I recently saw a very interesting video that claims that 1=2. I soon wrote an article for it to publish it on our school’s STEM magazine Lucidity. This article is an extremely simplified version of the Banach-Tarski Paradox. Please correct me if I am wrong in any way, and enjoy!**
Imagine a time where you wished you had one more, after finish eating a sweet, crunchy Godiva chocolate ball. While you would have walked away in hunger, mathematics indicates that you no longer have to; the Banach-Tarski paradox claims that a single sphere can be made into two, even with the exact same density.
Sounds intriguing? But before delving the specifics of the paradox, it is essential to have some, or at least a minimum amount of, understanding of “infinity.” Infinity, often delineated by Hilbert’s paradox of the “infinity hotel,” is described not as a number, but as a state of a constantly increasing number. Because there is no such quantity as the “biggest number,” (let n be the biggest number; then what about n+1?) infinity needs to be defined as a “state” rather than a numerical value.
Yet, this counterintuitive value can have a classification in size. Countable infinity, as the name suggests, is a kind of infinity that is “countable.” Though it may seem like a contradiction in terms, countable infinity refers to infinity that one can count the numbers in a finite amount of time. Take, for instance, an infinite set of natural numbers. Although the numbers in the set are constantly expanding, the amount of time that will take to count them is “finite,” as we know that natural numbers are 1, 2, 3, and so on. On the other hand, “uncountable infinity” accounts for a “larger infinity.” The number of all real numbers would perfectly exemplify the notion of uncountable infinity: we cannot even grasp the number of real numbers between 0 and 1. Even if we dive deeper and deeper into the chasm between 0 and 1, smaller and smaller real numbers will continue to emerge, meaning that it would take an “infinite” amount of time to count them, unlike that of countable infinity.
However, this monstrously tremendous number can be drastically reduced in size. As Ian Stewart named the idea, the “Hyperwebster” suggests that infinity can be divided into small numbers to manifest itself. Consider a dictionary that contains every single word (even the ones that do not exist) that can be made by the twenty-six Alphabets. It would certainly be huge, containing short terms such as “aa” from overwhelmingly interminable neologisms consisted of one million “z”s. But what if we separate only the words that start with “a” from the never-ending list, and named it volume A? Now, imagine removing the initial “a”s from the separated “a” words of the Hyperwebster. Surprisingly, we would again get the elongated list of words that can be made from the Alphabets but now shortened into a length twenty-sixth of the original work.
By the same token, it is possible to represent every single point on a sphere with relation to a standard point. By using the commands left, right, up and down, every single point can be represented. In this way, a sphere can be split into seven parts: its centre point; starting points; pole; and points that start from the command up, down, right or left. But here, if we get rid of the command “right” from the start of the set of points that start with “right,” it is possible to get a group of points that begin with “up” or “down.” In other words, it is possible to acquire the points that start with the command “up” or “down” by simply rotating the points that start with “right” to the left by 1 unit. Therefore, it will also be possible to collect the group of points that start with “right” or “left” by rotating the points that start with “up” down by 1 unit. But notice that putting the two rotated sets of points, the centre point, starting points and the pole results into a full sphere. As we started with 4 parts that have distinctive commands at the start of their commands, the process above can be repeated. The outcome is two spheres created by simply dissembling, rotating and reuniting segments of a sphere.
We can be more rigorous if we wanted to.
Though the paradox seems very counterintuitive, evaluations state that it might provide an insight on how molecules at the micro-level would act. As the pristine field of mathematics is intrinsically abstract, the paradox might be an important chance for scientists to expand their research on the microworld.
More articles available at: https://www.seanyoonbio.com/