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It’s a Beautiful Thing. But Why?


This is what beauty looks like:


We were cruising, surfing, when we came across a blog, The Wit of the Staircase or esprit d’escalier, inspired by “the perfect witty response you think up after the conversation or argument is ended”. It used to be run by a New Yorker called Theresa Duncan. (Both she and her boyfriend recently died, separately, under odd circumstances, though both are assumed to be suicides. (They claimed they were being pursued by the Scientologists.) She was a computer game designer and perfume afficionado. Her website seems to have a preponderance of Kate Moss images.

We stumbled upon this photograph on her website. We don’t think it is Kate Moss; in fact we have no idea who it is or who took it or for what. We don’t know whom to thank.

What we do know is that we find it a remarkable – an arresting – image.

Our initial response was that there is something about it that is “right”. Just exactly right. Balanced and harmonious. Why? Gradually the mind stirred and reminded us of the Golden Section (or Ratio, or Mean). Could that be it?

So we decided to do a little digging.

The Golden Ratio is roughly 1.62 : 1, or 1 : 0.62 or very roughly 8 : 5. (What they say is that it is “approximately” 1.61803398874989484820458… : 1)

So we applied those numbers to the picture and the result was startling.

Time after time, relationships in the picture seem to fit perfectly into the Golden Ratio. And ultimately the focus narrows down to the girl’s face. Did the photographer do this intentionally? Perhaps. Or perhaps the composition simply “felt right”, which is just as likely, because this Golden Section turns up in art, in nature, in science and in mathematics. The “rightness” of the ratio seems to be organic, built into how we see the world.

Here’s how it works.

Golden Rectangle Construction

a+b:a :: a:b.
Where the initial figure is a unit square, the ratio a+b:a and a:b, where b=1, is known as Phi (φ).

The unique solution to these equations is φ=(√5 + 1)/2 or 1.61803398874989484820458 . . .

Amazingly enough the inverse ratio is 0.61803398874989484820458 . . . that is, 1/φ = φ-1 . . .

On top of which φ2 = φ + 1 (2.61803398874989484820458 . . .).

I mean, for goodness sake, this is a magic number!

And the magic doesn’t stop with the maths.

Our sense of beauty seems to be determined by the Golden Section. Look at these beautiful women.

Our own Golden Cate Blanchett

Naughty Nude Vanessa Hudgens:

Even the Mona Lisa

Perhaps Matt Damon?

In fact, amazingly enough, the entire proportions of a “perfect woman’s body” (women in particular – don’t know why, maybe men tend to prefer studying women’s bodies) seem to be altogether in proportion to the Golden Ratio.

Even in the most subtle ways the Golden Ratio seems to apply when what we see is “beautiful”.

Kate Moss - mouth and eyes in golden proportion

In a beautiful face, such as Kate Moss’s, the ratio of the width of the eyes to the width of the mouth is 1 : 1.6…

But there’s more.

The Golden Section and the Fibonacci Sequence

Now, the Golden Ratio is related to the Fibonacci Sequence and we didn’t know that, so we’re still in shock at just how closely they are related.

If you create a spiral with the Golden Ratio you get a curve which is almost identical to the famous Fibonacci spiral.

Fibonacci spiral

The way things are ordered in nature is often based on the Golden Ratio/Fibonacci sequence.

Things like the way a pineapple’s skin is organised.

Or a Nautilus shell

Or a sunflower.

The Fibonacci sequence is generated by adding the two preceding numbers to create a third number. So you get 0,1,2,3,5,8,13,21,34, etc.

Extraordinarily enough, the two curves, Fibonacci and Golden Ratio, which are generated by entirely different operations turn out to be almost identical.

Fibonacci and Golden Section compared

And most amazingly of all, the ratio between numbers in the Fibonacci Sequence – generated quite differently from the Golden Ratio, remember – very quickly approaches the Golden Ratio. After only 26 operations it is already 1.618033989… (compared to φ=1.61803398874989484820458…). This strikes us as being extraordinary, but what do we know. There’s probably some perfectly simple explanation.

But it doesn’t stop there. They say if quantum physics makes sense to you you’re not getting it.

Well check this. A fairly recent test to see, as we understand it, whether quantum theory is correct. So nothing too serious, right?

Hardy’s test of quantum mechanics
Dan Styer, Oberlin College Physics Department

Hardy’s experiment is in many ways the cleanest test of quantum mechanics ever performed, because it is a sensitive probe of the most vulnerable aspects of the theory. In Hardy’s experiment a particular outcome never occurs according to local determinism and sometimes occurs according to quantum mechanics. The experiment, a variation on the Einstein{Podolsky{Rosen idea, was proposed by Lucien Hardy in 1993 and performed by Paul Kwiat and his coworkers at the Los Alamos National Laboratory in early 1999. The results were a thrilling conformation of quantum mechanics:¹ the outcome that would never happen under local determinism did in fact sometimes occur, and the measured probability of occurrence was exactly the probability predicted by quantum theory.

Aside: One thing that intrigues me about Hardy’s test is the mathematical origin of the probability 0.09017. . . . The number is g5, where the constant g is equal to (√5 – 1)/2 = 1.6180… and is called the golden mean”.
¹Our emphases

I mean, it’s even there in quantum probability? That’s preposterous, surely. Does this mean that the Golden Ratio/Section just is the very fabric of SpaceTime?


When applying a magnetic field at right angles to an aligned spin the magnetic chain will transform into a new state called quantum critical, which can be thought of as a quantum version of a fractal pattern. Prof. Alan Tennant, the leader of the Berlin group, explains “The system reaches a quantum uncertain — or a Schrödinger cat — state. This is what we did in our experiments with cobalt niobate. We have tuned the system exactly in order to turn it quantum critical.”
By tuning the system and artificially introducing more quantum uncertainty the researchers observed that the chain of atoms acts like a nanoscale guitar string. Dr. Radu Coldea from Oxford University, who is the principal author of the paper and drove the international project from its inception a decade ago until the present, explains: “Here the tension comes from the interaction between spins causing them to magnetically resonate. For these interactions we found a series (scale) of resonant notes: The first two notes show a perfect relationship with each other. Their frequencies (pitch) are in the ratio of 1.618…, which is the golden ratio famous from art and architecture.”

OUT-OF-DATE DELETEDAnd there’s just one more thing. If there were beauty in politics it would be a nicely proportioned swing of about 7.8%. Majority of 38 : Labor 93, Coalition 55, Independents 2.

It’s a beautiful thing.



[tags]Golden Section, Golden Ratio, Golden Mean, Scientology, religion, Naughty, Nude, Vanessa, Hudgens, Quantum Mecahnics[/tags]

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Comment from Nicholas Gruen
Posted: 23 December, 2007 at 4:58 am

Completely fabulous piece – thank you very much.

Just as a small bit of corroboration, I went to the linked site before I read your piece and my eye hit on the very picture you highlight in this post – though its beauty didn’t strike me nearly as much as the affected strangeness of the images.

But one request. I think Cate Blanchett is a fine actress but I don’t think she’s uncommonly beautiful. But that’s just my opinion I know. Similarly no doubt lots of people disagree with me but Matt Damon doesn’t look anything too special to me either. Reasonably symmetrical but not much more than that.

But it is true that there’s a lot of alignment, even across cultures, as to what makes a beautiful face. Could you please take a few ordinary faces and show how they don’t measure up in your method. It seems that you have some room to play with as to how you draw the golden ratio rectangles, and I’d be more convinced if I was sure that only or mostly those who are widely regarded as beauties were the ones that come up on your goldenmeanoscope.

Comment from roger migently
Posted: 23 December, 2007 at 2:30 pm

I think you’re right that Cate is not extraordinarily beautiful in the ordinary way but there’s something special about her, to me at least. I included Matt Damon for balance more than anything. Apparently he’s attractive to women. He just looks like a nice guy to me.
I will do as you ask when I have time. My initial feeling is that it may be difficult to demonstrate convincingly, given the admittedly “roomy” way I interpreted the ratio, as you noticed.
On the other hand, while I was writing the post I had in mind a documentary I had seen with Clive James John Cleese exploring the geometry of beauty with Elizabeth Hurley as his model. So that’s some sort of support. I’ll try to find a reference to it.

Comment from Kevin Sullivan
Posted: 2 April, 2008 at 2:18 pm

Yes there is a golden mean. I was obsesed with such things as a child. constantly drawing ever decreasing circles.
As an adult i became an artist and sculptor and utilised those meathods to much success. but these days i am beginning to suspect that it has something to do with the retina of the eye. The focal point is placed by a mono viewing photographer to avoid the blind spot.
If you placed retinal scans as overlays to your images you would see similarities.

Comment from David Byrden
Posted: 2 October, 2008 at 2:18 pm

Well, this doesn’t impress me at all.
The Golden Mean is probably the most uninteresting of all ‘special’ numbers. It’s very easy to see how it works and to calculate it.
As for the human faces? I checked one. You drew horizontal lines through Cate Blanchett’s chin, lips and eyebrows. The ratio between them is 1.8, which is not even close to the Golden Mean.
You drew additional lines through her nose and hair, but I don’t take those lines to be meaningful because there is no exact feature to pin them down precisely.

Comment from Biblio
Posted: 10 January, 2010 at 9:13 pm

A number is either irrational, or it isn’t. Binary thing. Fact.
—However— the golden mean is the “most irrational number”, in the sense that it is the most difficult to aproximate using rational numbers. Go ahead, search it up, you’ll find it.

The golden mean has a TON of weird properties. I will not list them, they are legion, and there are excellent math sites online.

—BUT— I’ll say this: in the context of order, chaos, and criticality, being the “most irrational number” is THE property you want to pay attention to.

In total order, nothing happens, nothing grows.
In total chaos, nothing grows either.
Things grow somewhere *in between* those. See “edge of chaos”, “(self-organized) criticality”. You will see.

And in there, you’ll always find the golden mean.
Because it is the most durable, resilient structure. The one that does not collapse on the onset of chaos. (technically, the last KAM tori to collapse on the onset of chaos)

Comment from Biblio
Posted: 10 January, 2010 at 9:19 pm

As for why sometimes, when searching for fibonacci/golden mean in nature, things do not turn out totally exact, the answer is simple: things TEND towards golden mean. A tendency, you see? Like a limit.

For example, when you see plant seeds, sometimes you do not quite find the fibonacci sequence. But you find the lucas sequence, instead…which is almost the same.

Once again, go ahead, dig it up, you will see.

Comment from roger migently
Posted: 10 January, 2010 at 11:13 pm

Thank you for this food for thought and learning

Comment from Ed Nash
Posted: 5 February, 2010 at 7:18 am

There is a highly interesting connection between science and religion which comes out as a by-product of El Naschie’s discovery of golden mean geometry and quantum golden mean theory. For a few hundred years facts and values were separated. Science is the world of facts. Morality and art is the world of values. Since Islamic science and later on Francis Bacon, the scientific method dealt only with facts. Anything dealing with values was called philosophy or religion even when facts are also involved. Needless to say art, music, emotions and beauty are all things related to values. It follows then that the golden mean which was used by the Egyptians, the Greeks and the Renaissance artists is a fundamental tool in the world of values. It is fundamental to art and consequently emotions, morality and religion. It has something to do with good and bad. You cannot give a definite objective reason for something being beautiful. Never the less we know and feel the beauty of a golden mean composition whether it is in the visual or audio arts. Here comes then the surprise. To discover the golden mean as the basis of quantum mechanics is tantamount to discovering that there is a bridge between the world of values and the world of facts. The experimental confirmation of the golden mean in quantum mechanics and the basic correctness of Mohamed El Naschie’s theory of Cantorian spacetime, which is based on the golden mean, is an irrefutable proof of what people sensed long ago but were not able to demonstrate in a mathematical way, namely that art, religion, mathematics and natural science are somehow deeply interconnected. The experimental work at the Helm Holtz Inst. in Germany and the theory of El Naschie and his colleagues is a radical departure from an old paradigm which lasted for over a thousand years. We can no longer claim that there is no connection between love and reason or emotions and facts. We do not yet have a complete proof. We are not even at the beginning. However with El Naschie’s theory experimentally confirmed, we may be at the beginning of the beginning of a new era in science, philosophy, art and religion.

Comment from Adam
Posted: 7 February, 2010 at 7:19 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.

Comment from Erez
Posted: 15 February, 2010 at 5:09 am

You have a typo in your article:

“One thing that intrigues me about Hardy’s test is the mathematical origin of the probability 0.09017. . . . The number is g5, where the constant g is equal to (?5 – 1)/2 = 1.6180… and is called the golden mean”

TRUE: g is equal to (?5 – 1)/2
FALSE: (?5 – 1)/2 = 1.6180…
TRUE: 1.6180… is the golden mean

The constant g is equal to (?5 – 1)/2 = 0.6180… = 1/? = ?-1.

(I assume you quoted Dan Styer incorrectly, and that Dan Styer’s original text does not contain the typographical error. If I’m wrong there, you might want to add a note correcting his text.)

Comment from roger migently
Posted: 15 February, 2010 at 9:37 am

I have checked that I quoted Styer correctly. You can see his paper here:
I’m not sure (being somewhat ignorant, if nevertheless enthusiastic, in these matters) what the question marks in your last line are showing.
I think the misunderstanding may occur because 0.6180… is the exact inverse of 1.6180… so in a sense either can be thought of as the golden mean (1:1.6180… :: 0.6180…:1). But you’re right. 1.6180… is phi.
Thanks for pointing this out.

Comment from Annon
Posted: 23 February, 2010 at 7:24 am

Let’s be more specific so that we are not lost in generality. If I may refer to the following paper of El Naschie which I am sure you have seen (On the exact mass spectrum of quarks. Chaos, Solitons & Fractals, 14, 2002, p. 369) then I could say the following. Four dimensional fusion algebra shows that it leads to the conclusion that we have four fundamental values. Two correspond to unity and the other two correspond to one plus the golden mean which is equal to the inverse golden mean. El Naschie identifies the ratio of the constituent masses of the down and up quarks with unity while the current masses correspond to the inverse golden mean. You can see the detail of the calculation in this 2002 paper dedicated to the 85th birthday of Nobel laureate Ilya Prigogine. When you read a little bit more you realize that the four dimensional fusion algebra comes out from topological quantum field theory when obtained from sub-fractals. El Naschie showed long ago that this is equivalent to his E-infinity theory. I beg to differ with you profoundly and insist that everything follows naturally from E-infinity theory. This is the simplest theory that there is from which E8 as well as Klein’s modular curve and all these holographic boundaries follow. The golden mean is in E8 but El Naschie is talking about another E8 which has the golden mean manifesting itself as a fuzziness. Let me be precise. The normal E8 with its hidden golden mean has a dimension 248. The fuzzy E8 of E-infinity theory obeys an additional simplictic requirement. It is equal to 248 minus half k square where k is equal to the golden mean power 3 multiplied again with the golden mean power 3 minus 1. This is exactly equal to 0.016261. Consequently the dimension of the fuzzy E8 is 247.983739. As I said this small difference follows from the requirement of the area preservation, that is to say, Hamiltonian dynamics with no physical friction. The only friction in the system ensuring stability is the maximal irrationality of the golden mean. E-infinity theory is simplicity par excellence. All what you do is to start designing your geometry from elementary random Cantor sets. By a well known theorem elementary random Cantor sets have a Hausdorff dimension equal to the golden mean. Nothing more is required. Everything else follows in the most elementary fashion. It is fascinating to see how E8, sub factors, topological field theory and noncommumative geometry follow without needing any sophisticated mathematical apparatus. The only comparable simple road to quantum gravity is Penrose tiling when compactified. Everything is there and nothing more is needed. The E12 of Munroe follows in the same way. I am afraid the malicious propaganda against Mohamed El Naschie, whether he survived it or not, worked as brain washing for anyone who wanted to have an open mind and look at this theory on the basis of its mathematical and physical merits. The experimental work in Germany shows that the ratio of two quasi particles termed meson is the golden mean. You just need to look at the equations of El Naschie of the said paper on page 373. You will find that the ratio between the mathematical masses of the up and down quarks is equal to the golden. That means that the ratio between the down and up quarks must be the inverse of the golden mean. Whether the equations are numbered or not, things are so clear that there is no room for misunderstanding. The result found experimentally in Germany is a natural consequence of the fundamental findings of E-infinity theory. In this particular paper I mentioned, four dimensional fusion algebra was used. I guess El Naschie tried in vain to bring his theory nearer to those who may be more familiar with sub factors. It is clear that science does not use only scientific criteria in judging the work of scientists. I am sure that even Munroe himself has a story or two to tell about this from his own life experience.

Comment from roger migently
Posted: 24 February, 2010 at 12:08 am

All very enjoyable I’m sure, Mr Sokal. In the interest of transparency, we feel it is our duty to put el Naschie in perspective. He is widely considered a hoax, or a crackpot and self-aggrandizer as comments at Slashdot make clear:

“It is well known among scientists that the impact factor of a scientific journal is not always a good indicator of the quality of the papers in the journal. An extreme example of this was recently uncovered in mathematics. The scandal is about one El Naschie, editor in chief of the ‘scientific’ journal Chaos, Solitons and Fractals, published by Elsevier. This is one of the highest impact factor journals in mathematics, but the quality of the papers in it is extremely poor. The journal has also published 322 papers with El Naschie as (co-)author, five of them in the latest issue. Like many crackpots, El Nashie has a kind of cult around him, with another journal devoted to praising his greatness. There was also a discussion about the Wikipedia entry for El Naschie, which was supposedly written by one of his followers. When it was deleted by Wikipedia, they even threatened legal actions (which never materialized).”

The explanation of el Naschie’s “work” above does put us in mind for some reason of the Sokal Hoax:

In 1996 Social Text published Alan Sokal’s article, “Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity“, apparently not realizing that it was actually a send up of much of the writing — ideologically based, relativistic, jargon-laden — that had become popular in certain academic circles. The article is almost unreadable and certainly nonsensical and though Social Text‘s editors later professed they had not been all that enthusiastic about it they embraced and published it without questioning it.

Comment from Annon
Posted: 24 February, 2010 at 5:28 am

It is interesting that everyone is now talking about the golden mean and the golden mean E8. Mohamed El Naschie was the first to intimate the relationship between E8 and the golden mean and subsequently introduced a fuzzy E8 based again on the golden mean. He work does not only apply to the Ising model which was tested experimentally in the Helmholtz Center in Berlin but to everything else as well. You could say he found a general theory based on the golden mean for high energy physics. Some call it quantum golden field theory. You can read much more about it in details from a very accessible short review The theory of Cantorian spacetime and high energy particle physics (an informal review) published in Chaos, Solitons & Fractals, 41, 2009, p. 2635-2646.

Comment from Jonathan
Posted: 24 February, 2010 at 5:37 am

To Roger Migently There may or may not be a cult around Prof. Mohamed El Naschie. As a scientist I would be very reluctant to use this language. If students or colleagues their teacher or mentor, what is so wrong about that. If there is some loyalty towards truthfulness, I do not think this should worry anyone. What is wrong about writing love letters to someone you admire. Is that not better than the blind hatred and envy and the scores of threats including assassination issued on certain blogs against people who have done nothing except science. What about the cult of global defamation for which a certain person Jason who clearly acts on behalf of his paymasters. Have you seen his blog? If you are not easily nauseated, please have a look at Jason’s blog entitled Watching El Naschie. Instead of counting pages, why don’t you read El Naschie’s work and make up your mind for yourself. More importantly why don’t you let other people read it and make up their own minds. I am afraid your real name is Said or you are somebody related to him or John Baez or Renate Loll or Christoph Drosser or maybe even Quirin Schiermeier. These names pop up continuously on the net proclaiming the message of hatred. This is a phenomena worth studying. What is driving this hate? The Nobel Prize, racial discrimination or religious fanaticism? Take your pick. It is beyond me to understand.

Comment from Nick
Posted: 16 May, 2010 at 3:43 pm

I think that beauty is a matter of, at least to a certain extent, personal opinion and cultural preferences. I’ll agree that there is something ‘right’ about that picture, but I don’t get that feeling with Cate Blanchett, intriguing to look at though she is.

Comment from roger migently
Posted: 17 May, 2010 at 7:50 pm

There is merit in this view but perhaps the cultural preference is the layer above the mathematical one. If the face or figure conforms to the golden mean (that is, it is prima facie attractive and worth considering further) perhaps that’s when the cultural preferences come into play. You might prefer a darker or lighter skin, for example, straight compared to curly hair, heavier rather than fine bone structure. Just a thought. But I do think that some people just seem “right”, even if not what a lot of people would call typically beautiful.

Comment from Yansen
Posted: 3 July, 2010 at 7:38 am

Two very informative papaers explaining in layman terms E-infinity theory and the role played by the golden ratio in fiber wool and high energy physics are the following:
Hierarchy of wool fibers and its interpretation using E-infinity theory
Chaos, Solitons & Fractals,
Chaos,Solitons and Fractals 41 (2009)1839 –1841
Ji-Huan He, Zhong-Fu Ren, Jie Fan, Lan Xu

Again the same article with little modifications
Hierarchy of Wool Fibers and Fractal Dimensions

International Journal of Nonlinear Sciences and Numerical Simulation,9(3),293-296, 2008

The two articles are authored by a great official organ of the E-infinity group who is Prof. Ji-Huan He.
It is a true pleasure reading these articles which puts everything into perspective. In fact the golden ratio was recently discovered to be enjoyed by sheep through their fiber wool and this is deeply rooted on the basis of conformal quantum field theory. This is all apart from its fundamental connection on polylogarithm. This is a beautiful theory with applications in quantum sheep’s wool and wooltex and related subjects. These recent discoveries should silence the critics of E-infinity theory provided this criticism is truly scientific and not just politically motivated to disturb sheep market and wooltex stores.

For the seriously motivated readers these are the abstracts of the two wonderful papers
First abstract:
Why do wool fibers show excellent advantages in warmth-retaining and many other practical properties? The paper concludes that their hierarchical structure is the key. Using E-infinity theory, its Hausdorff dimension is estimated to be about 4.2325, very close to El Naschie’s E-infinity dimension, 4.2360, revealing an optimal structure for wool fibers.

Second abstract
Wool fiber shows excellent advantages in warmth-retaining and many other practical properties possibly due to its hierarchical structure. Its fractal dimension of wool fiber is calculated which is very close to the
Golden Mean, 1.618. The present study might provide a new interpretation for the reason why wool fiber has so many excellent properties.

Some suspicious readers would ask why we have two different Hausdorff dimension for the same fiber wool.
-The answer is simple and it is that the two papers measured the Hausdorff dimension in two different frames. As any body know the dimension depend on the reference frame. This reveals that even dimension is a relative concept.

Comment from Mike
Posted: 10 October, 2010 at 6:04 am

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 proceedings of a conference in Shanghai and 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.

Comment from roger migently
Posted: 10 October, 2010 at 3:18 pm

Not even wrong

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