A few questions about the golden ratio
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A few questions about the golden ratio
First, is there scientific evidence to back up the claim that the golden ratio is aesthetically pleasing? Have there been any experiments in which people who knew nothing about the golden ratio were asked to choose between several shapes, one of which contained the golden ratio? If so, did the majority pick that shape?
Second, are there any practical applications for the golden ratio outside of aesthetics?
Third, I know the golden ratio is said to appear in nature, and that some of the claims of their appearance are bogus, while others actually have been observed. I'd like to know which of these claims that happen to be true are most prevalent.
Do any of you have answers to these questions?
Second, are there any practical applications for the golden ratio outside of aesthetics?
Third, I know the golden ratio is said to appear in nature, and that some of the claims of their appearance are bogus, while others actually have been observed. I'd like to know which of these claims that happen to be true are most prevalent.
Do any of you have answers to these questions?
Re: A few questions about the golden ratio
Suppose a line segment is divided into a smaller part of lengh 1 and a larger part of length [imath]\phi[/imath], in such a way that the whole line segment (of length [imath]1+\phi[/imath]) is [imath]\phi[/imath] times larger than the larger part:
[math]1 :\phi = \phi : (1+ \phi)[/math]
Then [imath]\phi = (1+\sqrt 5)/2= 2 \cos (\pi/5)=1.6180339877...[/imath] and we say that the line segment is divided according to the golden section.
Just one application:
In a regular pentagon, a diagonal is [imath]\phi[/imath] times as long as a side and each diagonal divides any other diagonal according to the golden section.
I hope this helps, although I am not sure you find this a practical application and I have certainly not answered all your questions.
[math]1 :\phi = \phi : (1+ \phi)[/math]
Then [imath]\phi = (1+\sqrt 5)/2= 2 \cos (\pi/5)=1.6180339877...[/imath] and we say that the line segment is divided according to the golden section.
Just one application:
In a regular pentagon, a diagonal is [imath]\phi[/imath] times as long as a side and each diagonal divides any other diagonal according to the golden section.
I hope this helps, although I am not sure you find this a practical application and I have certainly not answered all your questions.
 MartianInvader
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Re: A few questions about the golden ratio
Reports of the golden ratio's awesomeness are often overstated, but there's still a decent amount of awesome in there.
One application is finding a closedform formula for the n^{th} Fibonacci number, which uses the golden ratio.
I've heard anecdotally that people have done scientific studies on what rectangles are the most aesthetically pleasing and the golden ratio won out, but I'm not sure whether it's true.
Of course, masters of aesthetics for centuries have praised the golden ratio for its aesthetics, so that could be taken as evidence I suppose.
One application is finding a closedform formula for the n^{th} Fibonacci number, which uses the golden ratio.
I've heard anecdotally that people have done scientific studies on what rectangles are the most aesthetically pleasing and the golden ratio won out, but I'm not sure whether it's true.
Of course, masters of aesthetics for centuries have praised the golden ratio for its aesthetics, so that could be taken as evidence I suppose.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
Re: A few questions about the golden ratio
Following up on MartianInvader, the Binet formula is awesome enough, and the proof is lowcalorie if you use power series/generating functions. It's a good linear algebra exercise to prove it using diagonalization of matrices.
It lets you write the Fibonacci numbers as a sum of two geometric sequences. One of those sequences goes to zero fast, so, a few terms out, each Fibonacci number can be obtained by rounding off [imath]\phi^n/\sqrt5[/imath].
Try a few. It will creep you out.
It lets you write the Fibonacci numbers as a sum of two geometric sequences. One of those sequences goes to zero fast, so, a few terms out, each Fibonacci number can be obtained by rounding off [imath]\phi^n/\sqrt5[/imath].
Try a few. It will creep you out.
 Xanthir
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Re: A few questions about the golden ratio
ianfort wrote:First, is there scientific evidence to back up the claim that the golden ratio is aesthetically pleasing? Have there been any experiments in which people who knew nothing about the golden ratio were asked to choose between several shapes, one of which contained the golden ratio? If so, did the majority pick that shape?
Second, are there any practical applications for the golden ratio outside of aesthetics?
Third, I know the golden ratio is said to appear in nature, and that some of the claims of their appearance are bogus, while others actually have been observed. I'd like to know which of these claims that happen to be true are most prevalent.
The golden ratio is approximately 1.6, so virtually any two things that have about a 3:2 ratio have been used as an example of "the golden ratio in nature". This is nonsense, of course.
The Wikipedia article on the golden ratio happens to be pretty good in describing what is actually based on the golden ratio and what isn't.
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))

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Re: A few questions about the golden ratio
I read a paper where subjects had been asked to choose rectangles they found to be pleasing, and there wasn't much to support the fact that they found "golden rectangles" to be particularly pleasing.
http://math.hws.edu/vaughn/math/110/lecture/goldenratio.pdf
(The section I was talking about starts on page 12).
http://htpprints.yorku.ca/archive/00000003/00/goldrev3.htm
This looks relevant too. I only read the abstract and conclusion, but it references some studies into the psychological/aesthetic properties of the golden ratio. It concludes that if they exist, they are fragile and we don't currently have a definitive answer.
http://math.hws.edu/vaughn/math/110/lecture/goldenratio.pdf
(The section I was talking about starts on page 12).
http://htpprints.yorku.ca/archive/00000003/00/goldrev3.htm
This looks relevant too. I only read the abstract and conclusion, but it references some studies into the psychological/aesthetic properties of the golden ratio. It concludes that if they exist, they are fragile and we don't currently have a definitive answer.
Re: A few questions about the golden ratio
Crashthatch wrote:I read a paper where subjects had been asked to choose rectangles they found to be pleasing, and there wasn't much to support the fact that they found "golden rectangles" to be particularly pleasing.
I heard (yes, it's another anectode, but it's amusing) that this was repeated more recently, and that for horizontal rectangles the most pleasing ratio is now closer to 1.8 than 1.6, and that the supposed reason for it was that people are getting more familiar with widescreen 16:9 ratio. This anecdote is probably bogus, but nevertheless aesthetics are probably very much influenced by what we are used to in daily life, and so are likely to be very culturally dependent.
Re: A few questions about the golden ratio
I was under the impression that the golden ratio's aesthetics had more to do with more complex shapes than a simple rectangle, for example the repetition of a golden rectangle and the resulting spiral.
 gmalivuk
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Re: A few questions about the golden ratio
With the rectangles in the picture, what I find aesthetically pleasing is the mathematics of it. Without the rectangles, is that particular spiral any more pleasing than some slightly different one?
Re: A few questions about the golden ratio
Here is a telecommunication application of the golden ratio. The so called Golden code consists of 2x2 complex matrices of the form
[imath]\frac{1}{\sqrt 5} \left(\begin{array}{cc}
\alpha(a+b\phi)&\alpha(c+d\phi)\\
i\sigma(\alpha)(c+d\sigma(\phi))&\sigma(\alpha)(a+b\sigma(\phi))
\end{array}
\right).[/imath]
Here [imath]\alpha=1+ii\phi[/imath], [imath]\sigma[/imath] is the automorphism of the field [imath]Q(i,\phi)[/imath] determined by [imath]\sigma(i)=i, \sigma(\phi)=1\phi,[/imath] and a,b,c,d are gaussian integers, i.e. complex numbers with integer real and imaginary parts. The key properties of the Golden code are that it is a free abelian group (=lattice) of rank 8 and that the absolute values of the determinants of nonzero matrices in this set are all at least 1/5. Furthermore, geometrically the free abelian group is isometric to the 8dimensional hypercube Z^{8}. Here the metric of the space of matrices coming from the obvious identification of 2x2 complex matrices and 8dimensional real vectors. I think that it is known that 1/5 is the best one can do here. Such properties are helpful when a multiantenna radio transmission signal set is carved from the lattice. See
http://www.tlc.polito.it/~viterbo/perfe ... _Code.html
for more information about the Golden code. Rank 8 is obviously the highest we can have, because otherwise we get accumulation points, and consequently determinants arbitrarily close to zero.
My (former) graduate students found a rank 8 lattice with a higher minimum value for the minimum absolute value for the determinant, and (to make the comparison fair) same center density and size of fundamental region as the Golden code. It was a nice application of class field theory. That lattice is not hypercubical so the telecommunication engineers shy away from it for some reason. Sorry about blowing my own trumpet here. Couldn't resist.
[imath]\frac{1}{\sqrt 5} \left(\begin{array}{cc}
\alpha(a+b\phi)&\alpha(c+d\phi)\\
i\sigma(\alpha)(c+d\sigma(\phi))&\sigma(\alpha)(a+b\sigma(\phi))
\end{array}
\right).[/imath]
Here [imath]\alpha=1+ii\phi[/imath], [imath]\sigma[/imath] is the automorphism of the field [imath]Q(i,\phi)[/imath] determined by [imath]\sigma(i)=i, \sigma(\phi)=1\phi,[/imath] and a,b,c,d are gaussian integers, i.e. complex numbers with integer real and imaginary parts. The key properties of the Golden code are that it is a free abelian group (=lattice) of rank 8 and that the absolute values of the determinants of nonzero matrices in this set are all at least 1/5. Furthermore, geometrically the free abelian group is isometric to the 8dimensional hypercube Z^{8}. Here the metric of the space of matrices coming from the obvious identification of 2x2 complex matrices and 8dimensional real vectors. I think that it is known that 1/5 is the best one can do here. Such properties are helpful when a multiantenna radio transmission signal set is carved from the lattice. See
http://www.tlc.polito.it/~viterbo/perfe ... _Code.html
for more information about the Golden code. Rank 8 is obviously the highest we can have, because otherwise we get accumulation points, and consequently determinants arbitrarily close to zero.
My (former) graduate students found a rank 8 lattice with a higher minimum value for the minimum absolute value for the determinant, and (to make the comparison fair) same center density and size of fundamental region as the Golden code. It was a nice application of class field theory. That lattice is not hypercubical so the telecommunication engineers shy away from it for some reason. Sorry about blowing my own trumpet here. Couldn't resist.
 silverhammermba
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Re: A few questions about the golden ratio
The most interesting aspect of the golden ratio, in my opinion, is that it can be thought of as the ratio of self similarity. Nature is full of selfsimilar structures so, in many cases, the golden ratio shows up by necessity. I think it's silly to say that it's more aesthetically pleasing than other ratios. What makes these numbers beautiful is that they arise from natural situations without needing to be specially constructed.
Re: A few questions about the golden ratio
jaap wrote:I heard (yes, it's another anectode, but it's amusing) that this was repeated more recently, and that for horizontal rectangles the most pleasing ratio is now closer to 1.8 than 1.6, and that the supposed reason for it was that people are getting more familiar with widescreen 16:9 ratio. This anecdote is probably bogus, but nevertheless aesthetics are probably very much influenced by what we are used to in daily life, and so are likely to be very culturally dependent.
Cultural dependency of aesthetics very likely enters this equation. When I lived in the US for 4½ years I recall finding the paper sizes (also letter but legal in particular) rather strange looking. The most likely explanation is that I was so conditioned to expect A4. A transatlantic survey might help us out here
The golden ratio is nowhere to be seen on either paper size standard. A4 is [imath]1:\sqrt 2[/imath]. Not for aesthetic reasons but simply so that you can fold an A4 to form two A5s (and keep the shape). Similarly you can fold an A3 to form two A4s (we hand out several of those to students taking exams in these parts).
Re: A few questions about the golden ratio
Interestingly, the largest region of inputs that require a lot of computational time for Euclid's algorithm (for finding the GCD of two numbers) is along the line [imath]y = \phi x[/imath].
2 is not equal to 3, not even for large values of 2.
 Grabel's Law
Talent hits a target no one else can hit; Genius hits a target no one else can see.
 Arthur Schopenhauer
 Grabel's Law
Talent hits a target no one else can hit; Genius hits a target no one else can see.
 Arthur Schopenhauer
Re: A few questions about the golden ratio
silverhammermba wrote:The most interesting aspect of the golden ratio, in my opinion, is that it can be thought of as the ratio of self similarity. Nature is full of selfsimilar structures so, in many cases, the golden ratio shows up by necessity. I think it's silly to say that it's more aesthetically pleasing than other ratios. What makes these numbers beautiful is that they arise from natural situations without needing to be specially constructed.
Interesting... Does the golden ratio appear in some of the wellknown fractals, such as the Mandelbrot set?
 silverhammermba
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Re: A few questions about the golden ratio
I don't know about the mandelbrot set, but the classic construction of the golden spiral creates a (somewhat trivial) fractal.
I suppose it's more like the ratio of recursion, anyway, due to its close relation to the Fibonacci sequence.
I suppose it's more like the ratio of recursion, anyway, due to its close relation to the Fibonacci sequence.
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