The Paradox of Infinite Cloning

0:00Imagine you have one apple. If you slice it into two halves, you end up with two pieces, but they're smaller, half the original apple each. This matches our everyday intuition. You can't create more matter out of nothing. But there is a paradox in mathematics that imagines taking a solid ball and slicing it into just a handful of pieces. Without stretching, shrinking, or adding anything new, you put those pieces together again, and end up with two identical balls, each as solid and complete as the original. Sounds absurd, doesn't it? Our intuition says that it should not be possible because it violates our fundamental understanding of volume that you simply can't create something from nothing. But in the realm

1:06of mathematical constructs, it is possible. It's a astonishing idea, is known as the Banitarsky Paradox. First described by mathematicians Stefan Banit and Alfred Tarsky in 1924. It is a theorem in set theoretic geometry, which more formally states the following. Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield to identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their original shape. However, the pieces themselves are not solids in the traditional sense, but infinite scatterings of points. This has come to be known by many names, most famously the P and the Sun Paradox, or that a P can be chopped up and

2:09reassembled into the Sun. If this got you thinking, you are in the right place. We are going to go into the crux of the paradox today and see if we can resolve it. The paradox was proposed in 1924 by Banitck and Tarsky, who were central figures of the vibrant elder war Warsaw School of Mathematics. Stefan Banach, born in Krakow, Poland, is famous for his pioneering work in functional analysis, which deals with translating abstract mathematics into powerful new tools. Alfred Tarsky, originally from Warsaw, specialized in logic and set theory, the Banit Tarsky Paradox sounds impossible because it contradicts basic common sense about volume and space. Normally if you cut something into parts and rearrange those parts, without stretching

3:13or adding material, you'd expect the total volume to remain the same. Yet Banach and Tarsky showed that by using rotations and translations, which means just moving and spinning pieces around, you can end up with twice the original volume. How is this possible? The theorem is a theoretical paradox, which means it contradicts basic geometric intuition but is not false or a self-contradictory. The key lies in the strange shapes created during this process. These shapes are so unusual that we can't assign them a normal volume. The mathematical argument relies heavily on an assumption called the axiom of choice, which allows mathematicians to construct these bizarre sets that do not have volume in the ordinary sense. Without the axiom of choice, the paradox doesn't hold.

4:20To understand Banit Tarsky, let's start simple. Instead of the set of all whole numbers, starting from 0, 1, 2, 3 and so forth, clearly this set is infinite. Now let's break it down into two subsets, subset of even numbers such as 0, 2, 4, 6 and so on and subset of odd numbers, 1, 3, 5, 7 and so on. Each subset is also infinite. Surprisingly, these subsets have just as many elements as the original set, even though they're only part of it, so you have taken one infinite set and created two infinite subsets. Contrast this with a finite set like the numbers 1 through 10. If you split them into two subsets, say 1 to 5 and 6 to 10, each subset clearly has fewer elements than the whole. Finite sets behave intuitively while infinite sets break our common sense rules.

5:25What's even more astonishing is that you can transform each subset of the infinite set into the original set by performing a mathematical operation. For example, just divide each even number by two and subtract one from each odd number and then divide by two. This perfectly matches each subset to the original set without adding or removing any numbers. In the infinite world, a part can be exactly as big as the whole without adding or removing any elements. Let's extend this idea to a sphere. Mathematically, this sphere isn't just one solid chunk. Instead, it can be thought of as an infinite set of points, with each point infinitely small and packed closely together. How many points would it take to make a sphere far more than you can count?

6:25In fact, mathematicians call this uncountably infinite, meaning it's equivalent to the infinite set of whole numbers we discussed earlier. Let's say you spin this sphere around like a globe, each rotation moves the points on the sphere to new positions. Banach and Tarski discovered something amazing. That in combinations of these rotations produce an endless variety of distinct positions for the sphere's points. By repeatedly applying carefully chosen rotations, you can generate infinitely many different positions for each point, creating infinitely many copies of the same set of points. Here is where things get interesting. Banach and Tarski divided the sphere into sets of points that are unlike anything we normally imagine. Remember, we had imagined our sphere to be made of an infinite number of points. So these sets of points aren't solid chunks, but intricate collections of points scattered and intertwined throughout the sphere

7:30in highly complex ways. The ordinary concept of volume simply doesn't apply to them. We can't assign them a volume because they're constructed in a way that defies our usual methods of measurement. How is such a paradox possible? Answer lies in a mathematical principle known as the axiom of choice. Simply put, this axiom states that, given a collection of sets, it's possible to select exactly one item from each set, even if you have infinitely many sets and no clear rule for choosing. Think of it as having infinitely many jars each filled with grains of sand and being allowed to pick exactly one grain from every jar without any specific rule guiding your selection. This enables mathematicians to construct these peculiar, volume-defying sets used in the

8:30Banach Tarski paradox. Remember our first example of splitting one infinite set into two equally infinite subsets? Essentially, Banach Tarski takes this idea further into three-dimensional space. Because the sphere consists of an uncountably infinite set of points, rearranging those points into new configurations can produce outcomes that violate our normal understanding of volume. The Banach Tarski paradox not only applies to spheres, but it extends into other mathematical structures too. Researchers have found similar paradoxes in other mathematical spaces, governed by something called non-amenable groups, which, in simple terms, allowed this kind of paradoxical rearrangement. Interestingly, certain mathematical spaces like two-dimensional planes

9:32or a flat tabletop are governed by amenable groups and do not allow such paradoxes. In these cases, no matter how cleverly you slice and rearrange, you cannot create extra volume. Another fascinating extension appeared in 1990 known as Tarski's circle-squaring problem. Mathematician Miklosh Lashkovich showed that, using the axiom of choice, it's possible to take a circle and rearrange it into a square of equal area. There are some important boundaries though. Hyalberts' third problem, posed in the early 1900s, revealed a limitation. When dealing with solid shapes made of flat surfaces called polyhedral, you cannot rearrange one into another of a different volume without adding or removing material. Unlike the Banach Tarski paradox, polyhedral follow our normal intuition about volume,

10:35because their decomposition must preserve both volume and something called den invariant, kind of a second measure for polyhedral after volume. This stops any Banach Tarski type paradoxical rearrangement without adding or removing material. So there you have it, how to create two spheres out of one in the world of Banach Tarski.