[Mathematics] Sum of angles of triangle [Updated]
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I know that. But If I create a triangle with angles 30, 60 and 90 it has to have sides in that ratio (courtesy trigonometry). If that is not possible, sum of angles cannot be 180.
Just because the square root of three can only be represented as an infinitely repeating decimal in base ten does not mean a line that is a multiple of that value cannot be draw. The number clearly exits, and a line of that length can also exist. You are confusing the representation of the number with the reality of its existance. 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. On the surface of a sphere, and many other non-2D surfaces, the sum of the angles of a triangle is > 180 ( 540 is possible on a sphere).
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d@nish wrote:
Since one cannot draw a line of length 3^1/2, this triangle is not possible which in turn means that sum of angles of a triangle is not 180 degree.
:confused: You can't draw it exactly sqrt[3] because space is not infinitely divisible, but that goes for all numbers.
d@nish wrote:
Long ago, I had a read a book which stated that sum of angles of a triangle is not 180 degree (it was proven through a triangle formed by centers of three stars). I guess it was non Euclidean or something geometry. Anyone aware of this? And does anyone knows nice book where I can read more about that geometry?
In spherical geometry, the sum of the angles of a triangle is strictly greater than 180 degrees, and in hyperbolic geometry, it is strictly less than 180 degrees. I have a hunch that you're thinking of when they formed a triangle using lasers in empty space, measured the angles contained and deduced the overall geometry of space-time.
Ravel H. Joyce wrote:
deduced the overall geometry of space-time.
You know just the other day I was sitting in a meeting, and while listening to someone drone on I started to workout the geometry of their head in space time. I was concerned I might end up with an imaginary number, so I took the usual precautions, but then the girl from Accounting walked in and I had to do a soft reboot. I hate when that happens.
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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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Ravel H. Joyce wrote:
You can't draw it exactly sqrt[3] because space is not infinitely divisible, but that goes for all numbers.
That's what my point is. So does this means that all the geometry we had read is not correct? :confused:
Ravel H. Joyce wrote:
spherical geometry
That's the word or words. Thanks. :) I will have to dig in and find that book when I go to my hometown (where my parents live). It had a lot of interesting things. Probability of finding matching DNA's lot of stuff related to geometry and space. It was cool.
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Ravel H. Joyce wrote:
You can't draw it exactly sqrt[3] because space is not infinitely divisible
:confused: 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.
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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.
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Ravel H. Joyce wrote:
You can't draw it exactly sqrt[3] because space is not infinitely divisible
:confused: 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.
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Just because the square root of three can only be represented as an infinitely repeating decimal in base ten does not mean a line that is a multiple of that value cannot be draw. The number clearly exits, and a line of that length can also exist. You are confusing the representation of the number with the reality of its existance. 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. On the surface of a sphere, and many other non-2D surfaces, the sum of the angles of a triangle is > 180 ( 540 is possible on a sphere).
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?
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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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