Jim Tolpin’s article “Secrets of the Sector” in the June 2011 issue has stirred up a lot of interest and discussion among our readers. A sector is a way to eliminate arithmetic, especially division (multiplication’s tricky friend).

To help readers understand the sector, I made a couple short videos:

Video: Jim Tolpin’s ‘Secret of the Sector’

Make a Sector From a Crappy Folding Rule

One of the questions readers are asking is: Why did Jim make his sector’s divisions go to 13? One reader offered this suggestion:

*Dear Jim Tolpin,*

*I got my copy of the June 2011 issue in Tuesday’s mail. I read the article “Secrets of the Sector” with joy. It twigged a memory of long ago when I studied with an old carpintero near Santa Fe. He had something similar hanging on his tool rack, but didn’t show me what it was. Wednesday morning, I explored the Popular Woodworking web page and found the video on “Make a Sector From a Crappy Folding Rule.” I had one of those “crappy” modern rules sitting in the toolbox, and went into the shop and made my sector.*

*In the article, Jim states that it should have 13 divisions, but doesn’t indicate why. I almost ignored that directive and used 12, but made mine with 13. It intrigued me as to why. To my surprise, I wasn’t able to research much info off the web. Then it hit me!*

*I’ve been doing a lot of experimenting with using phi, or the Golden Section, in my turning; using it to refine relationships. I’ve no idea if this is Jim’s reason for using 13 divisions, but the ration of 13 to 8 is insanely close to phi; 1.625 ~ 1.618…! Certainly for most woodwork, it would be accurate enough.*

*I REALLY like the sector; it simplifies so much shop math and layout. Thank you for publishing the article!*

*— Bear Limvere – Woodturner, Artist, Musician*

http://www.BearLimvere.com/ – http://www.StandingPeopleDesigns.com/

Jim responds:

The reason I went to 13 divisions is close to what Bear discovered: you can proportion to 8 to 13 which creates the most pleasing (according to actual research!) rectangle. (Regardless of whether that’s a “golden” rectangle or not – turns out that whole Golden Ratio thing is largely a Victorian-era fabrication!)

Also, you want to go to at least 12 so you can easily scale in thirds, fourths, sixths. There is rarely any need to go beyond 13 as all the proportioning you might want happens below 13!

*— Jim Tolpin*

Port Townsend School of Woodworking

Ok so I’m a bit behind on my reading, and just got to this article. LOVE IT! So much so that I ordered myself a divider from Lee Valley. I just got the divider. Having never used this tool before, I have a question. The end points on the divider are not exactly the same length when touching fully closed. Shouldn’t they be? Seems to me this might lead to some inaccuracy in measuring. Should I send these back or are they ok?

Well, I shouldn’t have called the Golden Ratio a “Victorian fabrication” as the section was actually fabricated by mathematicians prior to that era. The fabrication I was referring to was the Victorian thinking that imagined pre-industrial furniture makers, luthiers (and probably other trades including blacksmiths)employing the golden section in design and layout of their products. I have yet to see any documents or source literature of that era attesting to that–in fact, what documentation we have points towards whole-number proportioning systems and simple geometry, not mathematics. (Which makes sense from the working artisan’s perspective as it would have been much too tedious to construct or calculate sections compared to simply stepping out ratios with a divider. They were trying to make a living after all. Just my two cents worth (and worth every penny as my Dad used to say).

In Jim’s article he says that he has not found a period source on how a sector was used. I’d suggest “A Treatise of Mathematical Instruments” by John Robertson. Flower-de_Luce Books is reprinting the 1775 3rd edition. Information at http://www.orbitals.com/books/index.html. Hard read with the old English long “S” and style of technical writing.

Thanks Daver…I ordered it.

Since this tool is an outgrowth of one scale on the sector, I think that it should be called something else.

The divisions need to be identical, but not to a specific measure, so it is a scale rather than a ruler.

It is based on the sector’s “line of lines” so I propose we call it a “LOL Scale”

Tongue not quite firmly in cheek,

Mike

Well, I think the whole Golden Ratio and/or Golden Rectangle as it relates the sectors will now be used to generate gold or at the very least greenbacks. 😮 Jim, thank you for writing about this as I know it is going to help me. Fred

Can you point me to a concise book that lists all the ratios of furniture making? I have not found one yet but have picked up some ratios individually. Thanks.

– Bill

Actually the Golden Rectangle was long before The Victorian Era. As blacksmiths we use it for design in a lot of metal working. Our member Boyd Holtan in the Appalachian Blacksmiths Assoc. explains it best in this article that he wrote on it. http://www.appaltree.net/aba/design.pdf. Check it out and you will learn a lot about nature from Boyd, Fibonacci (the 13) and the Golden Rectangle. By the way Chris I am still working on that “Redneck Roubo”. John

You guys and your higher math. I assumed Jim was a “This is Spinal Tap” and wanted his sector to exceed 11.

Lloyd

While I would certainly agree that the Golden Ratio has been overhyped and overapplied, there’s definitely a lot more to it than a “Victorian-era fabrication.” Its mathematical properties are such that it shows up in lots of unexpected places.

The most recent discovery that I’m aware of is that the structure of quasicrystals (proposed in the early 1960’s and first observed in nature in 1984) is closely tied to the Golden Ratio.

There has been some recent research in perception and aesthetics that indicates that a larger ratio, somewhere between 2.5:1 and 3:1 (roughly equal to the square of the Golden Ratio) is a “more preferred” value, at least in some contexts.

-Steve