Extraordinary claims. Ordinary investigations.

Folding the Yoshimoto Cube, Feeding 5000

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.


“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.

y_cube_aniIn 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:



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’.” 445px-Banach-Tarski_Paradox.svg

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.

bttbigbangdsaPerhaps 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.

– – –

A great book with more on the Banach-Tarski paradox is “The Pea and the Sun – A Mathematical Paradox”.

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Posted in Science | 40 comments

40 Comments so far

  1. Boing Boing January 19th, 2009 4:55 pm

    Fun with the Yoshimoto cube…

    Video demonstration of a variation of the Yoshimoto cube, invented in 1971. Link includes a video on how to make one yourself out of paper, as well as an introduction to the Banach-Tarski paradox (“a pea can be chopped up and reassembled into the Sun…

  2. maeseele (Peter Maeseele) January 19th, 2009 8:54 pm
  3. a20261 (Tom Endicott) January 19th, 2009 9:29 pm

    Almost as awesome as a rubik’s cube

  4. veraobmana (veraobmana) January 19th, 2009 10:01 pm

    bez rubika kubika (rubik, tu mirsi) izr?d?s ir ar? yoshimoto kubiks http://tinyurl.com/9pvtl5

  5. Jonathan Buys » Fun with the Yoshimoto cube January 20th, 2009 1:13 am

    […] Folding the Yoshimoto Cube […]

  6. El cubo Yoshimoto « Las Nius (the news) January 20th, 2009 4:53 am

    […] Plegando el cubo Yoshimoto fuente:forgetomori.com via: boingboing.net Posted by Ricardo (Las Nius) Filed in Ciencia y tecnología […]

  7. Folding The Yoshimoto Cube | onelargeprawn January 20th, 2009 4:56 am

    […] more at forgetomori. Other PostsIf you like this post, try these:Jizz in my Pants, it’s a SongEvery Flight on […]

  8. Yoshimoto Cube selber basteln | I am Jeriko January 20th, 2009 6:53 am

    […] Der Yoshimoto Cube ist auch so ein What the fuck?-Spielzeug, bestehend aus acht miteinander verbundenen Würfeln, die man unendlich oft falten kann, und die sich auch noch in zwei “hohle” Cubes aufteilen lassen. Ehrlich, eine bessere Beschreibung dafür habe ich auch nicht, schaut euch einfach das Video da oben an. Jetzt kann man die Dinger für teuer Geld im Kaufhaus der Wahl kaufen, bei Metacafe gibt es aber auch eine Videoanleitung, wie man sich selber einen Yoshimoto Cube aus Papier basteln kann. Nicht ganz einfach, das Ergebnis sollte aber entlohnen. Video gibts nach dem Klick! (via Forgetomori) […]

  9. istevens (Ian Stevens) January 20th, 2009 3:35 pm

    My parents had one of these Yoshimoto cube variants when I was a child. Still fun to this day. http://tinyurl.com/9pvtl5

  10. properlofi (jontracy) January 20th, 2009 3:42 pm

    I have to make one of these http://tinyurl.com/9pvtl5

  11. chernobylheart (chernobylheart) January 20th, 2009 3:43 pm

    Awesome, I am totally doing this on the clock: http://is.gd/gA0U

  12. links for 2009-01-21 | Nerdcore January 21st, 2009 2:30 pm

    […] Folding the Yoshimoto Cube, Feeding 5000 | forgetomori (tags: papercraft origami mathematics yoshimoto diy paper) […]

  13. vyaas (Vyaas) January 23rd, 2009 1:11 am

    Let your imagination run wild with the Yohimoto cube!: http://tinyurl.com/9pvtl5

  14. jamesjyu (James Yu) January 30th, 2009 7:09 pm

    behold, the amazing yoshimoto cube: http://tinyurl.com/9pvtl5

  15. Fun with the Yoshimoto cube February 24th, 2009 5:53 am

    […] Folding the Yoshimoto Cube […]

  16. abraaten (Aaron Braaten) April 21st, 2009 9:21 pm

    Way cool: The Yoshimoto Cube (found an awesome new blog to read). http://bit.ly/11DVe

  17. Timohfee May 4th, 2009 9:57 am

    Amazing. Not quite what i hoped, thought the video would show how to make an actually yoshimoto cube, but only took 5 minutes to make, easy to follow, and hours of fun afterwards!

  18. […] Folding the Yoshimoto Cube […]

  19. Brad June 17th, 2009 3:21 pm

    As a serious scientist you should check your statements before making serious allegations about honorable people based on hearsay. El Naschie was in the Inst. of Aeronautics and Astronautics, Tokyo University over thirty years ago. He was invited to give a key talk in IUTAM which is the Int. Union of Theoretical and Applied Mechanics. He sat in the office of Yoshimura and is pictured with his wife holding the Yoshimura buckling form. You can see his work on Yoshimura buckling in his book published by McGraw Hill. The book is called Stress, Stability and Chaos in structural engineering, an energy approach, 1990. El Naschie is not an astrophysicist. He is a universal genius with many interests including art, science and engineering. He got his Ph.D from University College, London in 1974 in structural engineering. He worked on the Banach-Tarski paradox and yes he published it in his journal Chaos, Solitons & Fractals of which he is the Editor in Chief. There is no editor in chief in the world who does not publish in his journal but Mohamed El Naschie was framed. The reason is he made a very major discovery in theoretical physics. He found that all the paradoxes of quantum mechanics can be reduced to the intricate geometry and topology of Cantor sets. He could resolve all problems of quantum mechanics and high energy physics by postulating the existence of fractal geometry for spacetime. A group of scientists plagiarized his work and published it in Scientific American. The leader of this group is Dr. Renate Loll, a German who works in the University of Utrecht in Holland. Loll used her position near to a Nobel laureate and connection to John Baez as well as her connection to the Max Blanc Inst. to orchestrate a viscous defamation campaign against El Naschie in order to justify using his work and not referring to him and pretending to have discovered Cantor sets and fractals for quantum mechanics on her own. This is the real truth and it is painful to see how propaganda works for villains.

  20. Chris June 18th, 2009 6:44 pm

    The work of Bruno Augenstein is extremely important both mathematically and physically. Interestingly as you pointed out many of his papers are published in Chaos, Solitons & Fractals as well as the New Scientist. Obviously they are two open minded journals. They do not adhere religiously to the almighty mainstream research. An important paper by Augenstein is entitled Links between physics and set theory, Chaos, Solitons & Fractals, Vol. 7, No. 11, 1996, p. 1761. El Naschie has an extremely interesting paper which he wrote during his time in Cambridge entitled Banach-Tarski theorem and Cantorian micro spacetime, Chaos, Solitons & Fractals, Vol. 5, No. 8, 1995, p. 1503. I hope you keep this blog and make it more known to other physicists working in quantum mechanics. It is very important for them. The work of Mohamed El Naschie and subsequent elucidation by Tim Palmer are probably the most important contributions to the foundations of quantum mechanics that there is. It is very unfortunate that the axiomatic structure of quantum mechanics which was introduced by von Neuman had a very destructive influence on the understanding of quantum mechanics as a realistic theory. Palmer and El Naschie have changed the situation radically and forever. Please look at the work of Palmer in the recent issues of the Proceedings of the Royal Society of London.

  21. […] Link: Via boing-boing Here’s another link. […]

  22. t'mara July 2nd, 2009 7:44 am

    i’ve seen his youtube postings, and its more ike, a man falls in love with himself.

  23. […] purportedly showing paranormal abilities I have ever seen. It’s not an extraordinary feat like the multiplication of the loaves and fishes, after all it’s just a piece of paper spinning. Hardly useful. But the many ways by which the […]

  24. Gary July 25th, 2009 4:13 pm

    The Banach-Tarski paradox isn’t as amazing as it first sounds. It’s only slightly more complicated than something trivial like mapping the unit interval to the three-dimensional universe.

  25. Mildred August 1st, 2009 8:30 am

    Two new pieces of info worth noting regarding Palmer and El Naschie. There is a new important paper entitled: Strange non-dissipative and non-chaotic attractors and Palmer’s deterministic quantum mechanics, published in Chaos, Solitons and Fractals, volume 42 (2009) 641-642. From this paper it is clear that Palmer and El Naschie are the source of a new revolution in theoretical physics. Particularly high energy physics and quantum mechanics will be affected. Regarding the credibility of Mohamed El Naschie, those on the right side of common sense will be pleased to know that Elsevier are publishing dozens of new papers by El Naschie. In addition, Nature has withdrawn its allegation. The German Journalist Schermeier who has written the defamatory article in Nature is facing criminal charges in the High Court in England. You can check all these facts on the web of the High Court.

  26. S.Q. Chen August 12th, 2009 5:32 pm

    The Yoshimoto cube is a fantastic device so could somebody tell me if this has anything to do with the Hilbert cube? A computer scientist and physicist, Prof. Ji-Huan He from Donghua University in Shanghai just published a remarkable paper on the Hilbert cube. This is an infinite dimensional cube which has a finite Hausdorff dimension equal to 4.23606. This reminds me of the Hausdorff dimension of E infinity spacetime of Mohamed El Naschie. Indeed Ji-Huan He gives credit to El Naschie regarding this cube. He even shows a sculpture of this cube which is in the University of Berkeley, California, USA. He’s paper is published in Chaos, Solitons & Fractals and I found it on Elsevier’s Science Direct. A second question is the following: Could the Hilbert cube be the missing link between El Naschie’s Cantorian fractal spacetime and T.N. Palmer’s invariant limit set? I mean could for infinite dimensional spaces the difference between spacetime and state space, i.e. Hilbert space vanish? If this is true and I think it is true then the VAK of Rene Thom, Cantorian spacetime and Palmer’s limit set are one and the same thing. Maybe Prof. Palmer could comment on this. I am a pure mathematician but I am extremely interested in physics. Any hints will be appreciated.
    S.Q. Chen

  27. B.M. Sidley August 13th, 2009 10:27 am

    I was discussing the disgraceful affair of Jan Hendrik Schön. My friend reminded me of the despicable media witch hunt which sometimes exceptionally good scientists are subjected to. False accusations in science are not uncommon. In fact most of the accusations of fraud in science turn out sooner or later to be tendacous or downright criminal. The false accusations against Mohamed El Naschie are very similar to those made against Thereza Imanisha-Kari who was a colleague of Nobel laureate David Baltimore. Mohamed El Naschie is a senior colleague of Nobel laureate Gerard ‘tHooft. Not only that but he is a very close personal friend to the entire family. Therefore I find it extremely disheartening to see that the media on the internet is not rehabilitating Prof. El Naschie with the same enthusiasm with which they defamed him. For instance the famous mathematician John Baez from the Dept. of Math. Of Riverside University in California, USA owes El Naschie a big apology. I have just read an article by Baez praising the golden mean to the sky. A few months earlier he was calling anybody who deals with the golden mean, like El Naschie, a crackpot. Occasionally it is very easy to find out the truth about things even if one is not a specialist. The citation index of Mohamed El Naschie is in the order of 4,000. By contrast the citation index of this man who made it his business to defame Prof. El Naschie, a certain mathematician from Croatia with a remarkable name, Zoran Skoda is only 10 or 12. Jealousy seems to be an affliction to which scientists and not only film stars are prone. I must say that Nature seems to be the exception. Despite the high profile and the high prestige of Nature as the leading scientific magazine in the world, they have withdrawn all their allegations and have conducted a thorough investigation to find out how they were wrongly led to write the defamatory article against El Naschie. There is no doubt that the damage done to molecular electronics by Schön’s deception is tremendous. However it is nothing compared to the damage wrongly caused to Imanisha-Kari and Mohamed El Naschie. I think Dr. Renate Loll from the Dept. of Physics, University of Utrecht, Holland also owes a big apology for the witch hunt El Naschie was subjected to. She, more than anyone else, knows the reason and the force behind it.

  28. Conrad August 26th, 2009 8:51 pm

    It is not the first time that the proceeding of the Royal Society cites the work of Egyptian engineer and theoretical physicist Mohamed Elnaschie but it is the most significant. The paper in question is by a rising star in the field of fundamental physics who is a Fellow of the Royal Society, Prof. T.N. Palmer. Palmer’s paper is titled: The invariant Set Postulate: a new geometric framework for the foundations of quantum theory and the role played by gravity. It is published in the Proceedings of the Royal Society. A published online 29 July 2009. doi:10.1098/rspa.2009.0080. Most people like to stand on the fence. They rather say wishy washy stuff rather than take a definite position when it comes to the foundation of quantum mechanics. In these few lines I am going to risk a true opinion and say without any qualification that Prof. T. N. Palmer’s paper is the most important paper ever published on the subject in the proceedings of the Royal Society since the pioneering work of Paul Dirac. His work is probably the first confirmation of the fundamental conjecture made by Ord, Nottale and El Naschie that spacetime is fractal. El Naschie made this statement more precise and stated that spacetime is an infinite dimensional but hierarchical cantor set. El Naschie’s work is a direct realization of Alain Conne’s non commutative geometry and Von Neuman’s dimensional function. Some of Palmer’s conclusions are however highly contentious. Why use black hole and gravity instead of resorting to Rene Thom’s and Mohamed El Naschie’s VAK? Vacuum fluctuation is more fundamental than gravity I would have thought. Another even more important weak point is the sharp distinction between state space and spacetime. Palmer’s early professional work was in meteorology. This may explain that he is not aware of the vast literature on the subject of fractal and cantorian spacetime. He should look into the work of Ji Huan He in Shanghai. A recent paper by He gives a clue about the unified picture which emerged in Cantorian spacetime where phase state space and spacetime are indistinguishable for infinite dimensional topology. The next few years will be very exciting years I am sure. We will witness a trend towards unification of the points of view of Palmer’s quantum mechanics, El Naschie’s cantorian spacetime and ‘tHooft’s dissipative quantum mechanics. I for one am looking forward to this development. This is very likely to be a new revolution any case because of new experimental evidence for Cantorian spacetime in the cosmic high energy data as found by American scientist Ervin Goldfain.

  29. Mark September 27th, 2009 12:24 pm

    I was browsing in a German blog although my German is not that good when I was accosted by an incredible ignorance of basic mathematics. Because the attacks on Prof. El Naschie have reached the level of diminishing return, those professional defamation experts turned their attention to Prof. Ji Huan-He from Shanghai University. Prof. He wrote a profound paper on the Hilbert cube. The blog master and some of the commentators took arms against the notion that a Hilbert cube actually a Hilbert Peano cube is a curve. They found it absurd that something could be a curve and a cube at the same time. They thought in their infinite ignorance that they have at hand the proof that El Naschie and his follower are talking nonsense. Little did they know. How can you start criticizing anyone so viciously when one is not familiar with the basic notions and fundamentals of fractal geometry? Respected colleagues a Serprinski casket is not a casket it is a curve. To be accurate it is a two dimensional curve. A Menger sponge is not a sponge, it is a curve albeit it is a curve in three dimensions. Similarly a Hilbert Peano cube is not a cube, it is a curve. It belongs to a well known category of paradoxical geometrical object called space filling curves. You should learn your Shakespeare before starting to criticize him. If the brains of those criticizing the theory of El Naschie would be as large as their ego and hatred, they would not have made these foolish comments and would have understood E-Infinity theory.
    G. Mark

  30. Anon November 5th, 2009 6:07 pm

    The Proprietor of this blog may like to have a look at Sarah Limbricks article dated 2 Nov 2009. It is clear from this article entitled Editor of Scientific journal sues Nature that El Naschie has taken serious legal steps against the subject matter of your blog. El Naschie has hired one of Englands leading libel experts and a well established firm Collyer Bristow of London. It will definitely be a long and costly legal battle. However it is now clear to any level headed person that El Naschie must have profound reasons to take this step in the High Court. I think it is the new culture of internet defamation which must be stopped. Without the internet the allegations made by N Category Cafe could not have been possible and consequently this entire regrettable affair.

  31. Anon November 6th, 2009 2:50 pm

    Sarah Limbrick would surely be interested to know what the leading libel expert in England had to say about the Nature article complained of. He said he is in a state of disbelief that the worlds most respectable scientific journal Nature should publish an article which bears all the hallmarks of the tabloid press. Another interesting point is the conspiracy theory linking the plagiarism of El Naschies work published in Scientific American with the Nature article as well as a far worse article published in Die Zeit. Interestingly all of these three publications are owned by Macmillan. I understand from confidential sources that a mega surprise will be released at the trial engulfing highly reputed names some of whom are Nobel laureates. The site is http://www.pressgazette.co.uk/story.asp?sectioncode=1&storycode=44545&c=1.

  32. Annoymous November 19th, 2009 7:24 pm

    Here is what a great Swedish scientist Tonu Puu said about Mohamed El Naschie or M.S. El Naschie in his beautiful book Art, Science and Economics published by Springer in 2006. The last paragraph of the preface says I was most honoured when some time ago Professor Mohammad El Naschie, Editor-in-Chief of one of the most exciting and successful journal publishing ventures (Elsevier) I have ever been involved in, suggested to publish this essay in parts in the shape of articles, though I naturally prefer the present connected publication form. This does not sound like a man who is a fake or imposter. I am sure people high up in the hierarchy of Nature will have to reconsider their position towards journalists working from outside British jurisdiction with its strict libel law and spreading unfounded allegations causing their Headquarters a great deal of embarrassment not to mention financial losses in legal fees.

  33. David November 30th, 2009 3:19 pm

    It is not just a matter of funding which is behind the defamatory article in Nature. I think prestigious prizes also play a fundamental role. I read that somewhere on the net but strangely it was removed. The prize in question seems to be the Nobel Prize. Some say that not even the devil could have thought of something as harmful to science as the Nobel Prize. They reckon it is not the prize money itself but the publicity which the Nobel Prize brings. In turn this is translated into money. Noting the recent discovery of the sleaze at the Ivy Leagues in the US I am not astonished that this cash is badly needed. If this is true for Harford why should it not be true for the Einstein Inst. in Berlin or the University of Utrecht in Holland. The names involved with Mohamed El Naschie are quite interesting. Somebody wrote yesterday that he would not be astonished if a best seller comes out of this horrific story in the next few years. It is alright for some. The same writer said try as hard as he can, he simply cannot fathom how the Editor in Chief of Nature could allow this tabloid piece to be published in his journal. He must have his reasons or he had a very deep snooze. He added that he may have had too much respect for Nature just as he used to have for the Nobel Committee. This implies that he has none any longer which is interesting. Finally the writer noted that Mohamed El Naschie had no vested interest and certainly no materialistic interest in publishing his papers because he was sufficiently rich and famous before turning to theoretical physics. The author of this remarkable comment closed by noting that any successful engineer who leaves engineering to be become a theoretical physicist should have his head checked. In other words, he doubts the sanity of theoretical physicists, Mohamed El Naschie included.

  34. Aphyr January 20th, 2010 3:35 pm

    I’m sorry to say, but this nifty little cube is not related to the Banach-Tarski paradox. BT relies on the Axiom of Choice, and states that through a finite number of rotation/translation steps on complicated subsets, you can construct two balls of volume V from one of volume V.

    The Yoshimoto cube transforms from one solid cube of volume V into two cubes with missing interior volumes, each a polyhedron with volume V/2. No AC required, no unmeasurable sets, no volume duplication–it’s just a neat folding trick.

    Still looks really cool. :)

  35. Never Yet Melted » Yoshimoto Cube January 22nd, 2010 10:34 am

    […] tip to Forgetomori, forwarded by Robert Breedlove. By JDZ Feedbacks on this entry via RSS 2.0 Please leave a […]

  36. Manny February 6th, 2010 9:36 am

    I am pleased that the truth has prevailed. Nature is now accused of trying to undermine Mohamed El Naschie deliberately. This accusation is not frivolous. How else can we explain the blind vicious attack by certain doubtful blogs on the golden mean work of El Naschie and how Quirin Schiermeier the journalist working for Nature utilized these vicious attacks to write a completely unacceptable article in Nature. Then came the heavenly justice when a German professor von Storch complained on his blog that the Nature article of Schiermeier deliberately misquoted him. He was gentle enough to say that the harm was not great. However in principle the harm could have been great. No one has the right to smear the reputation of anyone whether deliberately or recklessly due to irresponsible journalism. Now to the burning scientific question. How does the golden mean enter into quantum mechanics. The answer is as simple as it is ingenious. Mohamed El Naschie reformulates quantum mechanics in spacetime following the same concepts used by Richard Feynman as well the classical work of Einstein. Since the building blocks of spacetime are his elementary random Cantor sets and because these random Cantor sets possess the golden mean as a Hausdorff dimension, the golden mean slips into the fundaments of quantum mechanics. Nothing that quantum mechanics is the most fundamental theory upon which science is based, the golden mean could rightly be described as the basis of science. From this reasoning the ideas which Ed Nash expressed in his previous comment follows effortlessly.

  37. annonymous February 20th, 2010 3:20 pm

    Dr. Ray Munroe, a theoretical physicist with some understanding of El Naschie’s theory recently said on one of his comments I think in the Fqxi blog that only time will tell if El Naschie’s theory is correct. This is a sentence which is lovely and vague. It is a political sentence, not a scientific one. I guess Munroe knows very well that El Naschie’s theory is correct. He just finds it politically risky to identify himself with a theory about which he himself has written in Chaos, Solitons & Fractals when the author of the theory is under attack from Nature’s tabloid article. When science changes to politics then it is time for scientists to take arms against politics. The difference between politics and science is that in science you do not need to wait for time to prove it. All that you need is to compute it. A simple back of an envelope calculation would immediately show that a golden mean topology and geometry is the only logical solution for the two-slit experiment. If the building blocks of the geometry and topology of quantum spacetime is governed by the Hausdorff dimension of these blocks, then no wonder that this Hausdorff dimension will manifest itself in everything physical. The Hausdorff dimension is the golden mean. Consequently the most fundamental theory, namely quantum mechanics must be based on a golden mean spacetime. Such spacetime differs completely from the classical spacetime. It differs also from spacetime of general relativity. I cannot say that it also differs from the spacetime of quantum mechanics because classical quantum mechanics is not a spacetime theory. Therefore El Naschie’s E-infinity theory elevates quantum mechanics to a spacetime theory just like relativity and classical mechanics. Interestingly depending on the resolution you can obtain all geometry of all the three fundamental theories. You can move from 0.618033989… to a rational value of 0.5 and under circumstances unity and find all the limiting behavior you wish to discover and explore. It is that complex and yet that simple. In fact cellular automata is just another form of the above based on computers. Nature consists of very simple rules. You reiterate these rules trillions of times and then you obtain the complexity which is visible in the classical world. The simple rules discovered by E-infinity are that using the golden mean Hausdorff dimension which represents an elementary random Cantor set can produce the result of a cellular automata with infinite capacity. In other words you do not even need a computer. The universe possesses an infinite computer because it possesses the golden mean. You probably recall the work of Renate Loll popularized in a Scientific American article published in 2008. What Renate Loll, Jon Ambjorn and their colleague did in this remarkable paper was reproducing some aspects of Mohamed El Naschie’s work without using golden mean Cantor sets. They replaced this powerful mathematical tool by the best computer capacity they have at present. If they want to find precisely the same result found by El Naschie, then all what they need is to find an infinitely strong computer. I am sure the reader sees that this strategy is misguided and upside down. Instead of finding an infinitely strong computer, we should find a theoretical solution. This theoretical solution happens to be the golden mean geometry and topology of E-infinity theory.

  38. E-Infinity March 11th, 2010 1:42 pm

    A Slovenian scientist and mathematician following Mohamed El Naschie expand the idea of mechanical oscillators. Many papers have been published on this subject by L. Marek-Crnjac. Take a two degree of freedom oscillator. Two masses connected by two linear springs. Write the equation of motion. Set the value for the masses as well as the spring constants equal unity. The secular equation is then simply a quadratic equation. The Eigen values are golden mean related. The only positive real Eigen value is the golden mean. Imagine now that you have infinitely many such oscillators connected together. Consequently you can estimate the Eigen value using two well known theorems on Eigen values. These are the Southwell theorem and the Dunkerly theorem. They correspond to what we have studied in school about joining electrical resistance of Ome’s law. When they are successive you add the inverses and when they are parallel you add them. Eigen values are frequencies. Frequencies are energy and energy is mass. Extrapolating the whole thing to quantum mechanics as argued by El Naschie and Marek-Crnjac you have another plausibility explanation for why the golden mean will pop up in any accurate measurement in quantum mechanics phenomenon.

  39. Mike October 9th, 2010 3:58 pm

    If you want to unify relativity with quantum mechanics, you have to start by making quantum mechanics understandable. Compared to quantum mechanics, relativity is quite classical. It is a spacetime theory and there is nothing of the totally counter intuitive result of quantum mechanics in relativity. Yes we understand time travel and twelve paradoxes and so on. None the less this is nothing compared to the paradox of the two-slit experiment and particularly wave collapse. Probability wave – what is that? However there is something new on the horizon. A major breakthrough in understanding wave collapse. This is the least we can say about this new profound discovery. The most astonishing thing about it is why it was not discovered long ago. In a nutshell the essence of the argument is as follows: A quantum particle may be modeled as a point. However it is not any point. It is a Cantor point. That means it is a fractal point taking out of Laurent Nottale’s or Garnet Ord’s fractal spacetime. Consequently it is a point but much more than a point at the same time. Every Cantor point or fractal point is by virtue of self-similarity a point representation of the entire universe, i.e. the fractal universe upon sufficient magnification. This zooming process, as explained by Nottale, has no end. This is all well known stuff from the theory of fractals. Now comes the crucial point. Since this point is nominally a point we take it to be mathematically the zero set and physically to be a quantum particle. Now the boundary of the zero set is the empty set. The empty set has no element what so ever and is given in the classical theory a dimension minus one. Never mind all these numbers. The important thing is just to keep in mind that a Cantor or a fractal point represents a quantum particle and that the boundary of this quantum particle is the empty set. It comes as no surprise that El Naschie and his E-infinity group propose that the empty set is just the mathematical name for the probability wave function of quantum mechanics. Such a wave function is devoid of energy, matter and momentum to the extent that it mystified all physicists and led Einstein as well as Bohm to call it a ghost wave. There is even a theory by both men called the guiding wave theory. The guiding wave is nothing but the empty set. So far so good. Here comes the resolution of the wave collapse problem say the group of E-infinity researchers. Any attempt to locate the quantum particle will include interference with its boundary. Since its boundary is the empty set, then any interference will make the empty set non-empty. Consequently the empty set ceases to exist. On the other hand the empty set is our quantum wave function. It follows as a trivial result that when the empty set vanishes because it becomes non-empty, then the wave function also vanishes. The group of E-infinity did not stop at this disarming explanation of the wave collapse. Using the Menger-Urysohn and the Hausdorff dimension of the zero set and the empty set, they are able to make convincing calculations and derive the topology of the spacetime manifold which allowed such physics involving the empty set wave collapse. You can read about that in preceedings of a conference in Shanghai http://www.isnd2010.com and http://www.msel-naschie.com. With a theory like that we are in a much better position to start unifying quantum mechanics with relativity and produce a real theory of quantum gravity. At least there is more hope that way.

  40. G. Mahdi November 6th, 2010 8:56 pm

    Quite honestly I trust Prof. Mohamed El Naschie far more than I trust the judgement of Nature. I do not consider it outlandish to suggest a Nobel prize for El Naschie. Let me give you the rationale behind my conclusion. Surely you heard about the two genius Russian born scientists Andre Geim and Konstantin Novoselov. I was present at a talk involving the two lucky but deserving winners of the Nobel Prize of this year (2010). What they recounted speaks for the magazine Science and speaks against Nature. They said their paper was eventually published when they submitted it to Science. They added it was rejected twice by Nature. Something very unnatural is happening to Nature. Instead of publishing defamatory tabloid articles against El Naschie they should have at least published the paper of the two ingenious Nobel winners of this year. Nature also refused to publish the paper reporting the experimental discovery of the golden mean in quantum mechanics by the Helmholtz Center and Oxford University. It was again Science which published the article. A year or so later after the publication in Science, Nature overcame themselves to publish a short article about the article published in Science. When one of my colleagues who is aware of El Naschie’s papers and findings on the golden mean in quantum mechanics wrote a comment about El Naschie’s achievement and sent it to Nature, they refused even to acknowledge the bare facts. It is really becoming a personal war declared by Nature against El Naschie. The looser is the scientific community and science. In fact Nature is losing in a big way by publishing doubtful papers on climatology and ignoring fundamental and path breaking work because they will otherwise be forced to mention the name of El Naschie. It is unreal. Who thought Nature would behave in such a fashion? There seems to be something fundamentally wrong with the way Nature is being run at present. Of course Nature is the world’s most famous science magazine for everyone. The sooner they undertake self correction, the better it will be for everybody.

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