[Ted Kramer]

1.61803 is a relatively important number in nature.
I'm no expert, but it has to do with Fibonacci numbers and a ratio that happens to show up in nature often. It has been said that 1.61803 has to do with beauty in the Western culture.

Does anyone know if this "Golden Ratio" has ever been used or considered with regards to GCA?

I might be interested in doing a little research on this subject.

-Ted

[TEPaul]

Ted:

Sounds like you're talking about Nature's strongest angle. One might even say it's the angle of Nature's Darwinianism. It's the angle of tree roots, the most common angle of branches, the basic angle of your dog or cat's leg to its body  and it's the basic angle of repose of most good bunkers that endure. But who's this Fibonacci guy? He sounds like an Italian liar to me. And he's not the one who figured out the importance of this basic enduring angle in Nature, Bill Kittleman did.

[Jeff_Brauer]

Ted,

The 5 to 8 (or 8 to 5) ratio is discussed often in landscape design schools, and since so many of us are landscape architects by trade, I will assume that it has - knowingly or not.

[Michael Moore]

In 1921 the rules were amended so that the ball could weigh 1.62 ounces and be bigger than 1.62 inches in diameter.

[Ted Kramer]

Fibonacci came up with a series of numbers based on starting with 0,1 and then adding the preceding two together:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987

If you divide any of these numbers by the one before it you come up with 1.61803 - it works better with the bigger numbers for accuracy's sake.

This number series was used to answer specific math problems/questions and later the 1.61 ratio or "golden section" began to be explored in nature, architecture, and other areas.

I wonder if that ratio could be found in some fashion in certain sections of golf holes that are generally considered good or pleasing.

-Ted

[John Nixon]

Ted,

The 5 to 8 (or 8 to 5) ratio is discussed often in landscape design schools, and since so many of us are landscape architects by trade, I will assume that it has - knowingly or not.

That ratio is also used by designers/illustrators/etc - as a starting point on laying out a piece. I use it pretty much all the time.

Has golf ball/club technology made the application of that ratio to course design obsolete as well?  

[Tom_Doak]

Strangely, no one at Cornell ever talked about that relationship in landscape architecture.  The rest of my class weren't very math-oriented, I did all the arithmetic for the group.

So, I have never thought about the relationship in terms of golf, and all I can tell you for certain is that you can lay out a pretty good course without ever thinking about it.

However, as a photographer, I think that's the approximate L x W relationship for a 35 mm slide, which mimics the average person's field of vision ... so every time you do ANYTHING by eye you're kind of using the relationship.

[Tim Gavrich]

Fibonacci came up with a series of numbers based on starting with 0,1 and then adding the preceding two together:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987

If you divide any of these numbers by the one before it you come up with 1.61803 - it works better with the bigger numbers for accuracy's sake.

This number series was used to answer specific math problems/questions and later the 1.61 ratio or "golden section" began to be explored in nature, architecture, and other areas.

I wonder if that ratio could be found in some fashion in certain sections of golf holes that are generally considered good or pleasing.

-Ted

Ted--
Incidentally, the Golden Ratio is not quite the qoutient of consecutive Fibonacci numbers.  These quotients get closer and closer to the GR, but never reach it, as it is an irrational number (as is pi, 'e,' and most sines, cosines, and tangents).  8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615384 (the decimal portion repeats endlessly).  And so on.

But this is a very intriguing thought.  Now that I think of it, many fairly generically-sized bunkers are approximately GR.  I'd imagine that a lot of greens have approximately those dimensions. 34 yards deep, 21 wide seems reasonable (maybe most are a little wider).

It's definitely possible, and not surprising if there's truth to it.

[Mark_Guiniven]

Golfclubatlas favourite Tony Cashmore wrote his essay on this in Paul Daley's GA Vol. III

Apparently, Tony has a letter from Dr. Mackenzie to a guy named John Hoy, I think, in New Zealand, in which Mackenzie mentions Di Vinci and golden section.

[Mark_Guiniven]

From 'Visuals in the golfscape' by Tony Cashmore, GAV3 p169

But there is a quite fascinating subject here, which really deserves to be explored in a far greater depth than is possible in this essay – a little known aspect of Alister Mackenzie’s golf architecture. He had knowl-edge of, and continually exploited ‘the golden section’ rule when fashioning the visual impact of golf holes. In a letter he penned to John Hoy, a friend he had made in New Zealand, Mackenzie wrote: ‘... I have been re-reading [my emphasis] Leonardo da Vinci’s theorem on the three to five relationship for proportions, and how that arrangement pro-duces harmony in all we try to see ... ’

Will Tony now be asked to produce this letter in a Seth Raynor/Cypress Point-routing kind of way?

[T_MacWood]

Fascinating. I've never heard of Fibonacci. Who was he?

That letter from MacKenzie (and article) sounds very interesting too, I'm always interested in the Good Doctors interests and influences.

[Mike_Young]

Brandon OMahoney who works with me has written an entire paper on this in regard to golf courses.  Maybe I can get him to post it.
Mike

[Phil_the_Author]

1.61803   Isn't that what one needs to multiply the number of individual posts on Gca by to come up with the total that Tom Paul has posted!  

[Tony_Muldoon]

Brandon OMahoney who works with me has written an entire paper on this in regard to golf courses.  Maybe I can get him to post it.
Mike

Yes please

[Jim Nugent]

Fibonacci came up with a series of numbers based on starting with 0,1 and then adding the preceding two together:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987

If you divide any of these numbers by the one before it you come up with 1.61803 - it works better with the bigger numbers for accuracy's sake.

This number series was used to answer specific math problems/questions and later the 1.61 ratio or "golden section" began to be explored in nature, architecture, and other areas.

I wonder if that ratio could be found in some fashion in certain sections of golf holes that are generally considered good or pleasing.

-Ted

Ted--
Incidentally, the Golden Ratio is not quite the qoutient of consecutive Fibonacci numbers.  These quotients get closer and closer to the GR, but never reach it, as it is an irrational number (as is pi, 'e,' and most sines, cosines, and tangents).  8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615384 (the decimal portion repeats endlessly).  And so on.

But this is a very intriguing thought.  Now that I think of it, many fairly generically-sized bunkers are approximately GR.  I'd imagine that a lot of greens have approximately those dimensions. 34 yards deep, 21 wide seems reasonable (maybe most are a little wider).

It's definitely possible, and not surprising if there's truth to it.

A slight clarification for the mathematically curious.  I suspect this is what Tim meant, but...

The decimal portion of irrational numbers does not really repeat endlessly.  It never repeats and never ends.  Irrational numbers cannot be expressed as fractions, i.e. as p/q, where p and q are integers (whole numbers such as 1, 2, 3, 4, 5...their negatives and 0).  

There are a whole lot of irrational numbers.  Way more than there are rational numbers (numbers that can be expressed as p/q).  A branch of mathematics called set theory "proved" that irrationals are an order of magnitude greater than rationals.  Also, that if you count, say, by ten's (0, 10, 20, 30, 40...) you get as many numbers as if you count by ones (0, 1, 2, 3, 4...).  Seems like there should be one-tenth as many, yes?  Not so.  Of course, the proof requires that you're able to count to infinity, which is why I don't believe it.  

I didn't explain this well, but it is fascinating stuff if you have any interest in math.  

[Ted Kramer]

Brandon OMahoney who works with me has written an entire paper on this in regard to golf courses.  Maybe I can get him to post it.
Mike

I would be most appreciative.

-Ted

[Ted Kramer]

Fascinating. I've never heard of Fibonacci. Who was he?

That letter from MacKenzie (and article) sounds very interesting too, I'm always interested in the Good Doctors interests and influences.

Tom,

If you type his name into a google search, you'll come up with lots of info . . .

-Ted

[Evan Fleisher]

Fascinating. I've never heard of Fibonacci. Who was he?

That letter from MacKenzie (and article) sounds very interesting too, I'm always interested in the Good Doctors interests and influences.

Here are a few resources...

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html

http://math.holycross.edu/~davids/fibonacci/fibonacci.html

http://www.mscs.dal.ca/Fibonacci/

[John Keenan]

I just Googled Fibonacci and it seems that there is The Fibonacci Quarterly a modern journal devoted to studying mathematics related to this sequence.
 
As a foot note Fibonacci was also known as Bigollo, which may mean good-for-nothing or a traveller.

I expect to do a bit more research on this gentleman.

[Tony_Muldoon]

I'm surprised nobody has been as obvious as I am and mentioned its role in the Da Vinci Code.

And in a pathetic attempt to make this relevant: was there ever a lamer bit simile than when he observed that looking for a clue in there (not giving anything away am I?) was as hopeless as looking for the lost blade of grass on a golf course?  No golfer he.

[Marty Bonnar]

Living with a gal who possesses a degree in Pure Maths, I have been exposed to Sr. Fibonacci and his darn sequence way too much for one life!

Actually, I am a big fan of the whole mathematics-as-art-as-mathematics scene. Maths, it seems to me, is everything. For many years, I sought the answer, but as Douglas Adams kindly pointed out to me when I was 19 it is, of course, 42.

If this stuff floats your Newtonian boat, also check out Fractals and more interestingly, The Mandelbrot Set (cool name for a Band I always think!)

FBD.

[George Pazin]

Of course, the proof requires that you're able to count to infinity, which is why I don't believe it.  

I didn't explain this well, but it is fascinating stuff if you have any interest in math.  

Jim, my recollection of this branch of set theory is that the proof does not require you to count to infinity, which, of course, no one can do. If you are truly interested in this proof, I suggest you obtain a copy of my all time favorite non-fiction book, entitled Journey Through Genius, by Prof. Dunham. This book, pure gold for math geeks, covers the background history of interesting problems in math and a layman's version of the proof solving the problem, and then subsequent developments in the field for each problem. The last chapter of the book is devoted to the problem of irrational numbers, and it is beyond fascinating. Not as entertaining as the chapters on Gauss and Euler, but more interesting to me. (I think the guy who solved the problem was Paul Erdos, but it's been a good while - too long, in fact - since I looked at my copy of the book, so don't quote me on that. * D'oh - just checked - it was Cantor! Erdos didn't sound right to me, but I couldn't think of who it was.)

I enjoy all this golf stuff, but it's a bunch of lightweight fluff compared to pure math....

P.S. Here's the link to buy this book :

http://www.amazon.com/gp/product/014014739X/103-3743298-4202202?v=glance&n=283155&s=books&v=glance

Another good book for math geeks out there is Islands of Truth - A Mathematical Mystery Cruise. Neither of these books requires really advanced math, which was good for me, as I hit the wall learning linear algebra.

[Mike_Cirba]

Ted,

I assumed that was your course rating for ShoreGate.  

[Evan Fleisher]

Living with a gal who possesses a degree in Pure Maths, I have been exposed to Sr. Fibonacci and his darn sequence way too much for one life!

Actually, I am a big fan of the whole mathematics-as-art-as-mathematics scene. Maths, it seems to me, is everything. For many years, I sought the answer, but as Douglas Adams kindly pointed out to me when I was 19 it is, of course, 42.

If this stuff floats your Newtonian boat, also check out Fractals and more interestingly, The Mandelbrot Set (cool name for a Band I always think!)

FBD.

Martin,

Ask and 'ye shall receive... http://artists.iuma.com/IUMA/Bands/Le_Mandelbrot_Set/

Want a picture as well?...http://beehive.thisishull.co.uk/default.asp?WCI=SiteHome&ID=7921&PageID=42465

[Jamie_Duffner]

Fibonacci is a giant in the world of mathematics.  The Fibonacci sequence is used in technical analysis with startling statistical significance.  He is also credited with being the first person to present value cash flows, thus inventing the entire concept of discounting.  Modern finance is built on the foundation he created.

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