# Proportional Dividers - Golden Section and Generalized



## iowawoodworker (Mar 30, 2010)

*Phi (Golden Ratio) Rule (Ruler, Scale, etc.)*

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


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## oldworld124 (Mar 2, 2008)

iowawoodworker said:


> *Phi (Golden Ratio) Rule (Ruler, Scale, etc.)*
> 
> Hi,
> 
> ...


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


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## TopamaxSurvivor (May 2, 2008)

iowawoodworker said:


> *Phi (Golden Ratio) Rule (Ruler, Scale, etc.)*
> 
> Hi,
> 
> ...


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.


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## CaptainSkully (Aug 28, 2008)

iowawoodworker said:


> *Phi (Golden Ratio) Rule (Ruler, Scale, etc.)*
> 
> Hi,
> 
> ...


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.


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## iowawoodworker (Mar 30, 2010)

iowawoodworker said:


> *Phi (Golden Ratio) Rule (Ruler, Scale, etc.)*
> 
> Hi,
> 
> ...


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


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## iowawoodworker (Mar 30, 2010)

iowawoodworker said:


> *Phi (Golden Ratio) Rule (Ruler, Scale, etc.)*
> 
> Hi,
> 
> ...


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


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## iowawoodworker (Mar 30, 2010)

iowawoodworker said:


> *Phi (Golden Ratio) Rule (Ruler, Scale, etc.)*
> 
> Hi,
> 
> ...


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

Mark


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## woodspark (May 8, 2010)

iowawoodworker said:


> *Phi (Golden Ratio) Rule (Ruler, Scale, etc.)*
> 
> Hi,
> 
> ...


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


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## iowawoodworker (Mar 30, 2010)

*Proportional Divider (4 arms) - General Equation*

Ok, while I'm messing with the Golden Ratio and mathematics, I thought I'd also try to figure out how to make a simple cross beam proportional divider. It's been a while since High School Geometry, so I had to go back and reprove the relationships between proportional isosceles triangles to myself. I also couldn't find this specific formula for making a divider anywhere on the net in a reasonable amount of time, so I just re-derived it.

(I believe this is all correct, but don't use this for any mission-critical or potentially injurious tasks until you've tried it out! *grin*)

What we're talking about is essentially two arms of the same length, joined somewhere between their ends so that they can pivot around that point, and the distance between the arms at one end is some proportion of the distance between the arms at the other end. E.g. given a ratio (proportion) of 1:2, if I make the arms 2 inches apart at one end, the arms are 1 inch apart at the other end. The result is a divider having 4 arms in the shape of an "X"

(For those who are interested, this divider creates two similar isosceles triangles, and since the legs of the triangles are fixed, the law of cosines says that the bases of the triangles must be proportional however far apart they are set.)

This formula is generalized so that you can make a divider of any length and any proportion, including a cross-beam Golden Ratio divider. Just decide how long you want the overall divider to be, decide what proportion you want, and plug those numbers into the formula to determine how far from one end you need to put the pivot on the arms of the divider.

Enjoy!

Mark










BTW, if you use ratios, greater than 1, you are actually calculating the distance Y, not X…


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## TopamaxSurvivor (May 2, 2008)

iowawoodworker said:


> *Proportional Divider (4 arms) - General Equation*
> 
> Ok, while I'm messing with the Golden Ratio and mathematics, I thought I'd also try to figure out how to make a simple cross beam proportional divider. It's been a while since High School Geometry, so I had to go back and reprove the relationships between proportional isosceles triangles to myself. I also couldn't find this specific formula for making a divider anywhere on the net in a reasonable amount of time, so I just re-derived it.
> 
> ...


Therefore, you must be using .618 for R?


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## BertFlores58 (May 26, 2010)

iowawoodworker said:


> *Proportional Divider (4 arms) - General Equation*
> 
> Ok, while I'm messing with the Golden Ratio and mathematics, I thought I'd also try to figure out how to make a simple cross beam proportional divider. It's been a while since High School Geometry, so I had to go back and reprove the relationships between proportional isosceles triangles to myself. I also couldn't find this specific formula for making a divider anywhere on the net in a reasonable amount of time, so I just re-derived it.
> 
> ...


Actually, for the golden ratio… the angle is 31.7 degrees… If you set a protractor on this angle and extend the radius… THEN any x and y (ordinate axis) values at any point intersecting on the extended radius will be the sides of a golden triangle. Say 13 and 8 is the nearest whole value that will give the .618 ratio. The easiest solution, I used for this is using excel spreadsheets.. of course in the computer. However in a practical way.. 13 and 8 plotting (even without a protractor) will do.


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## iowawoodworker (Mar 30, 2010)

iowawoodworker said:


> *Proportional Divider (4 arms) - General Equation*
> 
> Ok, while I'm messing with the Golden Ratio and mathematics, I thought I'd also try to figure out how to make a simple cross beam proportional divider. It's been a while since High School Geometry, so I had to go back and reprove the relationships between proportional isosceles triangles to myself. I also couldn't find this specific formula for making a divider anywhere on the net in a reasonable amount of time, so I just re-derived it.
> 
> ...


TS,

Yes, if you're making a Golden Ratio divider, R = 0.618 = Phi^-1

Mark


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## iowawoodworker (Mar 30, 2010)

iowawoodworker said:


> *Proportional Divider (4 arms) - General Equation*
> 
> Ok, while I'm messing with the Golden Ratio and mathematics, I thought I'd also try to figure out how to make a simple cross beam proportional divider. It's been a while since High School Geometry, so I had to go back and reprove the relationships between proportional isosceles triangles to myself. I also couldn't find this specific formula for making a divider anywhere on the net in a reasonable amount of time, so I just re-derived it.
> 
> ...


Bert, you are correct!

To be clear, the angle you're talking about is the angle "a" formed by the x-axis (abscissa) and a line with the formula:

y=(R*x)+0.

The angle "a" can be found using arctan(y/x) or tan^-1(y/x).

In this case (for a Golden Rectangle), 
y/x = R = 0.618, 
and angle "a" = 31.7 degrees.










As the diagram shows, you can draw perpendiculars from any point on the line to each of the axes to determine the dimensions of a Golden Rectangle. (In fact, the rectangle in the diagram is a Golden Rectangle)

Your method of using a protractor and extending the radii is really useful! To extend your idea and generalize the protractor approach, simply calculate arctan( R ), where R is any ratio you want to obtain, and set your protractor to the resulting angle!

Thanks for pointing this out!

Mark!

(P.S. yes, I know that Phi, the Golden Ratio, is an irrational number, and all equal signs "=" should be construed as "approximately equals", but when was the last time you laid out a dimension to the closest 100th? 8^) )


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## BertFlores58 (May 26, 2010)

iowawoodworker said:


> *Proportional Divider (4 arms) - General Equation*
> 
> Ok, while I'm messing with the Golden Ratio and mathematics, I thought I'd also try to figure out how to make a simple cross beam proportional divider. It's been a while since High School Geometry, so I had to go back and reprove the relationships between proportional isosceles triangles to myself. I also couldn't find this specific formula for making a divider anywhere on the net in a reasonable amount of time, so I just re-derived it.
> 
> ...


Additionally, if you have a polar to rectangular calculator (mostly all scientific calculator has) then it would be easy to have values of x,y being the rectangular axis by converting in the calculator with given R, 31.7 degrees as the polar coordinate.

Thanks too Mark for the analysis….


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## TopamaxSurvivor (May 2, 2008)

iowawoodworker said:


> *Proportional Divider (4 arms) - General Equation*
> 
> Ok, while I'm messing with the Golden Ratio and mathematics, I thought I'd also try to figure out how to make a simple cross beam proportional divider. It's been a while since High School Geometry, so I had to go back and reprove the relationships between proportional isosceles triangles to myself. I also couldn't find this specific formula for making a divider anywhere on the net in a reasonable amount of time, so I just re-derived it.
> 
> ...


These dividers look like they would be infinitely easier to get perfect than Fibonacci Calipers ;-))


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## iowawoodworker (Mar 30, 2010)

*Revised General Equation*

A couple of peoples' questions prompted me to make the equation from my last post a little less general so using it might be a little easier…


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## iowawoodworker (Mar 30, 2010)

*Proportional Divider - 3-Arm*

Ultimately, my goal with all this mathematics was to figure out how to design an accurate 3-arm proportional divider like the one posted by David. Please visit his blog post for more information and links.

The first step was to understand the Golden Ratio (post 1), the second step was to figure out the geometry of a simple 4-arm divider (post 2 & 3), and the final step is to apply this theory to the 3-arm divider I want to make:










I just rearranged the two isosceles triangles from the 4-arm divider to produce a 3-arm divider. Now "Z" is not directly measurable, but it can be calculated. Once you select the X of your choice, and then calculate Z, you can use the second equation to calculate Y.

The cross piece (dashed line) is the same length as the line segment W, so you'll need to calculate that length as well (W = X - Y). Because A=B=C=W, once you know W, you know how far down on the legs to place the cross piece.

Remember that all these measurements are from fastener to tip (or from fastener to fastener for the cross piece). You'll need to add a little bit of length to accommodate the fasteners…

Now I have to do something with all this. I'll be putting together models for both the 4-arm and 3-arm dividers in Sketchup, and then I'll try to actually construct them! I'll post as I can…

Cheers!
Mark


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## iowawoodworker (Mar 30, 2010)

*Golden Divider Prototype*

Here's the prototype of my golden divider, made with 1/8" tempered hardboard. It'll do for now, but I have plans to make this from something a little nicer…

The longest legs on this one are 11 3/8" long, so it's a little bigger than the ones I've seen posted elsewhere.

You can get my Sketchup Model on the 3D Warehouse by searching for "Jawhorse-Sawdust"

Mark


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## PurpLev (May 30, 2008)

iowawoodworker said:


> *Golden Divider Prototype*
> 
> Here's the prototype of my golden divider, made with 1/8" tempered hardboard. It'll do for now, but I have plans to make this from something a little nicer…
> 
> ...


very cool. still want to make one of those, but the thing is - I do all the design in SU where it's just too easy to multiply/divide by 1.618…


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## iowawoodworker (Mar 30, 2010)

iowawoodworker said:


> *Golden Divider Prototype*
> 
> Here's the prototype of my golden divider, made with 1/8" tempered hardboard. It'll do for now, but I have plans to make this from something a little nicer…
> 
> ...


PurpLev,
I know… I "pre-build" just about everything I make in Sketchup first… sort of work out the construction process there…

but sometimes I like to improvise with a scrap or other piece of wood in the shop, and having a few design/layout tools out there helps a lot…

for instance, I'm planning to make a couple of boxes out of crown molding, and with the calipers, I can pretty much do it all in the shop without any math beyond addition or subtraction (for the box bottom grooves…)

M


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## TopamaxSurvivor (May 2, 2008)

iowawoodworker said:


> *Golden Divider Prototype*
> 
> Here's the prototype of my golden divider, made with 1/8" tempered hardboard. It'll do for now, but I have plans to make this from something a little nicer…
> 
> ...


Studie gave Todd and I the makin's for those. They are hard to get exactly on the money ;-( I was thinking that making them proportinal just using an X design as in one of your previous blogs would be the best way to go.


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## iowawoodworker (Mar 30, 2010)

iowawoodworker said:


> *Golden Divider Prototype*
> 
> Here's the prototype of my golden divider, made with 1/8" tempered hardboard. It'll do for now, but I have plans to make this from something a little nicer…
> 
> ...


TS,

You're right about the precision issue and the simpler X design; reduce the complexity to reduce the error. But remember that Phi (the golden ratio) is an irrational number that keeps going and going (i.e. 1.61803399…), so you can't ever get it exactly.

Besides, I'm only using this as a improvisational design tool, and I figure the mathematical error in the tool is all part of making a work piece uniquely mine! Once I figure a couple of rough dimensions with the divider, the rest is all rulers and stop blocks!

Mark


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## mafe (Dec 10, 2009)

iowawoodworker said:


> *Golden Divider Prototype*
> 
> Here's the prototype of my golden divider, made with 1/8" tempered hardboard. It'll do for now, but I have plans to make this from something a little nicer…
> 
> ...


Cool, I will copy the locking mecahnism and put this on mine.
Best thoughts,
MaFe


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