geometry

[Droogs]

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

 

[Phil Pascoe]

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

 

[xy mosian]

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

 

[Jacob]

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.

 

[Phil Pascoe]

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.

 

[Jacob]

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

 

[niagra]

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

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

Dario

 

[Jacob]

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!

 

[MusicMan]

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.

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

 

[AndyT]

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.

 

[MusicMan]

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

Keith

 

[AndyT]

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?

 

[Wuffles]

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.

 

[Jacob]

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.

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

 

[AndyT]

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!

 

[porker]

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

 

[Jacob]

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?

 

[Droogs]

excellent find tho porker and close enough for government work

 

[AndyT]

Agreed!

 

[Shultzy]

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