Golden Ratio Ppt Download
Jeffrey Dyer
The "Cleary group" $F_\tau$ is a version of Thompson's group $F$, introduced by Sean Cleary, that is defined using the golden ratio, and it's definitely of interest in the world of Thompson's groups. See An Irrational-slope Thompson's Group ( Publ. Mat. 65(2): 809-839 (2021). DOI: 10.5565/PUBLMAT6522112 ). Very roughly, where $F$ arises by "cutting things in half", $F_\tau$ arises in an analogous way by "cutting things using the golden ratio". There are lots of similarities between $F_\tau$ and $F$, but also plenty of mysteries, for example I believe it's still open whether $F_\tau$ embeds into $F$ (i.e., whether there exists a subgroup of $F$ isomorphic to $F_\tau$).
First, in a certain sense, $\phi$ is the hardest number to approximate with rational numbers. What do we mean by that? if you have an irrational number $\alpha$, and you want to approximate $\alpha$ with rational numbers of the form $\fracnd$, then you can get approximations as good as you want by making $d$ larger. However, suppose you are interested in getting as close as you can and wanting to know how bad a price you pay in terms of increasing $d$, it turns out that by multiple ways of making this precise, $\phi$ is the worst possible. This is closely related to the fact that it has continued fraction $[1,1,1,1...]$ and so the best possible approximates actually have numerator and denominator Fibonacci numbers.
In a related note, there's a recent paper in the Journal of Number Theory by Roswitha Hofer which generalizes the golden ration to fields of formal power series. Hofer's paper can probably be used as a jumping off point for generalizing some of the results mentioned above to other contexts.
There are several exponential-time algorithms (in theoretical computer science, arguably a subfield of mathematics), whose running time can be bounded by an expression of the form $n-k \choose k$ where $k$ is an integer between $0$ and $n$. Plugging in the $k$ that maximizes this term, we get an upper bound $\sim \phi^n$ where $\phi = \frac1+\sqrt52$, the golden ratio. This is often in contrast to the naive approach whose running time would be $\sim 2^n$.
edit Dec 2022. It seems it has not yet been recalled the relevance of the golden ratio and Fibonacci numbers in the theory of continued fractions, due to the Hurwitz's theorem.
It should be noticed that in both the examples above, and, it seems to me, in many examples quoted in these answers, the role of the golden ratio is not really due to a certain property in itself, but rather, to the fact that it is the best (or worse) case in a given context, so that it actually gives optimal bounds. I think this explains (and in some sense resizes) the "ubiquity" of the golden ratio and Fibonacci numbers in mathematics and in nature. It is not that simple models like the golden section, or "$x_n=x_n-1+x_n-2$", or the most popular "Fibonacci spiral" really explain so many facts (although it seems that out there most people really thinks so); it seems more true that they give first approximations and sharp bounds to more complex situations.
In an associative context, if the operation $*$ is the concatenation operation, then the binary words $t_n(0,1)$ are called the Fibonacci words. For example,$t_7=0100101001001$. Unsurprisingly, the golden ratio arises from the Fibonacci words too.
Clearly, the length of the Fibonacci words are the Fibonacci numbers and the number of occurrences of $0$'s and $1$'s are also Fibonacci numbers. Therefore, ratio of the number of 0's in a Fibonacci word to the number of 1's in the same Fibonacci word approaches the golden ratio. The golden ratio also arises from Fibonacci words in many different ways. For example, the $n$-th bit in any Fibonacci word except for the first one is $2-[\lfloor (n+1)\phi\rfloor-\lfloor n\phi\rfloor]$ (here we start at $1$, so $(abcde)[1]=a$).
The Fibonacci terms in a non-commutative context arise from large cardinals when obtaining the composition of rank-into-rank embeddings from the application operation on rank-into-rank embeddings. I don't think there are yet any known interesting relations between rank-into-rank cardinals and the golden ratio.
Golden ratio is relevant in most places where you consider Fibonacci numbers. One occurrence where it is particularly visible is the proof by Bugeaud, Mignotte and Siksek of the fact that the largest perfect power in the Fibonacci series is 144 (arXiv version here).
There's an identity for chromatic polynomials of planar triangulations called the "golden identity" found by Tutte, giving a quadratic relation between the values of the chromatic polynomial at $\phi+1$ and $\phi+2$. In a fairly recent paper Slava Krushkal and I extended this identity (or rather its dual for the flow polynomial of planar cubic graphs) to the Yamada polynomial, an invariant of spatial graphs. We also have a conjecture that this identity characterizes planar cubic graphs in terms of the flow polynomial.
The golden mean also crops up in matroid theory. A matroid $M$ is a golden mean matroid if it can be represented by a real matrix such that every non-zero subdeterminant is $\pm \phi^i$, for some $i \in \mathbbZ$, where $\phi$ is the golden mean. Somewhat surprisingly, Vertigan proved that a matroid is a golden mean matroid if and only if it is representable over both $GF(4)$ and $GF(5)$, where $GF(q)$ is the finite field with $q$ elements.
The golden ratio is closely related to the Zeckendorf representation which is one of the simplest examples of a numeration system (other than the usual base-$b$ represenations). As such, it's an important testing ground for new ideas. For instance, in this paper Drmota, Müllner and Spiegelhofer showed that the parity of the sum of digits in the Zeckendorf representation does not correlate with the Möbius function. Such considerations are interesting for their own sake, and are also connected with morphic sequences thanks to the work of Rigo: One can think of a morphic sequence as an automatic sequence in a non-standard numeration system.
A more surprising application may be in symbolic dynamics, where the "golden mean shift" consists of bi-infinite binary sequences the avoid adjacent ones. Similar to $a(n) = a(n-1) + a(n-2)$ being one of the most foundational recurrence sequences, this golden mean shift is a basic shift space in the younger field of dynamical systems.
One of the most clever and amusing, yet deep, presentations I've seen on the interdisciplinary import of the golden ratio and Fibonacci sequences is the video sequence by V. Hart "Doodling in Math: Spirals, Fibonacci, and Being a Plant" and, of course, the comments and refs in the OEIS entry A000045 on the standard Fibonacci sequence and the golden ratio contain a lot of applications.
The number $n(g)$ of numerical semigroups of genus $g$ is easily shown to be finite and is asymptotically given by $C \omega^n$ for some constant $C$ with $\omega=(1+\sqrt5)/2$ the golden number. This was conjectured by M.Bras-Amoros, Fibonacci-like behaviour of the number of numerical semigroups of a given genus, Semigroup Forum 76, No 2, 379--384 (2008), proven by A. Zhai, Fibonacci-like growth of numerical semigroups of a given genus, Semigroup Forum 86, No 3, 634--662 (2013). A different and hopefully more comprehensive proof (sorry, self-promotion) is contained in .
Golden ratio appeared in the recent breakthrough A constant lower bound for the union-closed sets conjecture of Frankl's Union Closed Set conjecture by Justin Gilmer, as well as the subsequent optimization Chase and Lovett - Approximate union closed conjecture of the lower bound constant. It relies crucially on the following fact:
The golden L is a translation surface that has received much attention recently in the theory of billiard dynamics. It is built out of a $1 \times 1$ square with $1 \times \phi$ and $\phi\times1$ rectangles glued to it along their length-$1$ sides. See for example the preprint Davis and Lelievre - Periodic paths on the pentagon, double pentagon and golden L.
Let $\rho$ be the binary substitution defined by: $$\rho(00)=\textempty word\quad\rho(01)=1\quad\rho(10)=0\quad\rho(11)=01.$$Let $R$ be the self-map of $[0,1]$ associating to every $x=(0.w)_2$ the number $R(x)=\left(0.\rho(w)\right)_2$, taking the binary expansion ending in $1^\infty$ in case of dyadic rationals.
While plenty of people will tell you that it "depends," we're willing to commit to a specific ratio: 30/ 60/ 10. It's how we run our social media and content marketing, and we think it's a good way for you to run yours as well.
It has many properties of the golden ratio, mirrored. This forum from 2011 is the only literature I could dig up on it, and it explains most of the properties I also found and more. -theory/17605-imaginary-golden-ratio.html
Since the only other forum which I could find that has acknowledged the existence of the imaginary golden ratio (other than the context of it as a special case imaginary power of e) I'd like to share my findings and ask if anybody has heard of this ratio before, and if anybody could offer more fine tuned ideas or explorations into the properties of this number. One specific qustion I have involves its supposed connection (according to the 2011 forum) to the sequence
In dynamic magnetic resonance imaging (MRI) studies, the motion kinetics or the contrast variability are often hard to predict, hampering an appropriate choice of the image update rate or the temporal resolution. A constant azimuthal profile spacing (111.246 degrees), based on the Golden Ratio, is investigated as optimal for image reconstruction from an arbitrary number of profiles in radial MRI. The profile order is evaluated and compared with a uniform profile distribution in terms of signal-to-noise ratio (SNR) and artifact level. The favorable characteristics of such a profile order are exemplified in two applications on healthy volunteers. First, an advanced sliding window reconstruction scheme is applied to dynamic cardiac imaging, with a reconstruction window that can be flexibly adjusted according to the extent of cardiac motion that is acceptable. Second, a contrast-enhancing k-space filter is presented that permits reconstructing an arbitrary number of images at arbitrary time points from one raw data set. The filter was utilized to depict the T1-relaxation in the brain after a single inversion prepulse. While a uniform profile distribution with a constant angle increment is optimal for a fixed and predetermined number of profiles, a profile distribution based on the Golden Ratio proved to be an appropriate solution for an arbitrary number of profiles.
760c119bf3