The Hypercoin

What is the hypercoin? The hypercoin is a 3-dimensional attempt to envision a 4-dimensional object. What would that object be? None other than the hypersphere. This link, in fact, is a link from the Thinking 4-D topic Visualizing the Hypersphere. A hypersphere, if you didn't already know, is a sphere with one extra dimension added. In other words, a hypersphere is to a sphere, what a sphere is to a circle. Let us discuss aspects of the hypersphere that make it of use to our discussion of the hypercoin.

Our planet - the earth - is a 3-dimensionally curved sphere. However, observers on its 2-dimensional surface perceive it as being flat - flat as if it were a flat 2-dimensional circle, that extended outward from the observer in all directions. The observer has no hint whatsoever that what he is standing on is curved - a 3-dimensionally curved sphere. He interprets, in essence, a 3-dimensional object as being 2-dimensional. However, what if we were to send a satellite up into space to look down upon the earth? What would we see? We would be able to detect the curvature of the 2-dimensional surface of the 3-dimensional object that we call the earth. By situating ourselves outside the surface of the earth, we are able indeed to see that it is round, not flat.

Assume that our universe is a 4-dimensionally curved hypersphere, possessing a 3-dimensional surface. Just as a person stands "on" the 2-dimensional surface of the earth, our universe equally so exists "on" the 3-dimensional surface of a hypersphere. Consider, if you will, that just as an observer on the surface of the earth experiences the surface of the earth as being completely 2-dimensional, our experience of being on the surface of a hypersphere is of a universe that is completely 3-dimensional. What this apparently means is that we cannot detect the 4-dimensional curvature of the universe's 3-dimensional surface because we are on that surface. By what means would we be able to document the curvature of this 3-dimensional surface? The answer is the same as with the earth: in order to be able to observe the 4-dimensional curvature of the universe, the point of view to which we would have to situate ourselves would have to be a point of view from outside the universe!!

Before attempting to describe the hypercoin, we will first familiarize ourselves in new ways with an object we come across in our everyday lives: the coin. What is a 2-dimensional object? It is an object, quite clearly, limited entirely to the dimensions of length and width. Given this, is a coin 2-dimensional? This is a tricky question. A coin is actually 3-dimensional: it is a 2-dimensional object possessing a tiny amount of 3-dimensional thickness. This is why a coin on the surface of a table, when picked up, remains solid.

Picture in your mind an elastic, bendable, hollow sphere, with 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 elastic sphere before the outward exertion of the strings was applied. In "collapsing", the elastic sphere loses its 3-dimensional curvature. The value 1.571 happens to be half of pi - and is related to the elastic sphere's "loss of curvature".

Picture in your mind a flat 2-dimensional plane. Within this flat 2-dimensional plane lies intelligent, thinking 2-dimensional beings. As part of the experiment we are to perform, assume now that we revert the above-described "collapsed" sphere to its original state. Next in the procedure, we line up the equator of the elastic sphere with the flat 2-dimensional plane. The equator of the sphere, take note, is the only part of the sphere that the 2-dimensional beings can detect: the "hemispheres" of the sphere lie above and below the flat 2-dimensional plane. The 2-dimensional beings, afterward, inspect the equator of the sphere from all sides. They experience the equator, in turn, as being the outer edge of a circle. Let us assume that we tell the 2-dimensional beings to attach strings, as before, to all sides of the outer edge of the circle they detect. We then tell them to pull the strings away from all sides of the outer edge. To the astonishment of the 2-dimensional beings, the circle briefly expands outward in all directions for a moment. When the 2-dimensional beings measure the circumference of the circle they perceive, they find that the circle now possesses a circumference 1.571 times greater than the circle as it was before the strings were pulled. In "losing its curvature", the elastic sphere gains 2-dimensional extension.

To express this in a way that we can relate to in terms of everyday experience, imagine that you have in your hands a grapefruit. Cut the grapefruit in half. Next, scoop out the contents of the inside of one of the grapefruit halves until it is completely hollow. Set this grapefruit half on a table, rounded side up. Put the palm of your hand directly on top of the hollow grapefruit half, and push down firmly. As you can see, the outer edges of the grapfruit half now make contact with the top of table over a far greater area than before the grapefruit was pressed down.

To give further meaning to the situation, we will perform another experiment. Imagine that your have in your possession a small dime-sized marble, and a quarter. The small dime-sized marble represents the elastic sphere before it was flattened. The quarter, in turn, represents the elastic sphere after being flattened. Let us take this a step further. We now place an ant on the quarter, and have it walk across the quarter's surface in any way it wishes. What happens when the ant reaches the outer edge of the quarter? It makes a 180-degree turn to the other side! As the result of further observation, one will find that the quarter gives the ant the same amount of surface area to cover as the marble does.

Let us return, then, to our discussion of the 3-dimensional object we call the coin: a coin is a 3-dimensional object possessing 2 "sides", each side being a 2-dimensional circular area. The existence of the 2 "sides", quite simply, is made possible by the coin's tiny amount of 3-dimensional thickness. Each of the "sides", in turn, are connected at the outer edge of the coin. What does this knowledge make possible? It gives us an opportunity to convey to one of the 2-dimensional beings in the flat 2-dimensional plane a version of the sphere that he can comprehend:

"A sphere consists of 2 "sides". Each side is a 2-dimensional circular area that overlaps the other. The circular areas, in turn, are joined at their outer edges. When on one of the sides, and you reach the common outer edge, you "switch over" to the "other" side, resulting in a reversal of direction."

This description, as you will agree, 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, was "smooth" - possessing perfect roundness, and being free of any kind of edge. Our description of a sphere to the 2-dimensional being, frankly, falls greatly short of what a sphere actually is. What our description of the sphere to the 2-dimensional being was, though, was a description that the 2-dimensional being could understand. It was a description "toned down" so that it could be grasped.

Possessing the knowledge that we now have, we can more effectively approach the concept that is the title of this writing: the hypercoin. Let us assume that extended into the fourth dimension lies a 4-dimensionally curved, 4-dimensionally flexible, elastic, hollow hypersphere. Analogy tells us quite clearly that if the equator of a sphere is a hollow circle, then the equator of a hypersphere is a hollow sphere. As with the 2-dimensional beings and their flat 2-dimensional plane, this hyperspherical equator is "lined up" with our own 3-dimensional plane. We experience this 'hyperspherical equator' as being the outer surface of a sphere. In a manner very similar to the attaching of strings earlier, we attach strings to several points on the outer surface of the equator of this flexible, elastic, hollow hypersphere. We then pull on the strings outward in all directions. The sphere, for a moment, briefly expands. The hypersphere - of which this sphere is an equator - is now 4-dimensionally flat. In "losing its curvature", the hypersphere gains 3-dimensional extension. To our astonishment, the circumference of the sphere is now 1.571 times greater than it was before the strings were pulled. What do we have before us? The hypercoin!

The hypercoin is a 4-dimensional object possessing 2 "sides", each side being a 3-dimensional spherical volume. The existence of the 2 "sides", quite simply, is made possible by the coin's tiny amount of 4-dimensional thickness. Each of the "sides", in turn, are connected at the outer surface of the hypercoin. This gives us an opportunity to present ourselves with a version of the hypersphere that we can comprehend:

"A hypersphere consists of 2 "sides". Each side is a 3-dimensional spherical volume that overlaps the other. The spherical volumes, in turn, are joined at their outer surfaces. When on one of the sides, and you reach the common outer surface, you "switch over" to the "other" side, resulting in a reversal of direction."

Let us envision the hypersphere using the hypercoin. We are to imagine that we are in a vast 4-dimensional universe in the shape of a hypercoin. We are in a spaceship. Which "side" of the hypercoin are we on? Choose whichever side you wish. The side that you have chosen is the side of the hypercoin whose spherical volume is a physical reality to us. We can travel anywhere we want within this vast spherical volume. What happens when we happen to reach the outer surface of the side of the hypercoin that we are on? We "switch over" to the other side of the hypercoin, and in doing so reverse direction. Having reverted to the "other" side, this side of the hypercoin new to us is now the side of the hypercoin whose spherical volume is a physical reality to us. It is the tiny amount of 4-dimensional thickness, take note, that makes each "side" of the hypercoin distinct. The precise manner in which each side of the hypercoin is separate from the other, however, is a manner of separation that no 3-dimensional means of reasoning can explain. This, quite simply, is because the hypercoin is a 4-dimensional object, beyond our direct comprehension.

As a coin possesses an outer edge, the hypercoin possesses an outer surface. This results in sharp, sudden reversals when the boundary (the outer surface) is met. Recall, once again, the dime-sized marble and the quarter. The ant, on the quarter, encountered these sharp, sudden reversals. On the marble, however, no such encounters were experienced. We are to use the quarter and the marble, then, as analogies toward our visualization of the hypersphere - and in doing so build upon what we have learned about the hypercoin. The hypersphere, in conclusion, consists of 2 "hyperhemispheres" - each a spherical volume - each 4-dimensionally curved towards each other, in a way that would result in nothing more, and nothing less, than what the marble possessed - a "perfect roundness" - pure curvature. With this thought in mind, let us prepare to apply what we've learned in The Hypercoin as we return back to where we were in Visualizing the Hypersphere.