[Mathematics] Sum of angles of triangle [Updated]
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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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Well since the square-root of 3 is a non-finite number, no you couldn't draw the line. Hence the figure drawn would not be a triangle at all since the two lines would never meet and the figure would not be closed. Ergo, the "point" were one side "doesn't meet" with the 3^1/2 side has no angle.
ahmed zahmed wrote:
the square-root of 3 is a non-finite number
It is finite. I think the word you are looking for is "irrational". Another irrational number is 3.14159...
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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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actually you do see a third one: the horizon is curved. (and not because there are waves)
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ahmed zahmed wrote:
the square-root of 3 is a non-finite number
It is finite. I think the word you are looking for is "irrational". Another irrational number is 3.14159...
no, I said the correct word: non-finite (or infinite). The number of digits is not countable, ergo it is infinitely long and yes, not rational, i.e., not complete comprehensible.
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I see it more like this: if X states Y is possible and I know Y is not possible, X has to be wrong.
well, I can draw a line of length 1 or of length SQRT(3) equally well as you can not prove I can't draw them. :)
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If we go for 100% accuracy, latter is not possible. If we go for anything less than that, latter wont be correct.
if you want 100% accuracy, then stick to math, and don't draw anything. :)
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if you want 100% accuracy, then stick to math, and don't draw anything. :)
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That's what I am saying with 100% accuracy, one cannot draw the 30,60,90 triangle. Never mind. I am off to sleep now. Good day. :)
Nor can you draw a 1 kilometer straight line with 1 micron accuracy. Good night to you. :)
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Surface of sphere "can be" considered 2-d if we are considering a part of its surface where sphere is of astronomical radius or we consider extremely small part of the surface. Otherwise I guess it has to be 3-d.
No. the surface of any 3d object is 2d. It's not a Euclidean plane, but that's not the definition of 2d; it just means you need 2 (no more, no less) variables to define any point on the surface; eg latitude and longitude.
3x12=36 2x12=24 1x12=12 0x12=18
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no, it just means you can't accurately measure the sqrt(3) side.
You can accurately measure sqrt(3) side as well as you measure the 1 side. The difficult is to measure the sqrt(3) side by means of 1 side. :)
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