Mathematics is seen by many as a mysterious and often unsettling subject. Answers often hide behind layers and layers of complicated equations, formulas and ciphers, the application of advanced concepts to real life is limited and I often find myself more confused after class than when I first entered. However, the real beauty of Mathematics is held by its ability to transform the world around you and provide evidence for what is inconceivable or completely against logic. It is this area of Mathematics that I find most intriguing and for my Math IA I will attempt to explain and prove one of these great paradoxes, that of Banach-Tarski. This mathematical exploration was conceived by two Polish mathematicians, Alfred Tarski and Stefan Banach, in 1924 and, in short, proves that it is possible to create a perfect duplicate of an object by simply decomposing it and then rearranging it. Same volume, same density, same weight, same everything (Banach and Tarski, 1924).
This mammoth of a paradox is very complicated and difficult to understand if presented raw. Therefore, in order to better comprehend it better, we must first explore some of the key concepts of infinity and mapping. These may seem unrelated at first glance, but the correlations will be revealed once we tackle Banach Tarski.
The first key concept we will look into is infinity and, more importantly, what is infinity? Infinity is a difficult concept to explain. Many would think of it as the biggest number. In fact, we often say “there is an infinite number of…”. This is untrue, infinity cannot be found on a number line, nor could we ever reach it or even hope to reach it. It cannot be the biggest number as this can never be reached. I prefer to think of it instead as the amount of numbers there are and not the greatest number. Infinity is the size of something that does not end.
This brings us onto our next concept of infinity, countable and uncountable infinity. Although it may seem counterintuitive or illogical, there are different sizes to infinity with some being “more infinite” or larger than others. The first and smallest form of infinity is countable infinity. This is the amount of natural and therefore whole numbers that there are (1,2,3…). Sets like these are unending but they can still be counted. In other words, one could count them from one element (e.g. 1) to another (e.g.2) in a finite amount of time.
Uncountable infinity on the other hand is too large to count or even begin counting. This can be visualized by the number of real numbers there are. This widens our net to not only all whole numbers, but all numbers between as well. One cannot even count the numbers between 0 and 1 within a finite amount of time. The uncountable nature of this set makes it so that, between 0 and 1, there are more numbers than there are numbers on the natural number scale. This may seem counterintuitive or wrong at first. After all, how can there be more than what is already infinite? This can be illustrated by Gerhard Cantors diagonal argument (Martin, 2019). Imagine we create a table in which all numbers between 0 and 1 are listed.
Since there is no starting point, we will use an infinite amount of randomly generated numbers with no repeats. If we assign to each of these numbers a natural number and the correspondence between them is 1-to-1 then there must be the same amount of numbers in each infinite set. This is known as constructive proof. This is untrue. Although both lists go on forever, this is not enough for the real numbers. If we go diagonally down our list and take the first number of the first row, the second number of the second row and so on as illustrated in the table and then add 1 to it (or subtract one if it is 9) we will have created a new number not found in the list. Based on this, Cantor’s diagram then uses contradictory proof to show that the set between 0 and 1 is uncountable. Since we have defined the set to have no repeats and therefore for all numbers to be different in at least one place, by changing a digit for each decimal point we will have created a new number. This means that although we have exhausted every possible whole number within the list, we can still create real numbers between 0 and 1 and so this set must be greater.
Whole Numbers Real Numbers between 0 and 1
1 0.77856439853…
2 0.67495738265…
3 0.23193348574…
4 0.32657829305…
5 0.84732719357…
6 0.22667788992…
… …
No Correspondence 0.882638…
Another aspect of infinity that is true yet counterintuitive is that there are as many even numbers as even and odd numbers. Initially, one could say that there must be half the amount of even as even and odd, yet, when you use the same table concept, as illustrated above, we find that there is a 1:1 correspondence. We will never run out of even nor whole numbers.
This leads to the crucial conclusion that Infinity will always remain infinity. In other words, infinity divided by two is still infinity. Infinity minus one is still infinity. Add 1, subtract 200000 or multiply by a factor of 10 and you are still left with infinity. A good way to illustrate this point is through Hilbert’s Paradox of the Grand Hotel (Ferrere, 2017).
Imagine a hotel with a countably infinite amount of rooms, each with a guest inside. Seemingly this hotel is fully booked and if a new guest shows up, the hotel would have to decline this invitation. Infinity, however, is more complex than this and so, for the hotel to accommodate the new guest, all they would have to do is move the guest in room 1 to room 2, the guest in room 2 to room 3 and so on. Because infinity is never ending, we will never run out of rooms and therefore can accommodate any new visitor. Infinity minus one works the same. Say the guest decides to leave, the hotel would only have to shift all guests backwards to create a fully booked hotel once again.
Now that we have made these numerical concepts clear, it is time to move onto shapes and geometry. Hilbert’s Grand Hotel can also be applied to a simple circle. We can think of points around the circumference as guests. If we were to remove one guest, and therefore one point from the circle, this point would be missing, an empty space. Infinity, however, tells us that this doesn’t affect the circumference at all. The formula for calculating a circle’s circumference is radius multiplied by 2π. If one begins marking points on the circumference equal to the radius starting from the missing point, they will discover that no two points will ever meet. This creates a never ending set, yet it is still countable just like guests in the rooms at Hilbert’s Hotel
Similarly, the missing point can be considered as a guest checking out and so , to fill the gap, all we must do is shift the points counterclockwise. Point 1 fills in the missing gap, point 2 takes the place of point 1, point 3 the place of point 2 and so on. Since we have a never ending amount of points available, there will be no missing gap. The irrational nature of a circle’s circumference allows for this lack of repetiiton or overlapping. Our initial missing point has been forgotten. We apparently never even needed it in the first place.
The final concept of infinity that will be discussed before tackling Banach Tarski is that of the hyperwebster (Stewart, 1996). This is a dictionary proposed by Ian Stuart which would include all possible combinations of letters that can be formed with the 26 letters in the English alphabet. It begins with the letter A, then AA, then AAA and so on for an infinite amount of As until AB which then continues with ABA, ABAA… This process continues infinitely until an infinitely generated combination of Zs.
A, AA, AAA, AAAA, AAAAA, …
AB, ABAA, ABAAA, ABAAAA, ….
AB, ABB, ABBB, ABBBB, …
….
Z, ZA, ZAA, ZAAA,…
Z, ZZ, ZZZ, ZZZZ…
If created, this dictionary would contain every single thought of every single person who has ever lived, every truth, every lie. It would contain the full biographies of all people living and dead. Who killed JFK, where is Cleopatras tomb, what happened to the arc of the covenant could all be found within its pages, along with every other possibility that didn’t occur. This would, of course, be an immense task to accomplish but if the company publishing this book wanted to save money when printing they could take a shortcut. If they were to take all the words beginning with an A and place them in a volume titled A, they wouldn’t have to print the initial A as the readers would already know they were reading Volume 1: “A” and could therefore add the A in front themselves.
By removing the first A, the publisher is left with every A word sans the first A which has become again every possible word combination. Just 1 out of 26 volumes has been decomposed into the entire hyperwebster.
A, AA, AAA, AAAA, AAAAA, …
AB, ABAA, ABAAA, ABAAAA, ….
AB, ABB, ABBB, ABBBB, …
….
Now, the time has come to tackle Banach Tarski using the concepts we have learned above. In order to do this, we must first understand how to take a 3D object and transform it into a hyperwebster. For this investigation we will be using a uniform sphere.
The first step in decomposing a sphere into different parts is to give every single point on the surface one name that is unique to it. To do this we will name them according to the way they can be reached given a single starting point. If we move this starting point across the sphere in steps, and at a length of arccos(1/3), it will never reach the same point twice so long as we do not backtrack. In order to achieve this and number all points on a sphere, all we need is to move in the directions Left, Right, Up and Down along two perpendicular axes. Since there are an infinite number of points on a sphere, we are going to need all the possible combinations of these 4 directions as shown in the diagram below so long as we do not backtrack as this would mean they cancel out (no LR, RL, UD and DU). It is also important to note that when conducting this experiment, one must write the letters from right to left so that final rotation is the leftmost letter.
It soon becomes clear that the amount of combinations is countably infinite. But if we apply each of these combinations to a starting point on the sphere, we can name each point by the combination that brought us there and then have mapped out every single point. We can also color code each of these points according to the final rotation. For example, let’s take the combinations I have highlighted above (RR, DD, DL). The first combination entails rotating right and then right again. This brings us to the point illustrated above and, since the final rotation is right, we can color this point red. The next combination ends in a downward rotation so for that we shall use the color blue. The next one ends with a leftwards rotation which we will represent using purple. Rotations that end with U will be illustrated with orange. If we apply this same process to every single rotation we will have a countably infinite amount of red, blue, orange and purple points on our sphere, each of which has its own unique name. To do this successfully would be an incredible achievement but, sadly, not enough. There are an uncountably infinite amount of points on a sphere’s surface.
The solution to this is surprisingly easy. Pick any point on the sphere that we have missed, make it into a new starting point by coloring it green and then run the calculations all over again using this as the initial point of rotation. Complete this an infinite number of times, using an infinite number of starting points and we will have filled in every missing point on a sphere’s surface just once with the exception of poles. Every sequence that we have applied to the sphere utilizes two poles of rotation. These are points on the sphere that, when rotated, return to their initial standpoint. For example, for every R and L rotation, the poles are north and south. The issue with poles is that they can be reached by more than just one combination and more than just one final rotation. For example, let us take a sequence that brings us to the North pole of the sphere. At this point, any subsequent rotation R or L will simply lead us to the same exact location. In order to rectify this, we shall just remove them from our scheme and color each of these in yellow. Every sequence has two and so there will be a countably infinite number of yellow poles on our sphere.
We have now finally achieved giving every single point on a sphere one name and one of six colors and can move onto decomposing it into parts.
In this 3D model, every single point on the sphere corresponds to a line of points below it, connecting it to the center of the sphere which is necessary to map out an entire sphere and not only its surface.
Our first step is to cut out and remove all the yellow points and therefore the poles of the sphere. Next come the green starting points, the red R (Right) points, the orange U (Up) points, the purple L (Left) points and the blue D (Down) points. If done correctly, the final product should resemble the image below.
This is our entire sphere, with just these six pieces, we can build the entire thing.
Now let us examine the Left (Purple) piece. It is defined as being the piece composed by every point that can be accessed by a sequence ending in a leftwards rotation. If we were to rotate this piece right, it would be like adding an R to the end of the point’s name. L and then R, however, is a backtrack and therefore cancel each other out. The remaining set becomes the same as a set of all points that end with L, but also with U, D and no rotation at all which means that all starting points are also included. Using a simple rotation, we have turned a sixth of a sphere into nearly ¾ of one.
If we add the right piece and the poles of rotation, we are left with an exact copy of our original sphere with components left over.
In order to make the second copy, let us take the U piece and rotate it downwards. Again, the DU’s cancel as it would mean no rotation at all and we are left with all the starting points, the U points, the L points and the R points. The problem arises when we notice that we still have not used our original set of starting points and therefore have no need for the ones we created.
What we can do is return to our original sphere by rotating it upwards again and remove every point that would turn into a starting point when rotated down, therefore every point whose final rotation is U. Take these points and put them in the down piece. However, after rotating, points named UU would simply become U and therefore give us a copy in the blue piece and in our original piece. To rectify this, we must take every single point that is just a string of U’s and place it within the down piece. Finally, rotate the remaining U piece down, making it congruent to the U, R and L pieces. Add the down piece along with some of the U points, the remaining starting points and we have almost created our second sphere sans the poles of rotation.
This means that in our second sphere there is a countably infinite number of holes where the poles of rotation should have gone. Around every sphere there is an axis around which, if rotated, every “pole hole” orbits around without hitting another. The path this missing point would take around a sphere is merely a circle with one point missing. Therefore, we can fill each of these as explained earlier and have therefore created our second sphere.
We have taken one sphere and turned it into two identical spheres, same size, same shape, same volume, without having added anything. It is thus possible, to take one object, decompose it and then rearrange it into more than the sum of its parts. 1 + 1 = 1.
The implications of this paradox are huge. Even today mathematicians, philosophers and scientists are still debating them. The greatest question is, without a doubt, whether this process could be applied to the real world. Mathematically, this is possible, and mathematics has been known to predict many future technological advancements and, in many cases, plays a lead role in their discovery. Whether Banach-Tarski will one day become reality, or whether it marks the point where mathematics and physics diverge is still unknown. History is rich with examples of mathematical theorems, discovered in a time when their applications were impossible to achieve and therefore had to wait decades, even centuries, for science and physics to catch up. Therefore Banach-Tarski is a possibility. Then again, the shapes created by decomposing an object could not just be a simple arrangement of dots and lines but would be infinitely complex and detailed. This is not possible to achieve on earth, as our measurements can only get so small and we only have a finite amount of time to achieve them. But that does not mean it is unachievable. Many scientists believe that this paradox is closely related to physics and many scientific papers have been written, such as the Hadron Physics and Transfinite Set Theory by B.W Augenstein, that draw a link between Banach-Tarski and the way subatomic particles collide, creating more particles than what we began with (Augenstein, 1984). The future regarding this paradox is still unclear yet I believe that the conclusions we determined throughout this exploration can be crucial in understanding more of our universe and the way we interact with it.