trivia - trisecting an angle & gradians

I'd long believed that it was impossible to trisect an angle. Started googling it recently and found that thats not quite true. You can do it if you have marks on the straight edge, the original "rules of the game" were pencil paper compass and unmarked straight edge.

http://en.wikipedia.org/wiki/Angle_trisection see 1/2 way down the page or attached snippett below

Am curious - why did 360 degrees survive decimalisation ? I have seen web pages advocating a decimal degree system but it never caught on. ie 400 units defining a full circle each quadrant being 100 units and abolishing minutes and seconds just have decimal places.

I've read on the internet that old European theodolites can be marked up in 400 Gradians and that the French Military used Gradians

https://books.google.com.au/books?id...circle&f=false

Bill

I'll put my hand up

at various times I've used all 5 of the main systems of measuring angles

Degrees - minutes - seconds

Degrees - minutes and decimal minutes

radians -- as in 90° = pi/2 radians

Gradians -- 400 gradians to a circle

Mils -- where 1 mill = 1 metre at a range of 1000m


I think the 360° system has survived because of it's "simplicity"

15° = 1 hour
0°15' = 1 minute
0°1' = 1 nautical mile, which makes working out distance relatively easy.

If you moved to any other system, these relationships break down.

Trisecting an angle with pencil, compass and straight edge.

I'm way too old to remember anything from 10th year geometry. But logic prevails.

Assumption is that the angle is drawn on a piece of paper and can be extended to the edges of the paper.

1 ~ Draw a line. Use a compass to mark off three equal sections on the line.

2 ~ Stretch the compass to the full distance of the three sections. Use the compass to mark on the angle lines where the distance fits, touching both lines of the angle. Draw a line between those two points.

3 ~ Stretch the compass to the length of one of the three equal sections of the drawn line in step one.

4 ~ Use the compass to mark transfer the length of the equal sections on the line drawn between the sides of the angle.

5 ~ Use the straight edge to draw lines from the point of the angle to the marks on the line in step 4.

The angle is now trisected.

Quote:

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Originally Posted by rrich

Trisecting an angle with pencil, compass and straight edge.

I'm way too old to remember anything from 10th year geometry. But logic prevails.

Assumption is that the angle is drawn on a piece of paper and can be extended to the edges of the paper.

1 ~ Draw a line. Use a compass to mark off three equal sections on the line.

2 ~ Stretch the compass to the full distance of the three sections. Use the compass to mark on the angle lines where the distance fits, touching both lines of the angle. Draw a line between those two points.

3 ~ Stretch the compass to the length of one of the three equal sections of the drawn line in step one.

4 ~ Use the compass to mark transfer the length of the equal sections on the line drawn between the sides of the angle.

5 ~ Use the straight edge to draw lines from the point of the angle to the marks on the line in step 4.

The angle is now trisected.

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rrich,

I think I have misinterpreted what you have written, is a sketch possible ?

Bill

I don't know if this will work as I drew it using Visio 2000. (15 years out of date)

It won't so I'll have to do it with pencil and paper and scan it in.

The real beauty of 360 is the large quantity of numbers it is exactly divisible by;
2,3,4,5,6,8,9,10,12 to name but a few.

Just to confuse things...in oil and gas drilling they use a decimalized foot. That is 10 inches to a foot. A drill pipe would be 40.7ft long.

Quote:

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Originally Posted by ian

I think the 360° system has survived because of it's "simplicity"

15° = 1 hour
0°15' = 1 minute
0°1' = 1 nautical mile, which makes working out distance relatively easy.

If you moved to any other system, these relationships break down.

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the relationship to nautical miles is only true for longitude at the equator

Quote:

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Originally Posted by Dion N

the relationship to nautical miles is only true for longitude at the equator

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true, but I'm sure there's a [relatively simple] function -- which I can't recall -- that adjusts for latitude

Quote:

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Originally Posted by ian

I'll put my hand up

at various times I've used all 5 of the main systems of measuring angles

Degrees - minutes - seconds

Degrees - minutes and decimal minutes

radians -- as in 90° = pi/2 radians

Gradians -- 400 gradians to a circle

Mils -- where 1 mill = 1 metre at a range of 1000m


I think the 360° system has survived because of it's "simplicity"

15° = 1 hour
0°15' = 1 minute
0°1' = 1 nautical mile, which makes working out distance relatively easy.

If you moved to any other system, these relationships break down.

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Correct - the direct relationship between time and angle measurement is why degrees have survived decimalization. Celestial navigation, geodesy, astronomy, spherical trigonometry and computations are far simpler using the derived unit / accepted unit of degree. The wikipedia article is also a little misleading as the degree is not only a measure of "plane angles" it is also a measurement of "spherical angles". To add a little more confusion the sum of the spherical angles in a spherical triangle always exceed 180 degrees.

Here is the sketch.

To me it looks trisected. I can't remember the theorem from Geometry but that was way back and long before JC was a choir boy.

Attachment 346044

if I read your diagram correctly, the line you have trisected forms a right angle with one side of the angle.

From trigonometry, the tan of an angle is the opposite over adjacent, so for the sketch you provided, and assuming (for convenience) that the angle to be trisected is 30°, then
Tan(30°) should equal 3 x tan (10°) = 3 x 0.1763 = 0.5289, but tan(30°) = 0.5774


The only way I can see to do a trisection, is by trial and error
Draw an arc across the angle, then by repeated trials find the compass setting that divides the arc into three

Quote:

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Originally Posted by rrich

Here is the sketch.

To me it looks trisected. I can't remember the theorem from Geometry but that was way back and long before JC was a choir boy.

Attachment 346044

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Rrich,

Thanks for taking the time to post the sketch explaining your method.

Bill

Shown here are a few methods for trisecting a line, and they are reasonably neat:
http://www41.homepage.villanova.edu/...ing%20segment/
(press the "play" button on each of the diagrams)

The last one has a very oblique angle of intersection between the circle and the straight line, and so too much error would be introduced (even just the pencil lead thickness changing from drawing the first few lines would introduce error).

But that won't allow us to trisect an angle (I don't think). To trisect an angle you'd have to trisect an arc of a circle first.

Quote:

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Originally Posted by FenceFurniture

To trisect an angle you'd have to trisect an arc of a circle first.

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:whs:

This sort of vaguely fits in this thread

180 bisected is 90 then 45 22.5 etc etc

60 is created easily by stepping a radius around a circle 60 then bisects to 30 15 7.5 etc etc
60 trisected is 20 then bisected is 10 5 2.5 etc etc

72 is created by standard pentagon construction

72 bisects to 36 18 9 difference between 9 and 10 from above is one degree

or

72 trisects to 24 then 8 then bisects to 4 2 1

But to arrive at 1 degree accurately you'd probably need a circle with a very large (4 metre ?) radius scratched onto a very flat piece of something. Any thoughts on this ?

How did the ancients do it - there seems to be universal agreement that 360 degree concept is thousands of years old and was inspired by 12 months @ 30 days each = 360

Yet I have never seen discussion about or pictures of a clay tablet or lump of stone or wood with an inscribed circle and 360 divisions - did the ancient builders and engineers have such things ?

Bill

Quote:

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Originally Posted by steamingbill

How did the ancients do it .....

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CNC.

Cuniform Number Counting.



This diagram is about as accurate as Excel will permit (and scuzi the tpyo).

https://www.woodworkforums.com/attach...5&d=1430796796


You can see that the middle double arrow line touches both red lines (which are the tri-sectors of the 30° angle shown in black).

However, the double arrow lines aren't long enough to complete the triangle for the outside two.

So, we have an isosceles triangle (2 equal sides, 2 equal angles) in the middle with two identical scalene triangles (3 unequal sides, 3 unequal angles) each side.

Therefore if the triangles are different, the base lines are different lengths, and the straight line has not been trisected, but the arc has.

Quote:

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Originally Posted by steamingbill

... to arrive at 1 degree accurately you'd probably need a circle with a very large (4 metre ?) radius scratched onto a very flat piece of something. Any thoughts on this ?

How did the ancients do it - there seems to be universal agreement that 360 degree concept is thousands of years old and was inspired by 12 months @ 30 days each = 360

Yet I have never seen discussion about or pictures of a clay tablet or lump of stone or wood with an inscribed circle and 360 divisions - did the ancient builders and engineers have such things ?

Bill

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Hi Bill

ancient builders and engineers learnt through an apprenticeship system

there was no need or desire to write things down -- the budding builder/engineer would be shown how do to it and having practiced it a few hundred time would know how to do it.
Like using a plumb bob and a piece of string as a level.

I remember reading a book about 40 years ago where the author speculated that the guys who built Stone Henge had played around with "circles" where pi would resolve to a rational number

as to large radius arcs, dirt can be got pretty flat
mapping the shadow of a stick stuck vertically in the dirt will give you true north (or true south if you're an Egyptian)
and, using a piece of rope, a circle of almost any radius can be drawn.

If you think about it, trisecting an angle means stacking three identical RA triangles on each other, with a ragged edge.

The all have the same base = 1 radius, they all have the same acute angle.

You could make each triangle non-RA, and connect a line across each of the three small chords.
Then the three new lines would still be equal in length, but you can see that they still wouldn't lie on a straight line.


Attachment 346310

This link says the Sumerians from 4000BC and the Babylonians from 2000BC-600BC used a base 60 number system.
http://www.math.tamu.edu/~dallen/mas...on/babylon.pdf

That's the heritage behind our degrees, hours, minutes, seconds ... and I think that the number of factors of 360 played a huge role in commerce. [1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180]

quoting ...
"In mathematics, the Babylonians (Sumerians) were somewhat more advancedthan the Egyptians.
Their mathematical notation was positional but sexagesimal.

- They used no zero.
- More general fractions, though not all fractions, were admitted.
- They could extract square roots.
- They could solve linear systems.
- They worked with Pythagorean triples.
- They solved cubic equations with the help of tables.
- They studied circular measurement.
- Their geometry was sometimes incorrect."



I have a book here on early mathematics and it says Ptolemy ~150AD produced a masterpiece called the Almagest, setting out astronomical models and mathematical tools, including a table of chords lengths of angles from 1/2 degree upwards in 1/2 degree increments(!)

It was written in base 60 notation and gives the cord length for each angle, assuming a radius 60 circle.

Quote:

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Originally Posted by pmcgee

This link says the Sumerians from 4000BC and the Babylonians from 2000BC-600BC used a base 60 number system.
http://www.math.tamu.edu/~dallen/mas...on/babylon.pdf

That's the heritage behind our degrees, hours, minutes, seconds ... and I think that the number of factors of 360 played a huge role in commerce. [1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180]

quoting ...
"In mathematics, the Babylonians (Sumerians) were somewhat more advancedthan the Egyptians.
Their mathematical notation was positional but sexagesimal.

- They used no zero.
- More general fractions, though not all fractions, were admitted.
- They could extract square roots.
- They could solve linear systems.
- They worked with Pythagorean triples.
- They solved cubic equations with the help of tables.
- They studied circular measurement.
- Their geometry was sometimes incorrect."



I have a book here on early mathematics and it says Ptolemy ~150AD produced a masterpiece called the Almagest, setting out astronomical models and mathematical tools, including a table of chords lengths of angles from 1/2 degree upwards in 1/2 degree increments(!)

It was written in base 60 notation and gives the cord length for each angle, assuming a radius 60 circle.

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Yep, I accept all of the above.

Am curious regarding the method employed to draw a circle with 360 tick marks around it thousands of years ago.. I'd find it very difficult to do that without pinning a huge sheet of paper to the floor of the spare bedroom.

.......... a bit of string with a sharp pointy thing on the end scratching a huge circle in the sand ?

.......... a straight edge and some super accurate dividers and compasses and a not quite so huge circle

........... a substantial log with a couple of trammel points in it

........... or simply an intellectual achievement all done in the head assisted by conceptual sketches on a piece of sheepskin or papyrus - after doing the brainwork an appropriate custom built triangle is created and given to Fred the Engineer whenever he has to build an awkward angle in the latest downtown stone tower.

Quote:

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Originally Posted by steamingbill

I am curious regarding the method employed to draw a circle with 360 tick marks around it thousands of years ago..

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When do you think they would have?
I could imagine a "high tech" sundial for a king, or a 180 degree protractor maybe for an engineer ... I assume that would be in super-computer territory back then ... maybe.

But just thinking about it ... with compass and straight-edge you can construct 90^o, 45^o, 60^o, ... and halve any angle ...

and with a ruler as you pointed out, trisect any angle ...

so if you treated it like a NASA mission ... or if your head was literally on the line ... with time and great care I'm sure it could be done. If you marked down to 5^o divisions, maybe you could treat the chord divided in fifths as close enough ...

Paul

I guess most people will know about the Antikythera device dating to ~200BC-100BC ...


https://www.youtube.com/watch?v=UpLcnAIpVRA

See this video at 24:00 regarding making a cog with a given number of teeth ...

https://www.youtube.com/watch?v=nZXjUqLMgxM

Oh ... and if you look at a chord across a 5^o arc, then working in Radians, if you call the angle (2.theta) then ...

half the arc is (R.theta) in length ... and half the chord is (R.sin(theta)) in length.

Attachment 346470


So the percentage error between the chord length and the arc is (theta/sin(theta)).

At theta = 0.0436 Radians (2.5 degrees), the error is 0.03%

and taking 10 degree sections (2 x 0.0873 Radians), the error is still only 0.1%

So really you just need to construct a big regular polygon.
Let me know how you go with that. :U

Cheers,
Paul

By my calculation,if you divided a circle into 15^o sections, and drew in the chord across each arc to get a 24-sided figure,

then somehow divided the chord into fifths, then in each 15^o section ...

the middle angle would be 3.02^o, the two on either side of that 3.01^o, and the outer two angles 2.95^o.

That seems a pretty reasonable sort of accuracy. Good enough for gummint work. :)

Cheers,
Paul

And.....


he's....


....gone.

Interesting video on trisecting with origami: https://www.youtube.com/watch?v=SL2lYcggGpc

Thanks for all replies.

Will check out the various things posted most recently over next couple of days.

Didn't know about the Amykethera thing, looks interesting

Thanks again.

Bill

Quote:

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Originally Posted by elanjacobs

Interesting video on trisecting with origami: https://www.youtube.com/watch?v=SL2lYcggGpc

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Thanks that was fascinating. Terrific video, will be looking at rest of that numberphile site for a while. anybody gets this far without watching the video then go back and do it.

Off to check out antyketheras now

Quote:

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Originally Posted by pmcgee

See this video at 24:00 regarding making a cog with a given number of teeth ...

https://www.youtube.com/watch?v=nZXjUqLMgxM

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Paul,

Many thanks for the fascinating links - I didnt know about that thing and clearly they were able to divide up circles and play with angles and geometry with a high levele of understanding and metalwork skills.

The manual cog cutting technique reminded me of something I'd read recently about dividing wheels where there could be (maybe ?) a way of estimating the dividing holes on a first attempt at a dividing wheel and then sticking it on the dividing head and then using the dividing head somehow to make the second one more accurate - I dont know enough about how that works but will find the book and post it here in case anyone is interested. Maybe I am raving - must find the book I was reading a few days ago and check it.


Found a link - I dont understand this idea yet but somebody may be interested. Probably discussed in many places.

http://www.homews.co.uk/page427.html
Bill

This is how I read it ...

Anyone can gauge a full rotation perfectly accurately, assuming the fit of the shaft and etc parts, with just a single mark.
If everything else is up to scratch, then 1 rotation will bring you to the same place every time,
and with eg a 40x reduction gear, each full rotation should give you precisely 9^o on your workpiece.

So eg to mark 40^o on your workpiece, you need to move 36^o by full rotations plus 4^o by a part rotation (of the divider).

The part rotation will be 160^o on the divider, which we assume is accurate to +- x degrees.

So the resulting movement of the workpiece will be 4^o +-(x/40) degrees ... total movement 40^o +-(x/40) and you've increased your accuracy 40x now.

Cheers,
Paul

Quote:

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Originally Posted by steamingbill

Thanks that was fascinating. Terrific video, will be looking at rest of that numberphile site for a while. anybody gets this far without watching the video then go back and do it.

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I've slowly worked my way through the whole lot of them and, on the whole, they're all fascinating. The same guy has channels on physics and chemistry as well, called Sixty Symbols and Periodic Videos respectively, for those interested.

Quote:

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Originally Posted by elanjacobs

I've slowly worked my way through the whole lot of them and, on the whole, they're all fascinating. The same guy has channels on physics and chemistry as well, called Sixty Symbols and Periodic Videos respectively, for those interested.

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And he's an aussie from ... South Australia (I think). Brady Haran.