Fibonacci Sequence In Theoretical Physics

[onionsoflove]

Considering that the fibonacci sequence shows up everywhere in nature I would think that it would be an important inclusion in theoretical physics. Has it been included anywhere? I was thinking it would be kind of obvious that since it's a mathematical pattern showing up blatantly all over nature it would be a hot topic for theoretical physicists to discover?

[Don Blazys]

Quoting onionsoflove:

Considering that the fibonacci sequence shows up everywhere in nature

I would think that it would be an important inclusion in theoretical physics.

Has it been included anywhere? I was thinking it would be kind of obvious that

since it's a mathematical pattern showing up blatantly all over nature

it would be a hot topic for theoretical physicists to discover?

Physicists are indeed discovering relationships between patterns

found in number theory and patterns found at the quantum level. Here:

 

http://www.physorg.com/news182095224.html

 

is but one example (which happens to involve the Fibonacci Sequence and

the Golden Ratio of which you speak.) What really impresses me here is

how clever some of these physicists are about inventing, developing and

manufacturing the gadgets required to make these incredible measurements.

 

Don.

[Qfwfq]

I can't find mention of Fibonacci sequence in that web page. :shrug:

 

Considering that the fibonacci sequence shows up everywhere in nature I would think that it would be an important inclusion in theoretical physics. Has it been included anywhere? I was thinking it would be kind of obvious that since it's a mathematical pattern showing up blatantly all over nature it would be a hot topic for theoretical physicists to discover?

This isn't exactly a good reason to undertake mammoth efforts in search of it. It reminds me of the kind of guidelines used by medieval philosophers.

 

If theoretical physicists some day find the Fibonacci sequence useful, no doubt they will use it.

[Jay-qu]

When doing calculations in theoretical physics involving the speed of light, c, and if one finds themselves in the US stuck with units of miles per hour, you can work out the approximate answer in km per hour by multiplying by the golden ratio. An amazing connection.

 

:D

[Don Blazys]

I can't find mention of Fibonacci sequence in that web page.

 

Maybe this will help you "find it"...

 

The OP speaks of the golden ratio and the golden ratio is approximated as

the ratio between any two succsessive Fibonacci numbers!

 

Quoting Jay-qu

When doing calculations in theoretical physics involving the speed of light, c,

and if one finds themselves in the US stuck with units of miles per hour, you can work out

the approximate answer in km per hour by multiplying by the golden ratio. An amazing connection.

It's easy to be "snooty", isn't it?

 

http://www.telegraph.co.uk/science/large-hadron-collider/3314456/Surfer-dude-stuns-physicists-with-theory-of-everything.html

 

Don.

[Jay-qu]

I fail to see the point of your link Don, this theory is known to be wrong.

[Qfwfq]

The OP speaks of the golden ratio and the golden ratio is approximated as

the ratio between any two succsessive Fibonacci numbers!

This is much less consequential than the other way around.

 

Suppose that a rectangle ABCD is such that, if you choose P and Q along the longer two sides (which are AC and BD) with ABPQ being a perfect square, then rectangle PQCD has the exact same proportions as ABCD, so that: AC/AB = CD/PC. Is it necessary to employ the Fibonacci sequence for constructing such a rectangle?

[Don Blazys]

Quoting Qfwfq

Suppose that a rectangle ABCD is such that, if you choose P and Q along the longer two sides

(which are AC and BD) with ABPQ being a perfect square, then rectangle PQCD has the exact

same proportions as ABCD, so that: AC/AB = CD/PC.

Is it necessary to employ the Fibonacci sequence for constructing such a rectangle?

 

No, but the OP was speaking of how the Fibbonacci sequence is ubiquitous in nature,

and that would clearly involve the Golden Ratio as part of it's "generating function".

 

Quoting Jay-qu

I fail to see the point of your link Don, this theory is known to be wrong.

Incomplete perhaps, but not yet proven wrong. Opinions are divided on that,

but you know what they say about opinions... they are like... you know...

everybody's got one! Last I read about it, it is still being worked on.

 

But the point here is that it is not an uncommon suspicion that things in nature

are patterned according to simple (meaning in principle) mathematical constructs.

 

The universe may or may not come to an "end" (whatever that means),

but its logical and mathematical underpinnings will go on forever.

 

Don.

[Qfwfq]

No, but the OP was speaking of how the Fibbonacci sequence is ubiquitous in nature,

and that would clearly involve the Golden Ratio as part of it's "generating function".

This doesn't support you in having said that the article involves both, you weren't talking about the OP there except for saying it was an example of what the OP was looking for.

[Don Blazys]

I should have said "implied both".

Sorry.

[tetrahedron]

The Fibonacci sequence shows up in a number (sorry for the pun) of near fundamentals. I myself discovered that if you map Fib numbers to the Periodic Table, their placement is neither random nor arbitrary. Up to 89, ALL Fib numbers, taken as atomic numbers, map to leftmost positions in half-orbital rows within the PT's block structure. Each orbital row is split in two. The first part contains singlet electrons in the lobes, the second doublets. You don't get doubling of electrons until singlets fill all lobes. Odd Fib numbers go with the first singlet position (s1,p1,d1,f1) in the table structure. Even Fib numbers go with the first doublet position (s2,p4,d6,f8). At 144 the perfect mapping gets off track.

 

Additionally, related LUCAS numbers also map similarly as atomic numbers. But instead of leftmost, they prefer rightmost positions in the half-orbital rows, where we have the last singlet or doublet electrons. Fib first, Luc last. Again the odd/even distinction also. But unlike the Fib mapping, which gets off track at 144, well beyond current elements, the Luc trend starts to go awry at 29, and stays there through 47 and 76 (123 will also be mispositioned). 29 and 47, copper and silver respectively, and coincidentally (?) in the same group, get around their mispositional problems with a simple 'fix'- they move one electron from an s2 filled orbital to the nearly full d9, ending up with s1,d10, both of which fit the trend in terms of electronic structure, if not table position. Osmium, 76, with a d6 electronic configuration, has altered behavior which resembles that of xenon, a noble gas with filled p6 orbital. Such 'imitations' are actually quite common in the PT, and account for many non-traditional periodic interblock relationships.

 

All of these mappings are related to the fact that the PT gets much of its graphical motivation from the Pascal Triangle, and so does the Fibonacci sequence. Pascal relations are also found in atomic nucleus and its filling patterns. For example half the (semi)magic numbers in the nucleus are doubled Pascal Triangle tetrahedral diagonal numbers. The other half are these minus doubled triangular diagonal numbers from directly above the particular tetrahedrals. A vertical relation! I just discovered this week that if one maps single tetrahedral numbers (half-magic) numerators divided by alternating Fibonacci and Lucas denominators, one generates the known curve of the nuclides (with x for protons, y for neutrons), though slightly offset near the origin. The offset is due to the difference between the strong and electromagnetic forces in terms of both strength and reach- the former only works at very short distances, the latter over long and short, so eventually EM wins out. Where the mapping of my Tet/Fib and Tet/Luc comes together with the known curve is where one sees the waning of the strong forces' reach. Other related numbers also generate similar curves- some have outliers, others don't. What is very interesting though is how well, near the origin, the outlines match, for dips and rises.

 

Jess Tauber

[Pincho Paxton]

I believe that the Universe is a fractal of seeds that happen from the structure of spacetime. And this is how the Universe gets its maths, because the maths are taken from a pattern. By using a pattern to create structures like a seed to create a diamond, you get less random events from the chaos. So you get lots of suns, and lots of Galaxies in a Universe that stops looking random. Not only that, but I think I also know the exact pattern that is the seed.

Edited April 21, 2012 by Pincho Paxton

[Don Blazys]

To: Jess,

 

Do you have a website?

 

Don.

[tetrahedron]

I have a discussion list: tech.groups.yahoo.com/group/tetrahedronT3, but I haven't posted anything there of note in some time- these days mostly I post to other blogs (such as this one) and private email circulations. Its really a matter of audience reach, not that it gets very high even in well known sites like this one. People are prompting me to go to a conference on the periodic table in Peru this fall to talk about all these mathematical findings, but for a variety of reasons I don't think that's really a realistic possibility.

 

As for these new curves that hug the nuclide chart, they obviously aren't exact- need more work, but clearly they do imply that there is some relation to the Fibonacci and Lucas sequences. Until now I hadn't seen one that matched nuclear data, as I had with the electronic patterns outlined in my first post. A different type of relationship, but then all the nuclear Pascal Triangle relationships are different from the electronic ones, though drawn from the same ultimate sources. A matter of hierarchical ranking or prioritization of some sort. There is also a possibility, one I've been mulling over for a long time now, that the shift from the P=N line which seems to dominate the nuclides early on to a curve that more closely matches one based on the Golden Ratio is somehow linked to different Metal Means. Why should nature only use Phi, one of many such? The general Metal Mean formula is (N+(sqrt((N^2)+4))/2. If N=0, the Metal Mean is 1.0000...., if N=1, Phi (sqrt5+1)/2, while N=2 gives the Silver Ratio, 2sqrt2. They are also based in continued fractions. I've seen evidence for all three in the electronic realm, so why wouldn't we also find them in the nuclear? Interestingly, the powers of the Metal Means all relate to the sister of the Pascal Triangle with sides (2,1), in that if one lines up their equations one above the other, then the numerical coefficients of the individual power terms are all diagonal numbers from this Triangle. And the powers marked by the terms all match the dimensional headings of the diagonals of the Triangle as well. A VERY intimate, and spooky, relationship, if you ask me. Remember also that this sister Pascal Triangle, when summing across shallow diagonals, gives both the Fibonacci and Lucas numbers, on opposite sides.

 

I've also started looking more closely at string theory, esp. with regard to symmetry. I've got a long slog ahead of me, clearly, to learn the requisite relevant math. But already I'm discerning, in the E8 pattern (thanks to very clear treatments I've been reading), certain hints of relationship to Fiblike numbers. It had already been announced, several years ago, as you're aware, that the Golden Ratio was showing up in certain resonance patterns in magnets, all based on E8 symmetry. But I don't think anyone noticed that the 248 dimensional definition itself showed a pattern. That is, if you divide 248 by 4, you get 62, which is the approximation of 100phi, rounded to the nearest one, where phi is Phi-1, or (sqrt5-1)/2, etc. 62 is found in a variety of natural phenomena, and important math. For example Nickel-62, with 34 protons (Fib!), is the most stable nucleus. It is the internal angle of a 20-gon, which like the penta- and deca-gon shows the Golden Ratio in its trigonometric relationships. You can get 62, part of a series that includes ...38, 62, 100, 162, 262... a variety of ways by combining Fib, Luc, and other related numbers using different operations. This series is also approximately 20Fib, and 2xFib numbers may be related to the actual dimensions within string theory: 4, 10, 26 etc. Every other 2xFib? There may be much more here, so stay tuned.

 

Jess Tauber

[email protected]

Edited April 21, 2012 by pascal

[tetrahedron]

Well, maybe, just maybe I've finally begun to discern a kind of generalized Fibonacci patterning within the nucleon filling system for atomic nuclei. As I've written here and elsewhere the idealized tabular organization of both the electronic and nuclear systems appears to be very strongly influenced by Pascal Triangle math. Patternings involve values from traditional deep diagonals, sums across shallow diagonals (giving Fib and its relatives), squares, etc.

 

These then tend to be organized along columns and rows within the left-step quantum number-based depictions of the electronic and nuclear filling patterns. Yet we find variation as well- what gets mapped along a row or column changes depending on how the quantum numbers themselves pattern.

 

In the electronic system it has been known for a long time that there are 'diagonal' periodic relations. Some rise to the right, others rise to the left. They are often ignored, so explaining them and what they do has not been a top chemist's or physicist's priority- even so there are specialists who do deal with them.

 

In the electronic system I had previously discovered the left-half orbital leftmost mapping for odd Fib numbers, and right-half orbital leftmost mapping for even Fibs, as well as the left-half orbital rightmost mapping for odd Lucas numbers and the right-half orbital rightmost for even Lucas. The mapping is imperfect for both, eventually getting off track in the table positionally. At least for the Lucas numbers 29 and 47, there are electronic-configurational 'fixes' involving redistribution s electrons so that s ends up half-full and d completely filled, both meeting the configurational facts of rightmost mapping, even though the element atomic number's position within the larger tabular structure is off by one move leftwards.

 

For osmium, element 76 we end up with a different 'fix', allowing d6 to act as if p6, the latter a full noble electronic configuration.

 

All these mappings take place within (half) orbitals within periods, so if there is any larger pattern it is visually elusive, at least for me.

 

Now, in the nuclear equivalent of the periodic system (dealing with one type of nucleon only), I seem to have found a diagonal-based pattern mapping generalized Fibonacci numbers, through fractions and multiples, to numerical differences between equivalent positions in spin-orbit split orbitals. Thus the mapping deals with a more derived structure than that found in the electronic system. I've extended the tabularization to almost 1000 nucleons, and the mappings hold true. Whether they are systematically mapped remains to be determined- there are a large number of combinations to deal with (I would imagine a computer program could do this very quickly) Differences may be either rising up left or rising right in the table. Both work.

 

But I'm a bit disturbed- combinations of multiples or fractions of generalized Fib numbers can generate most (all?) natural numbers, so it is still to be seen whether the differences I see mean anything or not. However if there is some coherent 'texture' to the whole, then that should quell my doubts. Also, this system is an idealized one- ignoring the actual mixing/switching of energy levels seen for real elements very often in the orbital filling patterns, due to spin-orbit effects, changes from sphericity, etc.

 

It would be interesting to know whether these real-world mappings make the generalized Fib mappings better or worse, or whether they might relate to the shift in Metallic Means that I've hypothesized characterizes the interactions between numbers of protons and neutrons in atomic nuclei. Each Metallic Mean has its own system of numbers sequences that are similar in flavor (if not in execution) to the generalized Fib numbers. Thus for the Silver Mean, which may or may not dominate for STABLE atoms of high atomic number (something we've never seen), there are Pell numbers, and thus generalized Pell numbers. All part of an even larger system of Pisot numbers. So a lot of work, and no end in sight....

 

Jess Tauber

[email protected]

Edited July 27, 2012 by pascal

[tetrahedron]

Turns out that this may work even better than I thought- the coverage of natural numbers by these generalized Fib-series sets by itself leaves many gaps at least for the early members, multiples and fractions do fill some of the gaps out- I guess there will always be primes, though.

 

Also it looks strongly like we are using both sides of the generalized Pascal Triangles whose summed samplings along the shallow diagonals, through the deep, classical diagonals, generate these sequences. Several months ago, when first looking at how the generalized Fibonacci sequences pattern in the electronic system to give periodic table parameters (including quantum numbers) I realized that one could extend the system in both directions, and that the mirror image sequences of those whose initials were negatives mapped as opposite side sisters on the same Pascal Triangle variants.

 

The electronic system didn't seem, in any way yet obvious to me, to used these negative-initialed sequence numbers. But the nuclear differences are using them. So, there may be some sort of symmetry relationship here still hidden within the overall distribution. Gets more and more interesting!

 

Jess Tauber

[email protected]

[Randy2021]

 

 

We use patterns to describe nature and if we look hard enough, we can even create a mathematical equation for the pattern. This does not mean that the pattern follows the equation. It’s the other way around, the equation follows the pattern. As our understanding grows, so is the need to come up with new and more powerful equations to describe the universe, e.g. from Newtonian Mechanics to General Relativity.

The equations we use to describe the patterns are mental constructs, it’s all in our mind. We create these mental constructs to make sense of what we see. Nature can work fine without the equations.

Let’s say there are 5 cats and 2 dogs. The cats and dogs are real but the numbers “5” and “2” are not. It’s only a numbering system that was invented to give meaning to what we see. A child can look at the cats and dogs but without the concept of numbers, he will not be able to group them as “5 cats and “2 dogs” or perform arithmetic operations.

So why is the Fibonacci sequence common in nature? Now that’s a more interesting question. It cannot be denied that it is observed in nature but for some reason, it is difficult to comprehend its importance. We observe it but we cannot quantify or give meaning to it using equations in physics.

The Fibonacci sequence and spiral is an outcome of a process of nature which is waiting to be discovered. There is no clear understanding on how the process works. However, even if we still don’t fully grasp the mechanics on how nature implements this process, I believe that it may have something to do with the “Minimum Energy” of a system.

One way to give a physical meaning or to find a scientific reason for this sequence and spiral is to derive an equation that describes a physical phenomenon which includes this sequence and spiral and then use the same information to describe other phenomenon.

---

“Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world.” — Albert Einstein

 

Refer to the below link for a physical application of the Fibonacci sequence.

Liber Abaci Revisited (1202 - 2021)