# geometry



## Phil Pascoe (16 Mar 2016)

Is there a way of dividing a circle into 5 and 7? I can't recall its ever cropping up. I'm making a dividing "table" to pattern and inlay turnery. I've used my indexing on the lathe to get 24 points, but for the purposes of aethetics I would like 5 and 7. i'm taking photos ... on a camera :shock: and if all goes to plan I'll do a WIP


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## AndyT (16 Mar 2016)

This might do - from 'The Modern Carpenter and Joiner and Cabinet Maker' volume III


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## Random Orbital Bob (16 Mar 2016)

Also I seem to recall from schoolboy maths if you set your compass to 1/5 the circumference and place the point anywhere on the circumference you can scribe an arc that bisects the circumference. Then move point to that bisected point and repeat until you have 5 equidistant points to which you draw in the straight lines. Voila, the sign of the devil!

But I'm sat outside hospital waiting for my Missus so can't try it and therefore it may be complete and utter rubbish! Always here to help


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## JSW (16 Mar 2016)

Use a protractor, 360 degrees divided by 5 = 72 degrees, or divided by 7 = 51.42 degrees, or am I missing something?


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## NickWelford (16 Mar 2016)

Printable 360 degree protractor


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## NickWelford (16 Mar 2016)

Or this


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## Phil Pascoe (16 Mar 2016)

JSW":2ftsxzj2 said:


> Use a protractor, 360 degrees divided by 5 = 72 degrees, or divided by 7 = 51.42 degrees, or am I missing something?


No, you're not missing anything.  I was wondering if someone would come up with a way of doing it with a compass/dividers that is so simple I'd always missed it. I might have to buy a protractor, although accurately marking off 51.42 degrees might be a little hit or miss.


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## JSW (16 Mar 2016)

Just print out the 360 degree protractor Nick linked to, pritstick it to your workpiece, strike thru the lines for whatever angles you need, remove template, join lines back up and then draw any size circle you want, already bisected.


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## AndyT (16 Mar 2016)

Phil, I've been reading up on this and feel I ought to warn you about not taking too many shortcuts...

According to the Science Museum, 

"Eighteenth-century craftsmen marked out the degree scales on scientific instruments with great accuracy, using basic geometrical principles, compasses and hand tools. However, this was a time-consuming task." 

until clever people like Jesse Ramsden invented the Dividing Engine to automate the job. So this is what you need to make:






then you can make a division plate with rings of 360, 192, 144, 120, 112 and 96 holes as described by Holtzapffel here.

It does look like an enjoyable project. When's the wip going to be ready? :lol: :lol:


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## Bm101 (16 Mar 2016)

51.42 degrees. Nailed it!  I just need to learn to cut at 90 degrees now.






God bless Moore and Wright ebay joblots! 8)

Andy, I'm pretty sure that's a time machine mate.


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## Jacob (16 Mar 2016)

phil.p":oikhyeb3 said:


> JSW":oikhyeb3 said:
> 
> 
> > Use a protractor, 360 degrees divided by 5 = 72 degrees, or divided by 7 = 51.42 degrees, or am I missing something?
> ...


Dividers are called dividers for a good reason - they are for accurate division.
To divide a line (or the circumference of a circle) by 5 - stride out an approximation of the division you want, as close as you can judge. 
There will be an error, plus or minus (unless you hit it spot on first go). 
Adjust the dividers to a point +/- 1/5 of this error (as close as you can judge) and try again. This will be closer. If not close enough carry on and divide the reduced error (as close as you can judge).
This is amazingly accurate for all practical purposes i.e. error will be at the limit of what you can see.
It's also the origin of the duodecimal system: it's easier to divide (and/or subdivide) a line by 2,3,4, and more practically useful, rather than than 5
A craftsman could easily mark out his own yard stick accurately in feet, inches, fractions. It'd be his own yard (non standard) but fully functional for his own project in hand.
The decimal system was based on crude counting of digits (fingers and toes) hence 5, 10, 20. The later duodecimal system was more sophisticated and met the needs of astronomers, navigators, precision engineers etc.


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## finneyb (16 Mar 2016)

I'm not fully understanding why you want to divide the circle, but if its for segmented turning why not use this http://www.woodturnersresource.com/extr ... gmentcalc/ to give you the internal angle values ? 

Brian


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## Bm101 (16 Mar 2016)

Jacob":1dazef2p said:


> phil.p":1dazef2p said:
> 
> 
> > JSW":1dazef2p said:
> ...



Yeh but not just anyone can cut a straight line at 3 or 4 different angles while maintaining the deception that it is to all intents and purpose a line of some variety. It takes hidden talent to do that. Very _deeply_ hidden talent.


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## Jacob (16 Mar 2016)

Bm101":1rdajd1l said:


> Jacob":1rdajd1l said:
> 
> 
> > phil.p":1rdajd1l said:
> ...


Not sure what you mean?


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## Bm101 (16 Mar 2016)

Ah, just mocking my saw technique Jacob. :wink:


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## transatlantic (16 Mar 2016)

I wonder if it would be easier to instead use a circle cutting jig to create a perfect circle of the intended radius.

Calculate the edge length using something like this to punch in your radius and number of edges http://planetcalc.com/92/

Pick any point on the circumference for your first point. Then use a ruler to measure out the edge length calculated above intersecting the appropiate part of the circumference. Repeat.

Seems as though it could be more accurate than trying to use a protractor, at least for big things


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## AndyT (16 Mar 2016)

Phil, if you want a method just using compasses and rule, this alternative from later in the same book might do for the pentagon. It avoids the step of dividing a straight line into five parts and looks a bit simpler. 






Not much help for the division into seven though.


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## Retire2004 (16 Mar 2016)

Hi phil.p
Try this one: www.cgtk.co.uk/metalwork/reference/divider
Not sure how to do the link

Regards
Tudor


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## Phil Pascoe (16 Mar 2016)

Brilliant - I think that'll do. In the fullness of time all will be revealed. I haven't a large enough dividers with a screw attachment, and huge one was proving to difficult to set finely enough although I'd probably get there in the end. I'm trying to keep it as accurate as I possibly can - I'm trying to make life easier, not harder.  
Thanks, everyone, some interesting comments and ideas.


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## Jacob (16 Mar 2016)

phil.p":2xgk03sc said:


> Brilliant - I think that'll do. In the fullness of time all will be revealed. I haven't a large enough dividers with a screw attachment, and huge one was proving to difficult to set finely enough although I'd probably get there in the end. I'm trying to keep it as accurate as I possibly can - I'm trying to make life easier, not harder.
> Thanks, everyone, some interesting comments and ideas.


You can replace dividers (or compasses) with a lath - pin at one end, notches at the other, new notches by way of adjustment.
Dividers work just as well around a circle as along a straight line


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## Droogs (16 Mar 2016)

just to let you save on time. you can not divide a circle into 7 equal parts as the number 7 although a prime number it is not a fermat prime number and therefore along with a few others the angle that is produces by 360/7 is not reproducable using division plates, dividers etc as per Gauss circle problem


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## Phil Pascoe (16 Mar 2016)

So you can't step off seven equal spaces with dividers? :?


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## xy mosian (16 Mar 2016)

Use something like Inksacpe, or Sketchup, both free.
Draw a shape with 5, or 7, sides.
Mark the centre, print, and stick on the wooden blank.

xy


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## Jacob (16 Mar 2016)

phil.p":2n498exb said:


> So you can't step off seven equal spaces with dividers? :?


No prob. You simply apply corrections of 1/7th of the error, as often as necessary. 2 or 3 steps will do it usually. The limit is in the adjustability of the dividers and/or your eyesight.
It's a bit like calculus.

It's the quickest way to set out equally spaced dovetails, amongst other things.


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## Phil Pascoe (16 Mar 2016)

I was being ... ahem ... a little facetious ...  
Yes, I'm trying to be as accurate as possible, but a thou or two here or there isn't going to ruin the thing. It's for machining a piece of wood, not a formula one car's crankshaft.


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## Jacob (17 Mar 2016)

Using dividers is a bit of a lost art. I hadn't given them a thought until I asked myself the question "why are they called dividers" and it took me some time to come up with the answer!
If you google "how do you divide with dividers" you draw a blank, which is surprising, unless you are into room dividers etc.
google "why are they called dividers" you get , er, me, on this forum a few years back!
why-are-they-called-dividers-t59439.html
You also get Peter Follansbee and others


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## niagra (17 Mar 2016)

You can do a 17 sided shape though. Here's a great video:

https://www.youtube.com/watch?v=87uo2TPrsl8

Dario


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## Jacob (17 Mar 2016)

niagra":a5xd5uwv said:


> You can do a 17 sided shape though. Here's a great video:
> 
> https://www.youtube.com/watch?v=87uo2TPrsl8
> 
> Dario


Fascinating!
Had to look up Euclidean construction of pentagon: http://www.cut-the-knot.org/pythagoras/pentagon.shtml

Using dividers by subdividing the error doesn't fit with Euclidean principles. Pity they didn't look into it they might have hit on calculus or something else.

Dividing by big numbers isn't easy with dividers alone, but dividing divisions by small numbers gets you there - from yard to foot to inch to half, quarter and you have divided by 144!


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## MusicMan (17 Mar 2016)

phil.p":23f6kkvo said:


> JSW":23f6kkvo said:
> 
> 
> > Use a protractor, 360 degrees divided by 5 = 72 degrees, or divided by 7 = 51.42 degrees, or am I missing something?
> ...



There is no geometrical construction that will allow you to do 5 or 7 in the way that you can do 6.

However, the successive approximation technique described by Jacob is very good, and as accurate as your patience.

Keith


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## AndyT (17 Mar 2016)

MusicMan":3foeq6bm said:


> There is no geometrical construction that will allow you to do 5 or 7 in the way that you can do 6.
> 
> Keith



That's what I thought I remembered reading too, but in what way does this method not count as a geometrical construction? It only uses the compasses and straight lines (actually only one, the diameter, so you can find points E and F.)







Ignore the stuff about the decagon and the Note.


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## MusicMan (17 Mar 2016)

Andy, I stand corrected, thanks! Now I must try to prove it ... where is this from?

Keith


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## AndyT (17 Mar 2016)

It's from 'The Modern Carpenter and Joiner and Cabinet Maker' volume III, published 1902 - was there ever a time when every tradesman really knew this sort of stuff?


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## Wuffles (17 Mar 2016)

AndyT":7103xi43 said:


> It's from 'The Modern Carpenter and Joiner and Cabinet Maker' volume III, published 1902 - was there ever a time when every tradesman really knew this sort of stuff?



From the olden IT days, it's why we kept books on the shelves, there was no way we could remember everything, but I can't imagine everyone had 'The Modern Carpenter and Joiner and Cabinet Maker' handy for reference. Dying arts.


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## Jacob (17 Mar 2016)

AndyT":1oqk39ge said:


> MusicMan":1oqk39ge said:
> 
> 
> > There is no geometrical construction that will allow you to do 5 or 7 in the way that you can do 6.
> ...


Seems to work in the Euclidean way (i.e. straight edge and compass/divider only, no measuring). Prof wossisname in the video said pentagon not possible.

Here's Euclid on the case; http://aleph0.clarku.edu/~djoyce/elemen ... pIV11.html

http://aleph0.clarku.edu/~djoyce/elements/Euclid.html


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## AndyT (17 Mar 2016)

I just watched the video - and he actually said that the division into *seven* (heptagon) is not possible with ruler and compasses. So we _are_ allowed to make pentagons! As was Euclid.

Personally, I find following the method hard enough. Understanding the maths behind it would be a lot harder.

Originating the maths by abstract thought takes genius - we're lucky to have had so many over the last few millennia!


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## porker (17 Mar 2016)

Being pedantic, the above method is an approximation (very close), so technically it is not possible. In practice it is extremely close. See here for an animation which also shows the "error"https://commons.wikimedia.org/wiki/File:Approximated_Heptagon_Inscribed_in_a_Circle.gif


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## Jacob (17 Mar 2016)

porker":3f6hwvov said:


> Being pedantic, the above method is an approximation (very close), so technically it is not possible. In practice it is extremely close. See here for an animation which also shows the "error"https://commons.wikimedia.org/wiki/File:Approximated_Heptagon_Inscribed_in_a_Circle.gif


Right. I did wonder! 
What about Euclid's method? http://aleph0.clarku.edu/~djoyce/elemen ... pIV11.html

PS I misread the above - heptagons no possibile but pentagons are A OK?

Who needs heptagons anyway? Er, except Phil?


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## Droogs (17 Mar 2016)

excellent find tho porker and close enough for government work


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## AndyT (17 Mar 2016)

Agreed!


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## Shultzy (18 Mar 2016)

I went to a Technical School between 1961 and 1969 and Technical Drawing was one of the subjects.

"How to Draw Polygons by the Universal Method of Polygon" http://www.engineeringdrawing.org/2012/03/universal-polygon-method.html is the method that was taught back then.

Strange how 55 years on I have found a use for it


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## Rob Platt (18 Mar 2016)

Draw a circle 573mm radius.
Make your 180 then 90 then 45
Then do your 60 by using your radius size from lets say the 90
any of the above can be divided again if necessary
using a plastic school rule with cms on it you can measure the divisions between the radii around the arc and every degree will be 1cm and then whatever portion there of.
Then any number of sides can be constructed
all the best
rob


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## bugbear (18 Mar 2016)

Rob Platt":3cluiwtw said:


> Draw a circle 573mm radius.
> Make your 180 then 90 then 45
> Then do your 60 by using your radius size from lets say the 90
> any of the above can be divided again if necessary
> ...



Well, that's another approximation - arc length and chord length are not (quite...) proportional.

BugBear


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## Rob Platt (18 Mar 2016)

where angels fear to tread


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## Jacob (18 Mar 2016)

bugbear":ggao2eid said:


> Rob Platt":ggao2eid said:
> 
> 
> > Draw a circle 573mm radius.
> ...


Arc and chord are not proportional at all, "quite" doesn't come into it Arc varies from pi x half chord to identity when chord approaches zero.


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## bugbear (18 Mar 2016)

Jacob":ygf8pu57 said:


> bugbear":ygf8pu57 said:
> 
> 
> > Rob Platt":ygf8pu57 said:
> ...



Better check your school textbook again; for Rob's number they're very close to proportional - but not quite.

BugBear


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## Droogs (19 Mar 2016)

oooh - Geometry the new sharpening cool


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## MusicMan (19 Mar 2016)

Much more fun!


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## Phil Pascoe (19 Mar 2016)

I should have known not to ask how to divide a circle into seven.


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## Rob Platt (19 Mar 2016)

What your making is effectively a protractor, the lines for 45,90, 180 and 60 are easily made using a pair of compasses, trammel, length of wood nail and pencil or whatever that can be fixed and maintained. 57.3 X pi = 180. So if you have a point on the arc that is lets say at the 90 radius every unit measured away from that point is 1 degree hence my use in this instance of a plastic rule and cms the less measuring you do around the arc the less likely you are to have an error hence the use of multiple radii. its irrelevant what you use as a unit the maths remain the same 1 unit is 1 degree. If you use cms it can be marked onto a standard sheet of ply or what i use is white painted hardboard
so if you want say a 27 sided figure. Divide 360 by 27 = 13.333. from a point where the radius crosses the arc take your plastic rule bend it so it follows the shape of the arc and mark off the 13.333 units. (easy when you use cms/mms). Take you dividers open them from to the two points and go around the circumference of the circle join up the dots and you will have a 27 sided shape. The actual size of which can be adjusted by reducing or extending the radii. It makes sense to me and I hope I`ve explained it so its understandable. Chords are what you are finishing with but what you are doing is taking your measurements around the circumference.
all the best
rob


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