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Playing with Infinity
by Kenneth W. Behrendt
A couple of weeks ago I visited a local Walmart department store with a friend and somehow found myself wandering around in their toy section. Out of curiosity I decided to do a short inspection of what the latest toys for kids were and see how they compared to what was available when I was a toy buying kid during the late '50's and early '60's.
I discovered that, with the exception of the introduction of high tech video games, things had not really changed that much over the decades. There were the usual dolls for girls and toy vehicles for boys. Now, however, these were becoming more and more complicated by the addition of microchips intended to make them more life-like. And, of course, practically all of the board games I had played as a youth were still available...only now they cost about five times as much!
While inspecting the game aisle, I happened upon a section that contained various puzzles and "brain teasers". My eyes soon fell upon an interesting little device that had a rather spectacular heyday back in the early '80's. I am referring to the little puzzle known as "Rubik's Cube". I had seen these over the years at many of the local flea markets I had visited and, on occasion, had picked one up and played with it for a few moments. Attracted by the bright colors of the new cubes at Walmart, I decided to buy one and see what I could do with it.
If it is possible that any readers have never heard of Rubik's cube, let me very briefly describe its history and features. It is a small plastic cube that measures slightly over two inches on an edge. Each of its six faces is a different color and each face is further subdivided into nine square pieces. There are six pieces that are at the centers of the six different faces. These pieces are each of a different color and always remain fixed in position with respect to each other. Surrounding the center piece of any of the six faces are four edge pieces that each touch two adjacent faces. Finally, there are the eight corner pieces that each touch three adjacent faces of the cube.
This gadget was invented in 1974 by a Hungarian Professor of Architecture and Design named Erno Rubik. By 1980 he managed to interest a toymaker in the design and within a year of its introduction it became one of the world's best selling puzzles. What makes it most unique is that it contains an internal cylindrical mechanism which both holds its 26 visible pieces together AND allows EACH of its six faces to be rotated about its center pieces INDEPENDENTLY of the other five faces! The cube feels very comfortable in the hands and the faces can move very smoothly so that each of its edge or corner pieces can be placed into any of their other allowed positions. Since the six center pieces can only rotate along with their faces, this means one can only move the remaining 20 visible faces around the cube.
The basic idea of this puzzle is disarmingly simple. When you first open the package it comes in, you will find that each of its six sides contains all nine visible pieces of the SAME color (note: On the one I bought the faces are colored red, yellow, orange, green, blue, and white). You can then grasp the cube in both hands, close your eyes, and proceed to randomly twist and turn its various faces. After a few moments of this, you open your eyes and discover that the colored pieces on each face are completely scrambled. Now, all you have to do to "solve" this little puzzle is twist and turn the faces again (with eyes open!) until you move the various pieces about so as to unscramble the colored pieces on each of the cube's faces and thereby restore the cube to its original apppearance of having only ONE color on EACH face. Sounds simple enough, right?
In reality, however, it is one of the most difficult puzzles ever invented! About 99+% of the people who purchased this item during the early '80's never did manage to solve it. My efforts almost came to a similar fate. After about and hour or so of fiddling with my cube, I did manage to complete one face which then became the top "layer". Over the course of the next few DAYS I attempted to move on to the middle layer and soon discovered that efforts to move the edge pieces into their correct positions on the middle layer tended to cause correctly placed pieces in the top layer to move OUT of their positions! After about a week of this annoyance, I was on the verge of auctioning off my cube on eBay when I finally managed to solve it.
Solving my cube required that I obtain a step by step solution. Fortunately, there are websites that offer solutions for the 3x3x3 cube which are based on several BOOKS that were written about the puzzle during the '80's. I won't go into the details of the solution here except to say that it requires one to work his way down from the top layer of the cube to the bottom layer using a variety of COMPLEX manipulations to move each piece one at a time and, if necessary, rotate it once it is in position so that its colors lie on the correct faces. If one makes a single mistake while performing these manipulations, then there is the very high probability that the cube will wind up so scrampled again that it will be necessary to start over from the beginning with the colored face one chooses to make the top layer! So, while I admit that I like the colors and feel of my cube, I must confess that I did not really enjoy it as a puzzle. It is just too complex an object for which to readily visualize a solution. One is, therefore, reduced to memorizing a variety of standardized manipulations of the various movable face pieces so that the real challenge of solving any scrambled cube is really a matter of having a good memory and the manual skill to twist the various faces about.
Of course, all of the drawbacks I cite above for Rubik's cube have to do with the enormous number of possible combinations into which its various faces can be configured. To be precise, a 3x3x3 cube with 26 visible pieces, 20 movable pieces, and six different colored faces can be configured into 43,252,003,274,489,856,000 possible combinations of pieces! Let's just round the number off to 43+ trillion trillion.
To give the reader an idea of the enormity of this number of different configurations, consider the following. If one was to spend one second looking at each of the possible arrangements of the 20 movable pieces in a Rubik's cube, then it would take over a BILLION years to view them all! If one could collect each configuration on a separate cube, then one would have enough cubes to cover the entire Earth, including its oceans, about 250 times over! If this same collection of cubes was placed side by side, it would stretch off into space for a distance of 250 LIGHT-YEARS! That's about 40 times the distance to the nearest star to our Sun. Is it no wonder that very few people ever managed to restore their cube to its ONE correct configuration?
Anyway, my experience with this interesting little invention did get me to thinking about large numbers in general. In particular, it got me to thinking about the CONCEPT of INFINITY which is what the remainder of this short article will deal with. First, however, let's very briefly review some basic mathematical concepts.
A "number" is a measure of the MAGNITUDE or size of a group of objects and is represented by a symbol called a "numeral". In the world we live in, we quite routinely assign numbers to two or more groups of objects and then use these numbers to decide which of the groups is bigger in membership. From the Arabic number system, we received the numeral "0" which is the number zero. It tells us that a group of this magnitude contains NO members. Sometime during the Middle Ages, "negative" numbers were invented. They tell us that the number of members in a group is LESS than zero. Many beginning students of mathematics are disturbed by the concept of groups of object with negative magnitude when they are first introduced to it, but it merely indicates the magnitude of DEFICIENCY of members of a group. The more deficient in members a group is, the more members that will have to be put into the group to INCREASE its membership to zero!
For example, if you had a bowl containing three apples, then you would represent the number of apples in it with the numeral "3". If you ate all three apples, then the number of apples in the bowl would be represented by the numberal "0". However, suppose over the course of the next three days, you borrowed three apples from a friend and ate them all. Now you could consider your bowl to contain negative three apples that you would represent by the numberal "-3". If you found three apples on a tree and decided to add them to your bowl, you might well find, perhaps a few hours later, that have zero apples in your bowl. Why? Because the friend whom you borrowed the three apples from days ago might come along and take back the three apples you owed him. Thus, we see that the 3 apples you found and placed in the bowl EFFECTIVELY added to the negative three already there to yield a total of zero apples in the bowl.
The above example is, of course, a trivial one. Negative numbers, however, are critically important in the science of physics. When two forces acting on an object are in opposition to each other, the magnitude of one of them can be represented by a negative number and this then can be used to find the NET force acting on the object.
It is also possible to use a straight horizontal line to represent the various numbers we use. On this line an arbitrary point is selected to act as the reference point and is said to represent the number zero (this point is also sometimes referred to as the "origin"). A fixed length is then used to mark off the regular numbers we normally use to the RIGHT of the zero point. These numbers are also referred to as "positive" numbers. On the left side of the zero point, we can use the same fixed length to mark off the negative numbers we use. Thus, the numbers we encounter as we move to the right of the zero point are 1, 2, 3, 4, 5, etc. and the numbers that we encounter as we more to the left of the zero point are -1, -2, -3, -4, -5, etc.
The above described horizontal line which is used to represent the numbers we use is referred to as the "real number line" in mathematics. It is also possible to construct ANOTHER line of numbers at right angles to the real number line and passing through its zero point. This second line is referred to as the "imaginary number line" and represents the imaginary numbers used in certain advanced forms of mathematics. We need not delve into these interesting types of numbers here, however.
The interesting thing about this way of representing the numbers we use is that the line used can be INDEFINITELY extended in BOTH directions. Thus, there is NO LIMIT to the magnitude of a group of objects or a number, whether positive or negative, that can be represented with these lines!
However, in the macroscopic world that humans normally inhabit and are LOCALLY aware of, we have no need to represent the magnitude of a group of objects by a line of infinite length. This is because practically EVERYTHING humans observe is finite in size, duration, or strength. For examples, we know that we own so many cars, houses, or dollars in the bank. We will live so many seconds between the time on our birth certificate and death certificate. That there are so many seconds, minutes, and hours in a solar day. And, that the Earth and the other planets in our solar system have a fairly constant size, mass, orbital period, etc.
The point of all this is that we are, from the first time we become aware of the limits of our playpens, conditioned to think in terms of FINITE numbers. After a while, it becomes natural for us to believe that everything in the universe is, in fact, finite in nature. Because of this, ancient peoples developed religions that described creations of humanity and our world which occurred at some specific time in the remote past and which predicted catatrophic ends to humanity and the world which would occur at some specific time in the future. Astrologers talked about how the "fixed" stars in the stationary vault of the heavens could determine the unalterable destiny of humanity. Even the early mathematics of geometry dealt with various shapes of fixed and finite volume.
In thinking these matters over in the last few weeks, an interesting thought occurred to me. What if, in reality, the TRUE nature of EVERYTHING is not finitude, but, rather, INFINITUDE! Let us now continue to pursue this idea to its ultimate and quite surprising conclusion.
In the science of mathematics there is a symbol used to represent infinity. It looks much like the Arabic numeral for the number 8, only it is laying on its side. In mathematics one learns that if one divides a finite number by another finite number which is allowed to dwindle in magnitude toward zero, then the quotient of the two numbers will increase to infinite magnitude. The implication of this is that any finite number divided by zero must equal infinity...and this can give one the impression that infinity is just another number.
However, the reality of the situation is that infinity is NOT a number at all, but rather a CONCEPT of limitlessness. It is a concept that is totally foreign to the way humans perceive the world around themselves. Yet, it is a concept which ultimately IS the FINAL reality of all that exists, ever existed, or ever will exist!
In a past article (see the author's article titled "An Ultimate Structure for the Multiverse") I suggested that we humans actually inhabit only one local universe that is an infinitesimal part of an INFINITE array of such universes that make up our present "multiverse". At approximately the same time, each local universe starts off as a "Cosmic Egg" which is a solar system sized Black Hole which contains a sphere of solid neutronium and its central "singularity".
These Cosmic Eggs grow by drawing in dust and gases from dying galaxies and, due to their immensely powerful gravity fields, crushing these materials down to create and add more neutronium to their surfaces. When they reach a certain critical size, the Cosmic Eggs become unstable and a process is set up whereby their gravity fields are suddenly weakened. At this point, due to the enourmous pressures inside of their cores, the Cosmic Eggs all "hatch" with a massive explosion that releases an expanding cloud of neutrons in all directions. After several billions of years, these clouds eventually form all of the galaxies in each local universe.
Over the course of, perhaps, a hundred billion years or more, the galaxies that form any single local universe move steadily outward from the location of the Cosmic Egg that created them. Eventually, all of the various energies stored in the outermost boundary layer of explanding galaxies in any local universe are dissipated and these galaxies begin to die. They finally begin to collide with the incoming galactic debris from other neighboring local universes and a NEW Cosmic Egg begins to form.
A point in time is reached at which ALL of the matter AND energy of the multiverse is again locked up within an infinite collection of Cosmic Eggs. For a period of, perhaps, tens of billions of years, the multiverse is completely dark. Then the infinite set of Cosmic Eggs again begin to simultaneously hatch and a new multiverse springs to life again.
The above is simply a Big Bang / Big Crunch model for the cosmos, but it postulates that the processes involved are infinite in BOTH space AND time! That is, the process is actually occurring throughout infinite space, has been in operation from the infinitely remote past, and will be in operation into the infinitely remote future. This is an entirely natural process and, in fact, is the ONLY possible process that can exist!
We see from this model that the concept of infinity is not just a theoretical subject, but rather the natural state of affairs everywhere throughout eternity. Since the multiverse is infinite in both space and time, one wonders what other strange concepts might also be possible for our infinite multiverse. Let us now explore some of these.
Upon contemplating these matters, one immediately comes to the conclusion that, with an infinity of local universes making up the multiverse, there must be a very large number of universes which are IDENTICAL to our own local universe. By identical, I mean that these other universes will have formed from a Cosmic Egg that was EXACTLY the same as the one that hatched to form our present local universe. Their Cosmic Eggs would have contained EXACTLY the same finite number of neutrons as ours and the motions of ALL of these neutrons would have been EXACTLY the same as those that formed our Cosmic Egg about 14 billion years ago. This would guarantee that when their Big Bangs occurred, the resulting galaxies that formed would be the same as the ones in our local universe and would contain exactly the same number of stars, planets, and living creatures.
In fact, not only is the number of these other universes which are truly identical to ours large, but it MUST be infinite! If this is the case, then as I sit here writing this article, there must be an INFINITE numbers of other authors with my name "out there" who are, at exactly the same instant, working on the exact same article that you are now reading! They all live on a planet that is identical to "our" Earth and which has the same countries, oceans, and peoples as does ours. Indeed, EVERY thought I am having or deed I am performing is, at the instant I have or do it, being repeated throughout the infinite distances that surround us! Not only is this true for the author, but it applies to every living creature throughout our local universe.
One then might wonder if it would ever be possible for a person to travel over to one of these other universes which is identical to ours so that he might be able to meet himself. Unfortunately, this possibility is very improbable for a variety of reasons.
First, because of the huge number of neutrons in the solar system sized black holes that compose a Cosmic Egg, the probability of two identical universes forming near each other (like within a few hundred BILLION light-years of each other!) is very low. Therefore, our nearest twin universe might be TRILLIONS or even HUNDREDS of trillions of light-years away. Even after we of Earth finally manage to duplicate the mass negating technology that allows UFO's to defy the laws of gravity and inertia, we may never get beyond the ability to travel to only the few hundreds of billions of galaxies in our own local universe.
Secondly, even if an identical universe to ours was very close so that we could just barely reach it with a massless spacecraft, we would be trying to meet people who had the exact same idea we did. As we landed on THEIR Earth to meet them, they would, at the exact same time, be landing on the Earth of yet another universe identical to theirs. We would find that they were not home just as they would discover that the people they were visiting were not home.
The implication of the above situation is that, although infinitely complex, there must be a subtle underlying SYMMETRY to the multiverse. While there is an infinity of DIFFERENT universes in the multiverse, the individual universes of EACH infinity of identical universes must be symmetrically arranged with respect to each other throughout the multiverse. The most logical arrangement would seem to me to be a simple cubic array of identical universes. Thus, if one was to represent a very large section of the multiverse by a large cube which was made up of many smaller cubes that were packed closely together, one could then imagine any infinite subset of identical universes to be represented by the points where the corners or vertices of the smaller cubes touch within the body of the larger composite cube.
Not only are any infinite subset of identical universes symmetrically arranged with respect to each other, but they must also be identically ORIENTED with respect to each other. We can further extend this analysis by postulating that each universe of an infinite subset of identical universes is also surrounded by the SAME local universes, each of which is a member of another infinite subset of identical universes. Thus, we see from this that while there is an infinitude of DIFFERENT KINDS of infinite identical universe subsets in the multiverse, they are all sort of "locked" into a particular, though enormously complex, relationship with respect to each other.
Each multiverse which arises between the hatching of its initial infinite array of Cosmic Eggs and its final infinite array of Big Crunchs to again form another infinite array of Cosmic Eggs is unique. If the Cosmic Eggs are arranged with respect to each other as was suggested above when they all hatch, then the expansion of any local universe will cause its expanding front of dying galaxies to finally encounter and begin merging with the similarly expanding material of 26 other local universes that surround it so as to create 8 new Cosmic Eggs that are arranged at the corners of a cube with respect to each other. Thus, each new Cosmic Egg that is formed during the Big Crunch phase of a multiverse will be different from the previous eight hatched Cosmic Eggs that contributed material to it. Interestingly enough, the total number of nearest surrounding universes that interact with any particular expanding local universe is exactly equal to the number of visible pieces found on a Rubik's Cube (i.e., 26)!
We see from this that each multiverse will be different from the one that preceeded it and will, in turn, give rise to yet another different multiverse. I suspect that, because each multiverse is spatially infinite and, therefore, contains an infinite amount of matter, this will guarantee that throughout eternity no two multiverses will ever be identical.
It is also important to remember that, while the cosmos consists of an infinite parade of multiverses, the major differences between any two of them are really just cosmetic. The numbers, positions, and shapes of galaxies in any local universe that helps form a particular multiverse will vary from one multiverse to another, BUT the fundamental laws of physics and chemistry will NOT vary from one multiverse to the next. Thus, the physics books that were valid in any multiverse will still be valid in the next multiverse and will remain so for eternity.
There are many perplexing questions that arise when one attempts to conceptualize the ultimate nature of our infinite cosmos.
For example, one may readily acknowledge that there CURRENTLY exist in our present multiverse an infinite number of local universes which are identical to ours and that, therefore, at this exact instant there are an infinite number of humans out there for EACH person now living on "our" Earth. One may further acknowledge that every thought and action of this infinite number of humans is also perfectly synchronized. But, is it possible that there are also OTHER infinite subsets of local universes wherein all of their Earth's humans are now (with respect to our time) a few minutes behind us OR a few minutes ahead of us in their thoughts and actions?
I tend to think that such "desynchronization" does not take place in a multiverse, even though it contains an infinite number of infinite subsets of local universes. The reason for my disbelief in this possibility is because, as was mentioned above, all of the infinite subsets of local universes are locked into a rigid relationship with respect to each other. For another desynchronized infinite subset of local universes which is structurally identical to our own subset to exist, its infinite number of Cosmic Eggs would have had to have hatched either earlier or later than the ones that formed our subset. This would require a multiverse wherein certain infinite subsets of Cosmic Eggs were hatching while others were still forming!
This situation would then ruin the concept of a multiverse "day" followed by a shorter cosmic "night" during which time interval the entire previous multiverse disappears into an infinite array of newly forming Cosmic Eggs. It would replace the neat day / night cycle of the mulitiveses with a perpetual twilight where randomness and chaos prevailed.
Such desynchronization between two infinite subsets of structurally identical local universes would require that some the subsets of growing Cosmic Eggs accumulated enough material to destabilize and hatch before others did. This would then require that the infinite subsets of Cosmic Eggs that formed the local universes surrounding the particular desynchronized infinite subset of local universes being considered would ALSO have to do the same thing. At some point this requirement would eventually relay itself back to OUR infinite subset of local universes. Thus, finally, IF a desynchronized infinite subset of local universes that was structurally identical to ours were to somehow form, its existence would IMMEDIATELY require that our infinite subset of local universes ALSO desynchronized by the SAME temporal interval. This requirement would, therefore, instantly synchronize the two infinite subsets so that they became a SINGLE subset. Thus, there can never exist a temporally desynchronized version of any infinite subset of local universes in any multiverse.
I hope that the reader has not been overwhelmed by the somewhat convoluted nature of the above reasoning. It is merely some of the logical implications of dwelling in the midst of a never ending series of spatially infinite multiverses.
Some readers may wonder if our multiverse allows for another infinite subset of local universes to exist wherein their lives, as they know them here on "our" Earth, somehow follow a different course. For example, could there be an infinite number of any of OUR currently living humans out there who, in their existences, manage to escaped some bad or tragic situation or who, on the other hand, managed to achieve some great thing by overcoming great odds against it.
The answer to this is, unfortunately, "No". Again, the reasons involve the previously analyzed concept of desynchronization and its strict prohibition in any multiverse...only in this case the prohibited desynchronization would not be a temporal one, but rather a STRUCTURAL one. IF that other infinite number of humans were truely identical to a person on our Earth now, then their local universes would have to be structurally IDENTICAL to our own. The EXACT same environmental conditions that caused problems for a particular human being here on our Earth would also have to exist for them too. Thus, one need never imagine that there is some "better" world out there wherein one's EXACT duplicate is now living a better (or worse) life.
Of course, this does not mean that there are not CURRENTLY an infinite number of humans out there who LOOK much like a particular human on Earth. Actually, any person's genome will be duplicated infinitely throughout a multiverse, BUT the vast majority of the humans with those genomes will be VERY different in personality, memories, and abilities from their version here on our Earth due to the widely structurally different infinite subsets of local universes that they will live in. One must think of these other humans, who are not members of our infinite subset of local universes, as mere "cosmic clones" who have destinies very different from their lookalikes here on Earth.
Another interesting aspect of the above reasoning is that if a person on Earth does, in fact, have a life filled with hardship, then that life will NOT be replayed in any of the infinite subsets of local universes that form in the NEXT multiverse. In fact, that person's particular unfortunate life will never again be played out in any of the infinite number of future multiverses. On the other hand, if the person had a spectacular life filled with accomplishment and weatlth, then that life will also never be repeated in any future multiverse. The lives of each of the uncountable identical humans that inhabit our infinite subset of local universes in THIS multiverse are unique and occur only ONCE in all eternity.
Again, it must be emphasized that, although no particular human's life will ever be repeated in any of the infinite subsets of local universes of any future multiverse, the GENOME of EACH of our Earth's humans WILL be infinitely repeated in EVERY future multiverse. While the humans that possess these genomes in future multiverses will APPEAR identical to their present versions on our Earth, they will be living quite different lives. Some will have lives more prosperous than those of their current versions on our Earth...and some will have lives that are less prosperous.
Even for those humans with less than average lives, there is always hope that things will improve for them...even if this is not their final destiny. This hope always exists because it is always impossible for us to know, with certainty, exactly what our future life will be like. And it is hope that keeps a person functioning despite the adverse circumstances in his life. Thus, this ignorance about the EXACT nature of our futures and the hope it engenders is a kind of blessing that I believe most people would really not want to give up.
This brings up a final metaphysical matter that I want to briefly treat before bringing this article to a close. Mainly, the question is whether or not it would be possible to change the future IF one had some sort of magical crystal ball that could show one exactly what that future would be.
The answer to this question is that such avoidance would NOT be possible! Surprise! The reason is that if one truly had a crystal ball that showed future events in perfect detail, then the picture it delivered would ALREADY have taken into account one's viewing of the situation and any subsequent efforts to change it! Thus, if the crystal ball showed a person being hit and killed by a bolt of lightening at a certain time and place, then somehow events in the person's life, over which he had NO CONTROL, would come into action in such a way so as to guarantee that he was at the designated place on time to fulfill his destiny. Fortunately, such crystal balls do not exist and probably never will. They would not change one's life in any way, but would only burden one with additional fears and worries.
Finally, in dealing with the concept of an infinite series of spatially infinite multiverses, we needed to develop a new kind of language that combined elements from the realms of philosophy and mathematics. With this new language, we were able to finally view the BIG cosmic picture and appreciate a few of its implications. The insights gained will, most likely, have little affect on one's daily life. However, it is somewhat comforting to know that each of us has an INFINITE genetic extension of ourselves that persists throughout infinite space and infinite time! In a sense, this is a kind of immortality granted to us by an also infinite cosmos...
(Note: this article completed September 24th, 2004)
COSMIC VAULT GIFT SHOP