What Is The Infinite Chocolate Paradox?

Recently, a chain of French supermarkets decided to sell the universally desired hazel chocolate spread, Nutella, otherwise known as God’s holy nectar sent from the blessed heavens, at a 70% discount. This astonishing slash in price resulted in what one could logically expect – a riot. However, I’m not using the word ‘riot’ in its metaphorical sense, I mean it in the literal sense.

The ‘riots’ spread across the supermarkets forced them to summon the police for help when customers resorted to fighting and jostling. While the hunt for discounted chocolate saw one woman’s hair being gruesomely pulled, another’s hand was profusely bleeding.

Nutella chocolate on bread

Nutella for breakfast. (Photo Credit: Janine / Flickr)

We love chocolate. We love it in all shapes, forms, sizes, colors, flavors and wrappers. Jerry Seinfeld joked about how the sole objective of his entire childhood was to acquire candy. His happiness knew no bounds when he was first introduced to the concept of Halloween, to which he remarked in disbelief — “Everyone we know is just giving out candy?” He promises he will “do anything they want” and after a short pause – “Yes, I can wear that,” he concurs.

If there is anything repulsive about chocolate, it is that it eventually ends. However, what if there were a way to produce infinite blocks of your favorite Hershey’s bar by simply breaking it and rearranging the pieces? Don’t believe me? Take a look for yourself.

What is this sorcery?

This phenomenon, known as the infinite chocolate paradox, spread like a plague and fueled outrage on social media. The procedure is simple – Take, let’s say, a chocolate bar constituting 5×5 blocks of chocolate. Cut the bar diagonally from slightly below the second block (from below) on the left of the vertical sides to slightly above the third block on the right.

Next, slice the newly amputated piece, which we’ll call piece A, vertically from the third block (from the left) on its upper end. Let’s call this piece B and what remains as C. Chop off a square (D) and a 1×2 rectangle (E) from the chocolate blocks constituting the first three blocks of piece B. The remaining piece from this partition is F.

Now, let the magic begin. Shift F to the right and fill the void left by F by placing C in it. Next, play the easiest game of Tetris by dropping the 1×2 rectangle over the 1×2 void above C. The 5×5 block is rearranged, with an additional block of chocolate to spare. Repeat this process to obtain infinite chocolate!

The Banach-Tarski Paradox

Of course, this isn’t plausible (duh). The gif illustrating the paradox is grossly skewed; it is only an illusion. Calling it an illusion implies that it is fake, which it is, because when the pieces are rearranged, the resulting bar isn’t the same bar we began with. Measure the vertical lengths of the two bars before and after performing the procedure and you will realize that this absurd way of cutting it in half has rendered it slightly shorter. The apparently extra block of chocolate comes at the cost of reduced size.

The infinite chocolate paradox is a crude representation of the Banach-Tarski paradox, which, by a notorious misinterpretation, allows the most daunting mathematical atrocity — 1=2. According to it, it is possible to divide a solid 3D sphere into 5 pieces and rearrange them to form two identical copies of the original sphere! There isn’t even stretching involved, only reassembling will produce replicas of the same density, same volume, same everything.

Banach–Tarski paradox ball

According to the Banach-Tarski paradox, it is possible to divide a solid 3D sphere into 5 pieces and rearrange them to form two identical copies of the original sphere. (Photo Credit: Benjamin D. Esham / Wikimedia Commons)

The math underlying the paradox, as you might have guessed, is extremely esoteric and therefore incomprehensible. It defies common sense and questions our intuitive perceptions of spatial concepts like ‘volume’ and ‘density’. It operates in the strange realm of infinity, a concept that has perplexed mathematicians since time immemorial.

The ridiculous phenomenon is possible only if the sphere or matter, in general, is assumed to be infinitely divisible, which it obviously is not. Matter is founded on rigid structures held in place by atoms. The concept is applicable only in the abstract, not the real world, because in the real world, matter is constrained by size. In the abstract world, however, where the paradox is feasible, matter can simply be viewed as a collection of points. In this case, infinite points.

Different Infinities

The paradox deals with measurable sets composed of immeasurable quantities. Consider the set of numbers 0,1 and all the numbers between them. This set is denoted by [0,1]. This measurable set can be further divided into uncountable, infinite real numbers starting from 0.000000000000000000001 followed by 0.0…2 and so on.

Even and natural number

Even though the set of natural numbers ‘seems’ denser, the set of even numbers can simply be ‘scaled’ and equated with them by placing both the sets one above the other. Because both are infinite, they will parallelly go on forever.

The length of these infinite numbers can be divided into two halves such that the points constituting both the halves have the same cardinality, because infinity divided by two is still infinity. This implies that there are as many even numbers as there are natural numbers!

Another way to magically conjure an additional set of infinite numbers from a given set of infinite numbers out of thin air is to refer to the distinction between ‘countable’ and ‘uncountable’ infinities.

People who believe that the number of natural numbers until infinity and the infinite number of real numbers between them are equal, such that each natural number can be assigned to each real number, are clearly wrong. This is because you can move down the real numbers diagonally and simply increment the numbers you progressively parse.

Infinite number table

Because all the natural numbers are exhausted, for they are ‘countable’, there are no new natural numbers to assign anymore. This implies that the infinity of real numbers is larger, larger to the extent of being ‘uncountable’, as compared to the infinity of natural numbers. We can now separate the newly created numbers to form another set of infinite numbers.

This, albeit by a grievous simplification, explains how the sphere can decompose into two identical spheres, as the density reduced to half is still infinite. Remember that this works only for mathematical points, not physical atoms. Furthermore, the five shapes the sphere divides into are highly eccentric, complex and distorted entities, unlike any ‘shape’ you’ve ever encountered. Enjoy your finite bar of chocolate.

References

  1. Harvey Mudd College, Claremont, California
  2. Wikipedia
  3. Maths.org – University of Cambridge

The short URL of the present article is: http://sciabc.us/hQwlC
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About the Author:

Akash Peshin is an Electronic Engineer from the University of Mumbai, India and a science writer at ScienceABC. Enamored with science ever since discovering a picture book about Saturn at the age of 7, he believes that what fundamentally fuels this passion is his curiosity and appetite for wonder.

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