Those numbers closely resemble those that determine ideal internal
dimensions of acoustic loudspeaker enclosures.
If the baffle board is 12 " wide than the depth of the box be will
7.5" and the height 19.2" or any ratio close to 0.625:1:1.6.
That is for a basic box.
They appear to share a pattern and a limit.
The number which is the limit of the successive ratios of the values in
"Fibonacci Sequence" is called the "Golden Ratio" is just a number
(approximately 1.618). You can play the same game (defining a
"recurrence relation") starting with values besides 1 and 1, and by the
same process you'll surely end up with a different limit, i.e. number.
The limit in the fractal case often lies in 2-D. A well-known one is
called a "snowflake".
The numbers e and Pi share a pattern and a limit in some of their
definitions too, maybe comparable to nature and evolution....
To get back on topic, some have suggested that a picture frame whose
length and width are proportion alto the Golden Ratio, 1: 1.618, will
look the most natural to the most people.
If this is so true, I wonder why the tv industry appears to have settled
on the aspect ratio 16:9 ~ 1.77 instead of 16:10 (which would be closer
to the Golden Ratio)?
Not true. Using "the same process", i.e., the same recurrence
relation, you will end up with the golden ratio no matter what the
starting values (as long as at least one of the two is non-zero). e.g.
8, -6, 2, -4, -2, -6, -8, -14, -22, -36, -58, -94, ...
At this point, the ratio of successive values has already converged to
1.62, and it keeps getting closer. Try to find a pair that will NOT
get you to within .01 of the golden ratio after ten iterations, and
you will quickly convince yourself of this.
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.
Well, he specifically said "defining a recurrence relation" which would
imply some other than the same one... :)
But, it isn't true that _any_ recurrence relation will generate a limit
as is implied, either...
I don't doubt it. IIRC, the trickiest part is showing the limit exists,
then the quadratic formula decides the value of the limit. Probably, the
more general result can be obtained by using the original one: If you
multiply the Fibonacci sequence by some any non-zero value c, then when
you take the ratio of successive elements the c cancels out. So the
sequuence 0, c, c, 2c, ... will yield the Golden Ratio too. Beginning
with -1, 0 instead we get -1, 0, -1, -1, -2,.. and this also yields the
Golden Ratio (discard the first term to see this). Taking the sum of
the 2 sequences
0, 1, 1, 2, 3, ... and 1, 0, 1, 1, 2, 3, ..., or equivalently any
zon-zero multiples of them
0, a, a, 2a, 3a, ... b, 0, b, 2b, 3b ...
basically helps establish your result. Minor technicalities omitted. If
a or b is zero, but not both, the conclusion is unaffected. So yes, I
agree. Score another one for the Golden Ratio!
There is a complex set of trade-offs involved.
Peripheral vision is more effective to the sides than up/down. This is in
part because, 'historically', threats were more likely to appear from the
sides. The 'range of vision', _vertically_, is typically about +/- 60
degrees from the horizontal. However, 'to the sides', it is typically 80+
degrees from 'straight ahead', and in a significant number of people it
can range to 90-95 degrees _and_more.
"Portrait" orientation (the long dimension vertical) is optimal -- in terms
of 'visually pleasing', that is -- at 1.618:1. This ratio occurs 'naturally'
in a bunch of aspects in the human body -- See da Vinci's figure studies.
"Landscape" is more natural, and 'panoramic', at a ratio that is 'wider'
and 'flatter'. You don't get much from the extra 'sky' in an exterior shot.
Similarly, for interiors, the floor-ceiling dimension tends to limit the
usefulness of a greater display height.
Also, realize that 'wide-screen' in the movie theater is typically _1.88_:1.
and the famous 'Cinerama' process from the 1950s, 1960s, and 1970s, was
*really* wide -- at _2.66_:1.
More in Mathematics but applications in Physics
and the Bio plants and animals. I've watched complex
presentations on the mathematics of Shell shapes.
Most interesting line of mathematics.
On 7/25/2011 12:36 PM, Han wrote:
Mathematician Mario Livio in his book: The Golden Ratio, examines
the so-called Golden Ratio as it applied to the arts.
While it is true that the ratio shows up in most unexpected places
(chambered nautilus being one), it is not so certain that artists
have consciously employed the ratio in their art. The Parthenon is
supposed to follow it, but it's not clear where to actually start
the measurement to coincide with the ratio.
Here's a quote from a web page:
"Furthermore, I should note that the literature is bursting with false
claims and misconceptions about the appearance of the Golden Ratio in
the arts (e.g. in the works of Giotto, Seurat, Mondrian). The history
of art has nevertheless shown that artists who have produced works of
truly lasting value are precisely those who have departed from any
formal canon for aesthetics. In spite of the Golden Ratio's truly
amazing mathematical properties, and its propensity to pop up where
least expected in natural phenomena, I believe that we should abandon
its application as some sort of universal standard for "beauty,"
either in the human face or in the arts."
The take away message is, don't get so trapped in following the
formula that you
overlook the art you are doing as a whole. Sometimes deviation from a
is more enticing, thrilling and original.
I never took the fact that the golden ratio is present in so many great
works of art to mean that the artists purposely employed the ratio in
the design of their art.
I've always seen it as a *description* of aesthetic pleasure, not a
*prescription.* Meaning, we have perceived certain works of art to be
beautiful or pleasing because, for whatever reason, our brains are wired
to like things that have this ratio.
"Playing is not something I do at night, it's my function in life"
Ah, but suppose many, many artists avoided the trap you postulate, or
more likely, never knew it existed. But yet their work product still
emerges so proportioned. THAT speaks volumes. I doubt that
sunflowers, ammonites, and myriad other works of nature got "trapped".
I'd put this tendancy right in there with Gibb's free energy.
That's what I contend. The great art isn't great because the artist
followed a formula. It's great because it has beautiful proportions. The
formula describes those proportions.
But the author at that link has a point worth discussing. Architecture
these days seems to be at one extreme or another; either absurd
grotesque or absurdly plain. The grotesque can be seen in Dubai where
everything is to a ridiculous excess... simply because it *can* be done
and someone will pay for it. The plain can be seen by the work that
comes from several generations of architectural teaching that says the
golden ratio is the perfect proportion. As the author implies, these
architects seems to just shoot for that formula without any forethought
for *art,* producing generic, homogenized structures that look almost
"Playing is not something I do at night, it's my function in life"
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