the (x,y,z) spherical surface in Equation 3a. But Equations 3b, 3c, and 3d have exactly the same "form" as Equation 3a and so the embedded circles on these other spherical surfaces behave exactly the same way. Then it's a very small step to realize that the circles with the w term intersect the circles without the w term at right angles. If all else fails, E-mail a query to me and I'll send back a little file that derives this relationship.]
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: 4 pi r2, as shown in Equation 2. 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 correspond to 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.
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: 2 pi2 r3 (Equation 3). 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).
This two-sphere configuation 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.
The most important thing to understand about the w term 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 configuation 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 (in one more dimension, of course): 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 look before our line of sight starts coming back.
The formal name for this unique position on the hypersphere is the "antipode": finding it is the second best piece of evidence that we actually inhabit a hypersphere. (If you're interested, the best evidence is to look or travel back to our original position after going all the way around the hypersphere and/or to see the same object near the antipode in two opposite directions.)
But what happens to our observations if an antipode is close enough to affect what we see? And how would we go about determining its most likely position?
This is the primary topic of Essay #2.
Go to essay 2: Perceptions in Hyperspace: How the Antipode Affects our Observations
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