[Mathematics] Sum of angles of triangle [Updated]
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when you choose three points on a sphere and connect them with straight lines, the angles will add up to more than 180 degrees; imagine two points on the earth equator and one on the North pole, the sum would be 270 degrees. See spherical excess here[^]. :)
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draw a line segment AB with length 2; draw two circles, one centered at A, one at B, both with radius 2 (or AB). Where they intersect, you got a third point C such that ABC is equilateral; and yes, you got a second solution for free. :)
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draw a line segment AB with length 2; draw two circles, one centered at A, one at B, both with radius 2 (or AB). Where they intersect, you got a third point C such that ABC is equilateral; and yes, you got a second solution for free. :)
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Luc Pattyn wrote:
you can draw it to any precision you like: start with an equilateral triangle, then split it in two halfs; you now have angles of 30, 60 and 90 degrees, and sizes proportional to 1, SQRT(3) and 2.
But you can't physically draw a line to an arbitrarily precise length. I know that mathematically it is quite trivial, but using atoms it is rather akin to trying to make a diagonal line out of Lego.
use more space, and more atoms or Lego blocks to create a larger figure, resulting in higher precision. if you concentrate on molecular particles, you won't be able to draw a line at all; everything is just gaps with some rare particles in between, Higgs or other. :)
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Rob Graham wrote:
On a 2D Plane surface, a 30,60,90 triangle can easily be drawn accurately, but you may not be able to precisely measure the length of the side that is a multiple of the square root of 3.
Which means sum of angles is not 180 degree. Right?
No. You not being able to do something does not prove or disprove something else. :)
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Rob Graham wrote:
On a 2D Plane surface, a 30,60,90 triangle can easily be drawn accurately, but you may not be able to precisely measure the length of the side that is a multiple of the square root of 3.
Which means sum of angles is not 180 degree. Right?
no, it just means you can't accurately measure the sqrt(3) side.
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Rob Graham wrote:
On a 2D Plane surface, a 30,60,90 triangle can easily be drawn accurately, but you may not be able to precisely measure the length of the side that is a multiple of the square root of 3.
Which means sum of angles is not 180 degree. Right?
Yes it is. In the real world, the distance between the two is 3^.5 to within the accuracy of your drawing the sides of length 1 and 2, creating a 90* angle between them, and drawing all 3 lines strait. In the abstract world of plane geometry the error of all 3 requirements is zero which means your third side is exactly 3^.5 long.
3x12=36 2x12=24 1x12=12 0x12=18
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use more space, and more atoms or Lego blocks to create a larger figure, resulting in higher precision. if you concentrate on molecular particles, you won't be able to draw a line at all; everything is just gaps with some rare particles in between, Higgs or other. :)
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I was just pointing out to him that the only time Sqrt[3] is not constructible - in reality - is when no other desired length is either.
Luc Pattyn wrote:
Higgs or other.
Almost entirely Higgs bosons and gravitons, I would wager. ;P
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See, here the triangle is not in two dimensions so whatever the calculations are do not hold true for a two dimensional triangle. Am I right or I need to read more?
It is all relative. The surface of a sphere is a two-dimensional object: you can draw lines on it, delimiting an area, etc. You could always read more about it, however you would never see the end of it. :)
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It is all relative. The surface of a sphere is a two-dimensional object: you can draw lines on it, delimiting an area, etc. You could always read more about it, however you would never see the end of it. :)
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Luc Pattyn wrote:
surface of a sphere is a two-dimensional object
don't think that's true since it moves through length, width AND height.
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I was just pointing out to him that the only time Sqrt[3] is not constructible - in reality - is when no other desired length is either.
Luc Pattyn wrote:
Higgs or other.
Almost entirely Higgs bosons and gravitons, I would wager. ;P
and "dark" matter/energy, i.e stuff we don't know or haven't imagined yet.
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My point is, that vertical line which I would draw as a median to the side of equilateral triangle, has to be of length 3^1/2. Now, since that cannot be drawn, everything goes void. Isn't it?
d@nish wrote:
that cannot be drawn
Can you draw a straight line of length 1 meter? If you accept your pencil/pen/whatever has a certain width and you are satisfied that lengths and widths should not be more accurate than said width, then you can draw it perfectly. Same for circles, and hence also for SQRT(3) and many more irrational numbers. :)
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and "dark" matter/energy, i.e stuff we don't know or haven't imagined yet.
dark matter is just some more Higgs, painted black to fool us all. :)
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It is all relative. The surface of a sphere is a two-dimensional object: you can draw lines on it, delimiting an area, etc. You could always read more about it, however you would never see the end of it. :)
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Luc Pattyn wrote:
surface of a sphere is a two-dimensional object
don't think that's true since it moves through length, width AND height.
in your Cartesian mind, yes. If you were the captain of a ship on one of earth's oceans, you would see longitude and latitude (hence two dimensions), and no third one, at least as long as you stay afloat. :)
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d@nish wrote:
that cannot be drawn
Can you draw a straight line of length 1 meter? If you accept your pencil/pen/whatever has a certain width and you are satisfied that lengths and widths should not be more accurate than said width, then you can draw it perfectly. Same for circles, and hence also for SQRT(3) and many more irrational numbers. :)
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no, it just means you can't accurately measure the sqrt(3) side.
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Luc Pattyn wrote:
Can you draw a straight line of length 1 meter?
Depends on the accuracy rate we agree upon. :)
OK, you choose the accuracy, we verify I can do 1 meter; next I will do SQRT(3) meter with the same accuracy, as outlined earlier. :)
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No. You not being able to do something does not prove or disprove something else. :)
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OK, you choose the accuracy, we verify I can do 1 meter; next I will do SQRT(3) meter with the same accuracy, as outlined earlier. :)
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