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Proportional Dividers - Golden Section and Generalized #1: Phi (Golden Ratio) Rule (Ruler, Scale, etc.)

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Blog entry by Mark Whitsitt posted 1348 days ago 4212 reads 2 times favorited 7 comments Add to Favorites Watch
no previous part Part 1 of Proportional Dividers - Golden Section and Generalized series Part 2: Proportional Divider (4 arms) - General Equation »

Hi,

I’ve been intrigued with the Golden Ratio (Golden Rectangle, Golden Section, Golden Mean etc.) for some time, and I’m getting ready to make a set of Golden Ratio (or Fibonacci) Dividers to assist in design and layout of some projects I’m thinking about.

I’m not going to review this topic in any great depth as there are excellent discussions of it right here on Lumberjocks (e.g. David's post). Suffice to say the Golden Ratio has many interesting features and appears to be significant in nature and aesthetics.

Specifically, the Golden Ratio is 1.618034, and is often given the generalized mathematical name “Phi”. One of the more interesting characteristics, to me anyway, is that the inverse of Phi is Phi-1. That is, 1/1.618034 = 0.618034
I think there is a mathematical categorization for such numbers, but I don’t know what it is.

Anyway, I found these “Phi rules” on Woodcraft earlier today, and thought I’d try my hand at generating my own template for such a rule. Here it is…

There are basic instructions in one of the PDF files. While the “English” scales are very close to being accurate, I DO NOT guarantee that 1 inch actually equals 1”. Even if the English measurements are off, the rule is internally consistent, so if you’re not into precision, you can still use this to lay out Golden Ratio dimensions.

Linked are a 24 inch Phi rule, which you’ll need a large format printer for; be sure to turn off scaling!

I also created a smaller 10 inch rule that you can print on standard letter size paper. This is the one with the instructions on it.

Enjoy!

Mark

-- -- "there are many good reasons to use old hand tools, but moral superiority is NOT one of them..."



7 comments so far

View John Ormsby's profile

John Ormsby

1276 posts in 2338 days


#1 posted 1348 days ago

Mark,

I have this Bridge City Divider and it has the Golden Mean (rule) capability. It is very accurate. I have played with the golden rule in the past and found it interesting. It does bring balance in certain situations.

http://www.bridgecitytools.com/Products/Dividers/PD-11+Proportional+Divider

-- Oldworld, Fair Oaks, Ca

View TopamaxSurvivor's profile

TopamaxSurvivor

14604 posts in 2276 days


#2 posted 1348 days ago

Mark, Your links ask me to sing into your Google account? Is that what it is trying to get done? Not quite sure, never ran into that before.

-- "some old things are lovely, warm still with life ... of the forgotten men who made them." - D.H. Lawrence

View CaptainSkully's profile

CaptainSkully

1190 posts in 2159 days


#3 posted 1348 days ago

Hey Mark. Just to clarify, I believe that Phi – 1 = 1/Phi is the only case in mathematics where that is true. For example 2-1 is not equal to 1/2. Yet another reason why Phi is so magical. It’s interesting that you called it an inverse instead of the reciprocal. The inverse is to the -1 power, which means 1/x^1, but I haven’t heard that since college. Nice writeup. Thanks for making me think.

-- You can't control the wind, but you can trim your sails

View Mark Whitsitt's profile

Mark Whitsitt

86 posts in 1580 days


#4 posted 1348 days ago

Hey, Survivor… the documents are indeed up on Google Docs, and you should be able to access these docs without having to log into my account. My testing seems to work, but if you want, send me your email address by private message and I’ll send them to you in an email.

Mark

-- -- "there are many good reasons to use old hand tools, but moral superiority is NOT one of them..."

View Mark Whitsitt's profile

Mark Whitsitt

86 posts in 1580 days


#5 posted 1348 days ago

Skully,
Absolutely magical… I haven’t actually seen any other instances where 1/x = x-1, and a couple of places on the interwebs say this is a unique characteristic for positive numbers, , so it’s pretty cool…

(I wonder if there are any positive real numbers where the more general case applies? e.g. 1/x = x – n where n is the integer part of x—e.g. 1/n.yyyy = n.yyyy – n)

and since x^1 = x, the inverse of x is 1/x (e.g. x^-1). I believe the term “reciprocal” is used when working with fractional numbers (x/y—> y/x) and “inverse” is used with real numbers (e.g. n.yyy—> 1/n.yyy). “Reciprocal” also works here since n.yyy = n.yyy/1

Cheers!
Mark

-- -- "there are many good reasons to use old hand tools, but moral superiority is NOT one of them..."

View Mark Whitsitt's profile

Mark Whitsitt

86 posts in 1580 days


#6 posted 1348 days ago

John,
That’s a pretty spiffy divider! and a pretty spiffy price!

Mark

-- -- "there are many good reasons to use old hand tools, but moral superiority is NOT one of them..."

View Div's profile

Div

1653 posts in 1541 days


#7 posted 1348 days ago

Mark, apart from the magic of Phi, I really like your tagline! It is so true….

-- Div @ the bottom end of Africa. "A woodworker's sharpest tool should be his mind."

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