trivia - trisecting an angle & gradians

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