Golden Ratio. Measure of appeal

Jimbo30

We are designing a new product at work so I was curious to see if this product met the Golden Ratio criteria. I was astounded to find that it did indeed measure close to the Golden Ratio. I am curious as to see if anyone of you have used the Golden Ratio concept in any of your work?.?.?.?

xanatos

I employ it in some of my web site design work, and occasionally when building various wooden things for use in and around my home. Not always a practical ratio for everything, though, but when the dimensions look close, I will usually try to trim it up to the actual ratio.

stamptrol

And, it's amazing how many things in nature follow the golden ratio: the best one is the ratio of belly-button height to overall height.

Cheers,

wass

The design of the HP "voyager" series of calculators uses the golden ration for the overall package width to height ratio. The HP-12C, one of the voyager calculators, is still being manufactured and sold in large quantities for the past 30 years. Some people credit this design for at least part of its amazing sucess, me included.

LoopyByteloose

The Golden Ratio is indeed a profoundly interesting mathematically concept. It included the Fibonacci series, which has the claim to fame of accounting for how bunnies breed, maybe including Playboy bunnies.

The other side of it is that a 72 degree angle crops up in the Golden ratio. This has given us the Penrose Tiles which are based on 72 degrees and 36 degrees. But the fun doesn't stop there.

The past few decades began to recognize that crystals were not all based on a rectangular system of tidy repetition. We now have new horizons in material science based on the quasi-crystal, which is a three dimensional representation of the the Penrose Tiles.

In other words, we are still finding that pesky Golden Ratio cropping up in new information about nature. It is a transcendental number, along with Pi. And there is just something I can't resist about transcendental numbers. They seem able to continually surprise us with new insights.

Now I indeed need to contemplate my navel positioning.

Whit

It is commonly taught in Architectural schools.

Tracy Allen

This is something I've been thinking about lately. My son and his fiance have a Fibonacci theme for their wedding, which will occur on August 13th -- 8 and 13 are Fibonacci numbers. To make 2011 fit, you can put the "20" above the "11" and color it so you see 21 vertically, and the bottom row horizontal is then 1,1, which seeds the Fibonacci sequence. Zero? It is the origin, I guess. Their invitation is a classical construction based on the series, a classical golden spiral, which mirrored makes a nice heart. Ahhhhhh...

The couple and many of their friends are musicians. I've been thinking about a way to bring the sequence to life musically when I give my little speech. Musical tones in Fibanacci ratios are particularly rich in subharmonics (beat frequencies), and especially so the golden ratio as the limit. Attached is a demo that works on the Parallax demo board, to play two mixed tones on each of the stereo audio channels. It does not do much yet, and I'm not sure where to take it as an entertaining allegory for musicians being married. But there you have it. It is set up to play a ratio of 8 to 5 on one channel, and a ratio of 13/8 against (1 + root(5))/2 on the other.

Tracy Allen

@byteloose, the golden ratio is not a transcendental number, far from it. The golden ratio is the solution of a simple quadratic equation: r^2 - r - 1 = 0 and the solution is r = (1 + sqr(5))/2. The sqr(5) is irrational.

A transcendental number is one that is not the root of a polynomial in any degree. A number that is the root of a polynomial of any degree with integer (or rational) coefficients is called an algebraic number. A transcendental number is one that is not algebraic. It is very difficult to prove that a number like pi or e is transcendental, despite the fact that it is known that most numbers are transcendental. That is the set of algebraic numbers are of measure zero on the number line and most of the "space" is occupied by transcendental numbers.

LoopyByteloose

@Tracy Allen
Humble pie once again.

Nonetheless, it is one of my favorite constants. Let's say it is a 'metaphysical constant' of sorts; along with pi and e and a few others that seem to resurface again and again in nature.

Tracy Allen

Math technical terms tend to be quite precise! Looking at Wolfram's site, I was interested to see that it can be proven that either (pi * e), or (pi + e), or both are transcendental. But if it is only one or the other, then the question remains of which one, still not proven.

I agree, the golden ratio is very special and appealing. Another of its many properties is that its continued fraction is the slowest converging of any real number.

That has interesting consequences for numerically controlled oscillators (as in the NCO mode of the Propeller counters). A frequency in that ratio to the clkfreq has the greatest amount of jitter, the most subharmonic content in its output.

LoopyByteloose

I realize that mathematics has very precise terminology, but part of me feels that the Golden Ratio should be a transcendental number. But some darned mathematician grabbed the term before I got to it.

I sailed off into contemplation of 3-D Penrose Tiles and Quasi-crystals after posting my comment. I suspect that there is something major to be discovered there.

So much of our present knowledge is based on Newton and his calculus is based pretty much of the summation of cubes and rectangles. One might say that they are the fundamental module of his mathematics and it is highly dependent on a Cartesian co-ordinate system. I do admit that there are also conversions to Polar notation and circular rotation, but it still starts out with the square and the cube to build its world.

But if you look at Penrose Tiles, you are presented with an entirely different fundamental modularity. Instead of one module (the rectangle or cube), you have two forms of tiles that provide fundamental modules to fill a plane or space.

Buckminster Fuller's geodesics are a spherical modeling based on the Golden Section and reflect a relationship with the 3-D Penrose Tiles. The icosahedral and dodecahedron have Golden Section relationships within them.


Could it be that the 3-D Penrose Tiles might be the foundation of a new better system of calculus, which is based on the Golden Mean? I dunno.


But if it is so. you heard it first on a Parallax Forum.

http://www.goldennumber.net/quantum-E8.htm

Mark_T

Penrose and some other aperiodic tilings are more simply described as slices through higher-dimension cubic lattices taken at irrational angle. For Penrose tilings it is a 5D lattice. The basic structure of aperiodic 5-fold pseudo-symmetry was known to the practioners of islamic art BTW.

http://www.geom.uiuc.edu/apps/quasitiler/

http://comdig.unam.mx/index.php?id_issue=2007.09#26855

Jimbo30

Wow, thanks guys. ...... Even a child in a womb harbors the golden ratio. This is why the structure of life is so beautiful.

localroger

If you just lay stuff out and fiddle with it until it "looks right" it's amazing how many Golden Ratio rectangles you will find in the result.

LoopyByteloose

http://goldennumber.net/universe.htm

Here is another page of profundity. While Mark T is able to use the jargon correctly, that jargon tends to go over the top of people's heads. Images seem to help people intuitively recognize its worth. And it isn't all about rectangles. There are spherical relations in the microcosm and the macrocosm. Something related to Physics on a profound level is going on.

Yes indeed, the Arab did something similar to Penrose Tiles, long before he copyrighted/patented his work. But it seems that was based on a 10 fold geometry, not Penrose's five-fold work. The fact that these tiles are aperiodic makes it hard for us to visually them extending infinitely in all directions, but they can and they may model interlocking/overlapping spheres of influence throughout the universe.