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5th May 2015, 12:47 PM #16
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Angle Constructions
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
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5th May 2015, 02:44 PM #17
CNC.
Cuniform Number Counting.
This diagram is about as accurate as Excel will permit (and scuzi the tpyo).
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.
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6th May 2015, 02:37 AM #18
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.regards from Alberta, Canada
ian
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6th May 2015, 09:18 AM #19
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.
trisect.JPG
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6th May 2015, 09:43 AM #20
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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6th May 2015, 03:54 PM #21
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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.Last edited by steamingbill; 6th May 2015 at 03:56 PM. Reason: Typo
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7th May 2015, 11:57 AM #22
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 90o, 45o, 60o, ... 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 5o divisions, maybe you could treat the chord divided in fifths as close enough ...
Paul
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7th May 2015, 12:17 PM #23
I guess most people will know about the Antikythera device dating to ~200BC-100BC ...
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7th May 2015, 12:49 PM #24
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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7th May 2015, 01:05 PM #25
Oh ... and if you look at a chord across a 5o 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.
trisect angle 2.jpg
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.
Cheers,
Paul
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7th May 2015, 10:40 PM #26
By my calculation,if you divided a circle into 15o 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 15o section ...
the middle angle would be 3.02o, the two on either side of that 3.01o, and the outer two angles 2.95o.
That seems a pretty reasonable sort of accuracy. Good enough for gummint work.
Cheers,
Paul
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7th May 2015, 11:47 PM #27
And.....
he's....
....gone.
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7th May 2015, 11:52 PM #28
Taking a break
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Interesting video on trisecting with origami: https://www.youtube.com/watch?v=SL2lYcggGpc
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8th May 2015, 07:54 AM #29
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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
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8th May 2015, 08:16 AM #30
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