The following document is mainly an effort to understand just what a hypersphere is like. The main technique used to approach this concept is to thoroughly observe aspects and concepts found within the second dimension. Based on the observations, one can use "analogy" to "fill in the blanks" for our own dimension, where which there would otherwise be no answers. The subject discussed in this document, however, is far, as of now, from complete.
Most people assume a coin to be 2-dimensional. But it is not.
If it was, it would have zero-thickness, and could not exist in 3-space. It would seem to me, that an object lacking 3-dimensional thickness would have to "paste" itself to a 3-D object, such as the top of a table. A coin, then, could be said to be a circle possessing a "tiny amount" of 3-dimensional thickness.
There is a certain hypothetical structure that is similar to the coin: picture an elastic, bendable, hollow sphere, with an equator and poles properly marked. Several strings are attached to the equator on all sides of the sphere. The strings are pulled outward from the sphere until both poles meet (that is, until the sphere is flattened). The equator is now the outer edge of a flat disc, which has a circumference 1.571 times greater than what the equator was on the original sphere. The number 1.571 happens to be half of pi - and is related to the sphere's "loss of curvature".
If the equator of the original sphere were "lined up" with a Flatlander plane, Flatlanders there could inspect the equator from all sides, but could only sense the equator as the outer edge of a circle. However, if the Flatlanders attached strings to all sides of this "outer edge", and then pulled away from the equator on all sides, to their astonishment, this "circle" briefly expands in all directions. When measured, the circle now has a circumference 1.571 times greater than that of the circle in its original state! This circle now contains no 3-dimensional curvature (that is, it can be contained within a tabletop-like surface). There is a grapefruit example below to clarify this concept.
A quarter, a dime, and a small (dime-sized) marble are simple objects that can be used as models: a quarter, as the "flattened sphere"; a dime, as the size of the equator before the sphere was flattened; and a small marble, as the sphere before it was altered. With the help of a small ant, these surfaces can be studied: one can have the ant walk all over the surface of a quarter. Even though the ant will often walk up to the coin's edge, this poses no problem: the ant can quickly shift over to the "other side", reversing direction in the process. Next, the ant can be observed, traversing the surface of the marble.
So how can this be applied? The sphere, in its initial state, and the sphere, in its "flattened" state, possess the same amount of surface area! That is, both the quarter, and the marble, give the ant the same total amount of area to walk on.
Everyone knows that a coin possesses 2 "sides". It can be stated, more fully: the coin is a 3-dimensional object having 2 sides, each side being a 2-dimensional circular area. It has been stated that the "flattened sphere" is a valid way to envision a sphere. And thus, it can be used to describe the concept of a sphere, that a "Flatlander" could understand. Assuming that the Flatland universe is a sphere, they could be told:
"Your universe consists of 2 "sides". Each side is a giant, flat circle. When one of your spaceships encounters the boundary of the side he's on, it "switches over" to the "other side", and reverses direction. The 2 "sides" are 2 totally separate circles, linked only at their outer edges."
This description is not an entirely accurate description of a sphere. For example, a sphere has no sharp, sudden reversals - reversals like what the ant encountered at the edge of the quarter. The marble, however, is "smooth" - perfect roundness, free of edges - which the above description was not. What the description is, though, is a description that a Flatlander can understand. That is, a concept "toned down" so that it can be understood.
Recall, if you can, the earlier description of Flatlanders "flattening" a sphere: through the power of analogy, one can perform the same task with a hypersphere. If the equator of a hypersphere (an equator which, by analogy, I assume we perceive to be the outer edge of a sphere) were "lined up" with a 3-D plane such as our own, one could inspect this "equator" from all sides, but could only sense the equator as being the outer edge of a sphere. As before: if we were to attach strings to all conceivable points on the surface of the sphere, and pulled the strings away from it in all directions, the hypersphere would "unfold" into a sphere with an equator 1.571 times greater than the "great circle" (equator) of the original hypersphere. This "unfolded hypersphere" results in a "3-dimensionally flat" sphere, with all 4-D curvature removed from it (that is, the space is not distorted from the way that we experience space normally).
This "unfolded hypersphere" can be expected to have properties similar to those of the "flattened sphere". And based on what has been discussed so far, the "properties" of this new structure shouldn't be very hard to study.
Indeed, this structure could best be called, then, the hypercoin. That is: like a coin, which is a 2-dimensional circle given a tiny bit of 3-D thickness, a hypercoin is a 3-dimensional sphere given a tiny bit of 4-D thickness. As stated earlier: though the "flattened sphere" has the same orientability as an unaltered sphere, its hypothetical physical structure is not fully accurate to that of an actual sphere.
Recall the description of a coin stated earlier: "a 3-dimensional object having 2 sides, each side being a 2-dimensional circular area". From this, and from all that's been said, we can attempt to describe a hypercoin as: "a 4-dimensional object having 2 sides, each side being a 3-dimensional spherical area".
Recalling the little ant on the coin, and the proposed "hypercoin", we can attempt to present a hypothetical description of the shape of our universe: our universe consists of 2 "sides". Each side is a giant sphere. If a spaceship encounters the boundary of the side it's on, it "switches over" to the "other side", and in the process reverses direction. The 2 "sides" are 2 totally separate spheres, linked only at their outer edges.
We know the "edge" of a coin to be where the outer edges of each circular side are "connected". The "edge" of a hypercoin is more than you might expect: where the outer edges of both spherical sides are "connected". Please note, that a real hypersphere would not possess attributes such as "sides", "edges", and "switching over". It would be totally smooth, possessing nothing less than perfect roundness.
A simple way to think of this is to imagine a grapefruit that has been cut in half. The inside of one of the grapefruit halves has been scooped out until it is completely hollow. Set this half on a table, rounded side up. Put the palm of your hand directly on top of it, and push down firmly. As you can see, the skin of the grapefruit has spread out farther than it was when it possessed curvature.
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