Watch as the cubes unfold and then, like magic, become two geometrical figures known as stellated rhombic dodecahedrons. Stranger than the name is that the dodecahedrons then fold again as two cubes the same size as the original cube. Wizardry? Miracle? Hellraiser? No, mathematics.
Keep reading for more about this black math and a bonus of biblical and quantum miracles.
THE CUBE THAT FOLDED ITSELF
“I never dreamed that a Christmas present my mom bought at the Museum of Modern Art would become such a big hit! Now, it’s popping up all over the Internet”, wrote Philip Brocoum, who in a few days has had more than 300,000 views to the video.
It’s a Yoshimoto cube, invented by Japanese Naoki Yoshimoto in 1971. Made up of eight interconnected cubes, it’s capable of unfolding itself in a cyclic fashion. That means you could keep folding, or unfolding it, indefinitely.
In the toy Brocoum’s mom bought him, the cubes were also cut into two identical polyhedra, each capable of forming a Yoshimoto cube containing a hollow space inside with the exact shape of another Yoshimoto cube “open” as as dodecahedron (several other shapes are also possible).
If that sounded somewhat complicated, the animated GIF on the right may illustrate the miracle of the multiplication of Yoshimoto cubes better. It’s simply that a solid Yoshimoto cube can unfold into two hollow Yoshimoto cubes.
They can be bought at the MoMa not so cheaply for $55, but you can also create one yourself using paper and glue. It will take a lot of work, and if you would rather spend just a couple of minutes, you can a have a simpler version of the folding cube using a sheet of paper, scissors and tape. It won’t multiplicate itself, but it will unfold, and it will be a Yoshimoto cube. Just follow the video below:
FEEDING FIVE THOUSAND
Speaking of multiplication of cubes, that certainly brings to mind a rather known Biblical tale, and a rather unknown but perhaps much more impressive mathematical theorem. It’s the Banach-Tarski paradox.
This seemingly innocent theorem actually questions our whole understanding of reality, or at least the mathematical understanding we have of it. It proves with mathematical rigor “that a solid ball in 3-dimensional space can be split into several non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. This is often stated colloquially as ‘a pea can be chopped up and reassembled into the Sun’.”
It may sound preposterous, but it was proven more than 80 years ago by polish mathematicians Stefan Banach and Alfred Tarski, derived from rules and axioms commonly accepted as true and underlying much more mundane ideas. Through them we can conclude that there is, in principle, a way of cutting a loaf of bread and reassemble the pieces to feed five thousand people.
The small detail, of course, is that as you may have guessed, this is in practice impossible. It has never been actually observed, despite some extraordinary stories that only happened in the distant past (we may have to wait the Second Coming to see it again). The problem is on how to make the cuts. They must be infinitely curved, with an infinite scattering of points, infinitely smaller than even the most fundamental piece of matter. And even if you could somehow accomplish that, you would in the process end up with pieces of no definite volume, again violating fundamental physical laws.
So, perhaps the math-visual trick of the Yoshimoto cube is the closer we may get in real life to the miracle of the multiplication of loaves – you can cut them into pieces to make a Yoshimoto cube. Solid loaves would unfold into hollow pieces of bread, not actually a miracle, but that would surely make a nice party trick.
Nevertheless, If actually multiplicating loaves is as far as we know impossible, some pretty impossible things happen in quantum physics.
In Yoshimoto’s toy, the duplication is just apparent. In the Banach-Tarski paradox, the multiplication of volume is a pure mathematics conclusion with no connection to the real world.
Or perhaps not. Illustrating beautiful polyhedra and rigorous mathematical theorems making reference to biblical stories may make some uncomfortable, but since we are here, let’s jump into even wilder things, with proportionally curious consequences.
In “The prescient power of mathematics”, science writer John Gribbin asks just “WHY is mathematics such an effective tool for describing the way the physical world works?”. Decades before Albert Einstein came up with his General Theory of Relativity, for instance, the mathematical tools to formalize it had been already created and explored… as pure mathematics.
Einstein “simply” – that’s no small feat – realized their usefulness in describing the physical world around us incorporating a series of observations up until then considered anomalous.
It’s as if someone invented the screwdriver before the screw, just as an intellectual exploration, and later discovered how that screw found lying in his garden fit perfectly well into the screwdriver. And didn’t think about how bizarre this series of events really was.
Well, the Banach-Tarski paradox may have a practical application, this sort of absurd multiplication of volume may actually happen in the real world, if only in very small scales. With the taunting title of “Hadron physics and transfinite set theory”, late mathematician and physicist (or vice-versa) Bruno Augenstein noted how:
“every observed strong interaction hadron reaction can be envisaged as a paradoxical decomposition or sequence of paradoxical decompositions. The essential role of non-Abelian groups in both hadron physics and paradoxical decompositions is one mathematical link connecting these two areas. The analogies suggest critical roles in physics for transfinite set theory and nonmeasurable sets.”
And if you read this text to this point you may perhaps understand what lies behind all that complicated talk. Hadrons are, as one large collider already made famous, particles such as protons in the nucleus of atoms. And as subatomic particles, they do some pretty amazing things including…
The incredible multiplication of protons. During collisions, a proton can turn into several exact copies of itself, depending on the energy of the interaction. And they do it in a way that, as Augenstein realized, can be precisely described by the Banach-Tarski (BT) theorem on how a solid sphere may be decomposed and then reassembled into multiple copies. The decomposed pieces are quarks. Curiously, just as the individual pieces in the BT decomposition may end up with no definable volume, we can’t find isolated quarks. They don’t exist, they always combine with other quarks to make definable, larger particles.
Are the subatomic particles, in scales smaller than those we can peek into, entering a strange world where transfinite mathematics and nonmeasurable sets are actually real? Well, another of Augenstein’s work on this idea was published in the properly titled and now unfortunately defunct scientific journal “Speculations in Science and Technology”. This is some wild speculation, as we warned. And it can go wilder.
An astrophysicist even published in another journal the suggestion that this BT magnification by which a small pea could be decomposed and reassembled into a ginormous volume could explain… the Big Bang. Or even a Big Crunch, as a ginormous volume could also be decomposed and reassembled into a small pea.
Perhaps we are going too far? Probably yes. The astrophysicist in question is Mohamed El Naschie, recently exposed as a not so serious academic. he published the speculation in his own journal, as well as more than 300 other papers of his authorship. Finding El Naschie by chance when looking into the speculations on the BT theorem, mixing miracles and geometrical toys may be a good sign that we already travelled a long way.
Back into solid ground, then, where the Yoshimoto cube is just a nice toy and the BT theorem just a puzzling mathematical theorem, I hope you have enjoyed the trip. Forgetomori Airlines thank you for choosing us, and hope you fly with us again for the next trip into imagination land.
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A great book with more on the Banach-Tarski paradox is “The Pea and the Sun – A Mathematical Paradox”.
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