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The Golden Ratio

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Blog entry by stiletta posted 1387 days ago 1456 reads 1 time favorited 17 comments Add to Favorites Watch

Being a macihinist as well as a woodworker, I was always intrested in the Golden ratio .

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

The golden ratio is an irrational mathematical constant, approximately 1.6180339887

I just happened to be reading exodos 25-10 in the bible and came across this passage.

The Lord said to Moses,
Tell the Israelites to make a Box out of acacia wood, 45 inches long, 27 inches wide, and 27 inches high.Cover it with pure gold inside and out and put a gold border all around it.12 Make four carrying rings of gold for it and attach them to its four legs, with two rings on each side.13 Make carrying poles of acacia wood and cover them with gold14 and put them through the rings on each side of the Box.15 The poles are to be left in the rings and must not be taken out.16 Then put in the Box the two stone tablets that I will give you, on which the commandments are written.

Build a box 45×27” !, this is so close to the golden ratio it actually works out to 1.6666666 I wonder which one is correct.

It seems to me that this ratio goes a little futher back than the renaissance or the Greeks.

So the question is… Who was the first architect ? And when was the ratio first used?

Thanks for sharing,

-- WWW.MACHINISTCHEST.COM



17 comments so far

View OttoH's profile

OttoH

862 posts in 1636 days


#1 posted 1387 days ago

Interesting, for when Noah was told to build the ark (Genesis 7:15) it was to be 50 cubits wide and 30 cubits tall, that too is 1.66666666666666. The Golden Ratio or a very similar measurement has been used for a very long time.

-- I am responsible for how I respond to everything in my life - - Deadwood SD

View SouthpawCA's profile

SouthpawCA

254 posts in 1859 days


#2 posted 1387 days ago

Aliens!!! Yep, it was aliens from Alpha Centauri or maybe Mars.

-- Don

View Eli's profile

Eli

141 posts in 1632 days


#3 posted 1387 days ago

Check out “The Golden Ratio” by Mario Livio. The golden ratio isn’t as prevalent as most people think. 1.666666… is just 1 2/3. While that IS a Fibonacci fraction, it’s not the ratio. “Approximately” the golden ratio is tricky because the level of approximation is subjective. If 1.6666666… is close, why not 1.7 or 1.5 or 2?

Anyway, the book is great and, beyond delving into some of the math-magic involved, it analyzes the various places the ratio can allegedly be found to determine whether it is legitimate. He looks for places that have evidence of the ratio AND evidence that the architect used it knowingly. As he points out, the ratio can be found all over, but if the designer didn’t use it on purpose, it’s not significant. Basically, he looks for the answers to your questions.

I vaguely remember that “most pleasing” rectangle business being very questionable as well. I can go look up Livio’s conclusions if you’d like.

In the end, the ratio has become important in design not because it was in the past, but because people think it was in the past. Perception is reality.

Eli

View Dark_Lightning's profile

Dark_Lightning

1690 posts in 1735 days


#4 posted 1387 days ago

1.6666666666 (hmmm, lots of 666 666 in there…) isn’t really all that close to 1.618033…etc. There are many texts devoted to the study of the Golden Mean. Some are complete booshwah, making all sort of esoteric scientific claims. Some have a thorough mathematical grounding. Professor Mario Livio has written a really good and interesting book in the mathematical vein entitled “The Golden Ratio”. In it, he discusses the history of the number. There, he discusses how the number seems to be a sort of “ideal” that people could design towards, sort of a “mean” (in the mathematical sense) that is approached, but not necessarily a hard and fast number rule, around which things are made and perceived as beautiful by mankind. Remember, though, that we are already part of nature, and as such, we are inclined to see this sort of beauty. This number can be found in places as far apart as snail shells seem to be from sunflowers. Quite interesting.

View Dark_Lightning's profile

Dark_Lightning

1690 posts in 1735 days


#5 posted 1387 days ago

Wow, Eli, I was writing while you were posting! Cool!

View sras's profile

sras

3813 posts in 1755 days


#6 posted 1387 days ago

Here’s a little Golden ratio math:

Every now and then I want to calculate the ratio just to make sure I have it right. Not that I worry that much about it – I just like to verify my memory. A rectangle with a Golden ratio has the following characteristic:

If you remove a square equal to the the short side, the remaining rectangle also has a Golden ratio.

If we call the long side A and the short side B, then (A/B) = [B/(B-A)]. The algebra from this works out to a value of 1/2 + [Square root(5)]/2 = 1.6180339887

I know, I’m a nerd

-- Steve - Impatience is Expensive

View Eli's profile

Eli

141 posts in 1632 days


#7 posted 1387 days ago

Could you go through that math? You can pm me if you like. I don’t see how you got numbers from that equation.

View Eli's profile

Eli

141 posts in 1632 days


#8 posted 1386 days ago

Thanks, Barry. I was wondering more about this part: “(A/B) = [B/(B-A)]. The algebra from this works out to a value of 1/2 + [Square root(5)]/2” Where did the 1/2 and sqrt(5) come from? I got as far as (a^2 + b^2)/b=a.

View sras's profile

sras

3813 posts in 1755 days


#9 posted 1386 days ago

Okay – it might be tricky to do in text form. I’ll give it a try. I have about an hour before company shows up.

-- Steve - Impatience is Expensive

View sras's profile

sras

3813 posts in 1755 days


#10 posted 1386 days ago

Lets start with a rectangle that has the Golden ratio:

I tried to draw one with text characters, but cannot get it to work.

xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x…......................................x
x…......................................x
x…......................................x
x…......................................x
x…......................................x
xxxxxxxxxxxxxxxxxxxxxxxxxxxxx

If you ignore all the dots in the middle, that looks about right. Lets call the long side A and the short side B. The ratio of these sides (A/B) will be the value of 1.61…
One of the properties of a golden rectangle is if you remove a square equal to the the short side, the remaining rectangle also has a Golden ratio. Something like this:

xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x…....................x….............x
x…....................x….............x
x…....................x….............x
x…....................x….............x
x…....................x….............x
xxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Sorry I can’t come up with better line drawings! The point is the box on the left is a square with sides of length B. The rectangle on the right has its long side of length B and the short side is now (A-B). Since it is a golden rectangle, then B/(A-B) has the same value of 1.61…

(( NOTE: I got the expression A-B backwards in my original post. That’s what happens when I skip steps!!))

Since both ratios have the same value, then (A/B) = [B/(A-B)]

Okay so far?

I’m going to break this into multiple posts to make it easier to point out where my description may not be working. We’ll call this step 1

-- Steve - Impatience is Expensive

View sras's profile

sras

3813 posts in 1755 days


#11 posted 1386 days ago

Now we need to work with this expression (A/B) = [B/(A-B)]

The first thing we want to do is set the length of side B equal to 1. That makes the length of side A equal to the Golden ratio. We need to replace the “B” in the above expression with a “1”. Then we get

(A/1) = [1/(A-1)]

Now we need to rearrange this a bit. First get rid of the “1” on the left side since A/1 = A

A = [1/(A-1)]

Now for a little bit of algebra. I will go through this one step at a time. Makes for a little longer description, but then I don’t skip any steps.

Multiply both sides of the expression by (A-1)

(A-1) x A = (A-1) x [1/(A-1)]

The part on the left side of the = sign simplifies to A^2 – A. I am using A^2 to describe A*A

A^2 – A = (A-1) x [1/(A-1)]

The part on the right side of the = sign simplifies to (A-1)/(A-1) which is equal to 1. That gives

A^2 – A = 1

Next is to subtract 1 from both sides of the expression

A^2 – A – 1 = 1 – 1

This simplifies to

A^2 – A – 1 = 0

This is known as a quadratic equation

We will call this step 2

-- Steve - Impatience is Expensive

View sras's profile

sras

3813 posts in 1755 days


#12 posted 1386 days ago

Now we need to solve the quadratic. I am going to just throw the soltuion at you at this point. If you recognize it great. If not, trust me ;)

The general form of this solution is when you have an equation like

a X^2 + b X + C = 0

Where X is the unknown and a, b, and c are constants, then X is found by the expression

X = [-b +/- SQRT ( b^2 – 4 a c )]/(2 a)

The +/- means that a solution is valid using + or – and SQRT means to take the square root

Our equation is

A^2 – A – 1 = 0

Here A is our unknown
the constant a is 1
the constant b is -1
the constant c is -1

Putting this in the solution equation, we get

A = [-(-1) +/- SQRT ( ( -1 )^2 – 4×1 x -1)] / (2×1)

This simplifies to

A = [1 +/- SQRT ( 1 + 4)] / 2

or

A = [1 +/- SQRT (5)]/2

In this case we are looking for the solution with the + sign and we get

A = [1 + SQRT (5)]/2

or

A = 1/2 + [SQRT (5)]/2

Or

A = 1.6180339887

We will call this the last step (3)

-- Steve - Impatience is Expensive

View sras's profile

sras

3813 posts in 1755 days


#13 posted 1386 days ago

stiletta,

I apologize if I drug your conversation in a direction you other than you were hoping for. Feel free to post a comment and get us back on track if you wish!

Steve

-- Steve - Impatience is Expensive

View stiletta's profile

stiletta

23 posts in 1839 days


#14 posted 1386 days ago

Steve,

No problem that I can see…Have at it!

John

-- WWW.MACHINISTCHEST.COM

View Eli's profile

Eli

141 posts in 1632 days


#15 posted 1386 days ago

Thanks.

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