Determining the golden section?

[polkat]

I'm looking for a simpler (faster) way to determine the golden section of a line. In Alvin King's paper of Cremonese F hole placement, he shows an example for a 100mm line (I assume millimeters) from A to B and places the golden section C at 61.8mm. If 61.8 is accurate on a base of 100, could it not be said then that 61.8% (percent) of any length line would represent the golden section?

[Dean_Lapinel]

That's a logical assumption that I wish I could verify. Unfortunately I have read way too much about the history (and theories) of violin geometry and I'm at the point of thinking that we are trying to cage a naturally wild animal.

In essence you are correct with regards to the technical term but as it applies to violins I think there may have been more freedom in design.

See this link though for a nice discussion on these wonderful mathematical phenomena.

Golden section

Dean

[Ken Pollard]

As Dean mentioned, the 61.8% is ok -- mainly being 2 / (1 + sqrt(5) ) -- but not very Baroque.

As an alternative, you could take a ratio of consecutive Fibonacci numbers

1, 1, 2, 3, 5, 8, 13, 21, 34,...

For some applications, a ratio such as 8/5 ( or 5/8) gets you pretty close, or if you want it to 3-place accuracy, you can use 13/8 (or 8/13). Ratios of integers appeals to my sense of the time-period. I really have no idea whether it was used in violin design or not. I used to know, but then I learned more. Now I'm just confused. Certainty is so much easier with fewer facts to worry about.

[polkat]

Interesting article Dean, if a bit confusing. Still, it seams to support .618 as a sort of golden number, and .618 is indeed 61.8% of 1 which is 61.8mm of 100mm. I will research it further.

Ken, you went a bit over my head there, but I understand the gist of it. Whoops! I feel confusion setting in!

[Ken Pollard]

Didn't mean to go over your head, Polkat -- just didn't want to go under, either.

The Fibonacci sequence is a maybe 12th-century idea, created by adding the two prior numbers to get the next.

1+1 = 2, 1+2=3, 2+3= 5, 5+3=8, and so on.

The Golden Ratio is at least Ancient Greece. So both of these things were in the air back in Strad's time. Which doesn't mean he used them.

But he could have.

Anyway, folks found that as you made ratios of 2 Fibonacci numbers, they converged to the Golden Ratio. In my math classes, I show the students how to create a golden ratio using a right triangle with one leg 1 unit long, the other 2 units long, and the hypoteneuse then square-root-of-5 long. The golden ratio, as I usually see it, is written as the reciprical of what you have, or 1/0.618 (approx). That's where the 'sqrt-5' came in in my previous post.

Anyway, ratios of integers are cool for the Baroque time period, as far as I know, and to get to a Golden Ratio, on could use dividers set to 8/5 or 13/8, and be good enough. At least that seems reasonable to me, for what that's worth. If you were looking for an 'easier' way to get a golden ratio, you could make some dividers with the appropriately scaled legs. Or just create a ruler with the appropriately scaled tick marks. Count off 5 for this, 8 for that.

On the other hand, it may in practice be easier today to just multiply your measurement by 0.618, which is what you may have been asking anyway.

[Wolfjk]

Hi Ken Pollard,

quote:

---

1+1 = 2, 1+2=3, 2+3= 5, 5+3=8, and so on.

quote:

---

Now you let the rabbits out of the bag!

[GMM22]

The simplest (and entirely acceptable) way to get golden ratios is by multiplying any initial length by .618. One can also multiply a given value by 1.618 to get the larger section.

[GMM22]

I think the Golden Ratio is one of the most interesting things in mathematics. I use it whenever I can.

[Ken Pollard]

Hi Wolfjk,

"Now you let the rabbits out of the bag!"

As a simple test question, I'd ask my students to state the connection between Ancient Greek architecture, rabbit breeding, and Baroque violin design (back when I was pretty sure it was involved).

Students who had attended lecture that day would write "The Golden Ratio" for full credit. Students who hadn't attended lecture that day, and who hadn't bothered to inquire, would write something inspired like "You can use math to describe them." That would be zero credit -- it was a math class after all.

And then there were a few kind hearted souls who would write "The Golden Rule". Worth a few points -- because I'd want someone to do that for me if the situation were reversed.

[David Tseng]

Perhaps the geometrical construction is easier to understand. Let's draw a right triangle ABC in which one side is twice the other in length. Take BC as the unit of measurement, i.e., BC=1. Take C as the center, CB the length and draw an arc BP. Now take A as center, AP the length and draw an arc PD. Draw a perpendicular line from D. ADC' and ABC are similar triangles and therefore the ratio of 2 is maintained in the smaller triangle. In a similar way, continue the golden section of AD, etc.

In drawing the violin mold, AB is the center line or the length of the mold. Do the GS starting from Top or B end, then do the same starting from A. The scroll can also be constructed using GS.

[Wolfjk]

Hi K Pollard,

quote:

---

And then there were a few kind hearted souls who would write "The Golden Rule". Worth a few points -- because I'd want someone to do that for me if the situation were reversed.

quote:

---

Perhaps you were over generous!

Thanks for your post. Now I know that the "golden section" would be OK if violins grew on trees!

Cheers Wolfjk

[polkat]

Thanks guys. I was thinking in terms of straight lines, so GMM22 seems to have verified my opening thread. It's been over 40 years since high school, and math was never my strong point, but I'm beginning to get the rest of it!

[pandora]

This is somewhat off-topic, but if you're following this thread because you're into wood, building things, and classic proportions, you might want to read Jonathan Hale's "The Old Way of Seeing" (if you haven't already). It wasn't just the ancient Greeks who built Fibonacci series and golden ratios into their buildings; I'm surrounded by them here in my mostly-18th c. New England village. Read it and you'll start seeing regulating lines EVERYwhere...it's kinda fun.

[Collin]

"http://goldennumber.net/images/goldenrulercard.gif" target=

"_blank">Golden Ruler

And:

"http://goldennumber.net/geometry.htm">Golden number

geometry

Fun stuff.

[Darren Molnar]

If your looking for a more accurate golden section mathematically

you just add more decimal places on to this infinite

number.

1.61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 28621

35448 62270 52604 62818 90244 97072 07204 18939 11374 84754 088807

53868 91752 12663 38622 23536 93179 31800 60766 72635 44333 89086

59593 95829 05638 32266 13199 28290 26788 06752 08766 89250 17116

96207 03222 10432 16269 54862 62963 13614 43814 97587 01220 34080

58879 54454 74924 61856 95364 86444 92410 44320 77134 49470 49565

84678 85098 74339 44221 25448 77066 47809 15884 60749 98871 24007

65217 05751 79788

 That`s it to the four hundredth decimal place, how much ,more

accurate do you want to be?

I think a lot of times the confusion with these kinds of numbers is

because they are,actually impossible numbers. There is no common

unit of measurement that is contained in the smaller line that will

fit into the larger line that is created when a line is divided at

the golden section.none. In other words,math does not fit. Modern

man has trouble trying to fit this in their head. It will always be

approximate if your using mathematics. To be accurate and not

approximate you have to use the realm of geometry.

[Bacon]

"For some applications, a ratio such as 8/5 ( or 5/8) gets you pretty close, or if you want it to 3-place accuracy, you can use 13/8

(or 8/13). Ratios of integers appeals to my sense of the time-period. I really have no idea whether it was used in violin design or not. I used

to know, but then I learned more. "

5/8 at the time of Strad was a perfect minor sixth

[Johnmasters]

quote:

---

Originally posted by:

Darren Molnar

I think a lot of times the confusion with these kinds of numbers is

because they are,actually impossible numbers. There is no common

unit of measurement that is contained in the smaller line that will

fit into the larger line that is created when a line is divided at

the golden section.none. In other words,math does not fit. Modern

man has trouble trying to fit this in their head. It will always be

approximate if your using mathematics. To be accurate and not

approximate you have to use the realm of geometry.

---

There is no rational number which is a comon divisor. That means the Golden ratio is not the ratio of two integers. Same statement. You meant to say "irrational number", not "impossible number." This ratio is the only one where in 1/X = 1-X and this X has an irrational solution. Other irrationals are pi or the sqrt of 2.

Modern mathematical man has absolutly no trouble with this concept............ It will always be approximate if you are using arithmetic, a special branch of mathematics.

For the antiquing people: natural damage should eventually approach a fractal nature. The rabbit problem which gives rise to the Fibbonacci series and is closely related to the golden ratio is relevant. It shows that one representation of the Fibbonacci is self-containing an infinite number of times. This leads to fractals. Bad antiquing is seen immediately because the brain tends to see patterns which violate this sort of thing. So you could discuss antiquing in this thread too.

[Darren Molnar]

 from

johnmasters:"  Modern mathematical man has absolutly no

trouble with this concept............ It will always be approximate

if you are using arithmetic, a special branch of mathematics. "

 If we have no trouble,then,why was this thread

started?

 You are correct, irrational, not impossible. This is also the

nick-name my wife gave me.

[Ken Pollard]

Irrational numbers, as well as terms such as imaginary numbers, mean different things to mathematicians than to 'normal' people. I've taught 'math appreciation' classes for awhile, and find the concept of precise definitions difficult to pass along to my students. Rather than working with a definition, they tend to want to argue with it, or interpret it.

The number pi is defined as the ratio of a circle's circumference to its diameter. It is also an irrational number, meaning that it cannot be expressed as a ratio of integers. So the circumference is not an integer (whole number) multiple of the diameter. The ancient Greeks knew this, and it seemed to have bothered them a great deal. Apparently they even tried to suppress this terrible secret.

Any truncation of the decimal representation of an irrational number, such as the Golden Ratio being 0.618, is a mathematical approximation. In the physical world, you just make sure to truncate at a point where it is inconsequential. Given my knife-wielding skills, a couple decimal places works for me. You may need another one, but after that, it's really not going to show up in a physical object obsevered with unaided human eyes.

Venturing into something which I know even less about, the idea of equal temperament has at its root the problems of irrational numbers -- how does one divide up an octave? The various ratios -- fifths, thirds, and so on -- don't add up correctly.

[Darren Molnar]

Have any of you picked up Francios Denis`

book on violin geometry? If your into this thread you`ll love it.

It covers all the ratios, irrational or not,  that apply to

instrument design. because of his work, I`ve abandoned copying for

now and am working on personal designs based on geometric,harmonic

and arithmetic ratios.

 

And they look more traditional than my old tracing

paterns.Its just another level of understanding for the

instruments. Cool stuff. And yes I paid a big bag of money for it

and yes, it was worth it.

[Wolfjk]

Hi Darren Molnar,

quote:

---

Have any of you picked up Francios Denis` book on violin geometry? If your into this thread you`ll love it. It covers all the ratios, irrational or not, that apply to instrument design. because of his work, I`ve abandoned copying for now and am working on personal designs based on geometric,harmonic and arithmetic ratios

quote:

---

If you follow the exact geometric design, are you not somewhat limited by the arching and ergonomic necessities of the instrument?

In practical terms, the golden section rectangle is not the most popular choice of shapes. It is the juxtaposition of shapes, lines, curves and the space they enclose or exclude that attract the beholder.

Regarding the irrational numbers; it is natures way of keeping those of us who are going roun the bend from bumping into each other!

Cheers Wolfjk

[Guest]

And for those that want to explore the number even more, don't

forget Mario Livio's book "The Golden Ratio".

I'm sure that there are tons of pages out there written on the

subject, but for the price (used under 10 bucks), it might provide

some interesting reading for some of you........

Johnmasters - Quote: "For the antiquing people: natural damage

should eventually approach a fractal nature."

I am very intrigued by this statement. Maybe you could start a new

thread explaining the practical applications/techniques...? Or PM

me and I'll give you my email.

Very curious indeed.......

(Thanks to all for the very informative thread !!!)

E.

[Fellow]

If you divide all numbers into two sets, A and B in such a way that

any number in A is less than any number in B (set) then either A has a maximum or B

has a minimum. The last statement is VERY profound. If your number system is not

true, (last statement) Einstein theory of relativity won't work neither. Don't you believe this?

(My professor told me when the question oF irrational numbers came up)

[Darren Molnar]

 

hi

wolfjk,  """"""""If you

follow the exact geometric design, are you not somewhat limited by

the arching and ergonomic necessities of the

instrument?"""""""""""""""""

The

same as if your following given molds,they do have to be playable

in the end.  Think of it as another tool in the toolbox.

Example; suppose a violist asks for a cutaway design and you`ve not

done one before. You could grab the lid from the coffee tin and

trace a cookie cut but it would look  just as described. If

you had the knowledge of compass skills behind you though,you could

maybe find a more cohesive curve.

[Johnmasters]

quote:

---

Originally posted by:

Darren Molnar

   If we have no trouble,then,why was this thread

started?

---

I didn't say "we", I said mathematicians.