[Mathematics] Sum of angles of triangle [Updated]
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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. :)
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
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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.
You're wrong: it is finite. The number of digits representing the number is an infinite one. According to your definition, even
1/3is an infinite number.If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
This is going on my arrogant assumptions. You may have a superb reason why I'm completely wrong. -- Iain Clarke
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). 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. 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? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
Geometry simply doesn't care you can't draw exactly a sqrt(2) line (and you can't draw exactly a 1 line as well). :)
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
This is going on my arrogant assumptions. You may have a superb reason why I'm completely wrong. -- Iain Clarke
[My articles] -
actually you do see a third one: the horizon is curved. (and not because there are waves)
You cannot perceive that: you're a bidimensional captain. :)
If the Lord God Almighty had consulted me before embarking upon the Creation, I would have recommended something simpler. -- Alfonso the Wise, 13th Century King of Castile.
This is going on my arrogant assumptions. You may have a superb reason why I'm completely wrong. -- Iain Clarke
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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?
d@nish wrote:
Which means sum of angles is not 180 degree. Right?
Absolutely NOT! you can accurately measure and construct the angles. The siomplest construction was mentioned by another poster (perhaps you failed to comprehend): construct an equilateral triangle using any convenient side length you wish. Locate the center of one side, then draw a line to the opposite vertex. You now have two perfect 30,60,90 triangles, whose short side and hypotenuse are lengths you know very accurately, but whose long side is some multiple of the square root of 3.
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). 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. 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? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
The sum of angles of a triangle is 180 degrees if and only if the edges occur within a surface having zero curvature; that is to say, if and only if the triangle is drawn on a flat surface. If the triangle is drawn on a positively curving surface -- such as the outside of a ball -- the sum of its angles will be greater than 180 degrees. If the triangle is drawn on a negatively curving surface -- such as the surface of a hyperboloid -- the sum of its angles will be less than 180 degrees. This topic is covered in depth in the mathematical field of topology. Cosmological evidence is mounting that the universe as a whole has a negative curvature; that is one of the explanations given for the increasing rate of the universe's expansion. That would mean that a triangle defined by any three points in the universe would, by definition, have angles that added up to less than 180 degrees.
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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.
You are incorrect: "number" and "infinity" are unrelated concepts. While the square root of three cannot be expressed precicely, it is possible to express it as an asymptotic convergence to a single value; that is to say, construct f(x) such that f(x) -> 3^(1/2) as x increases. That makes the number finite. The fact that there is no value of x such that f(x) = 3^(1/2) is what makes it irrational.
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Yes, they are called equilateral triangles :laugh: Here is the proof: Take three toothpicks of equal length. Define their length arbitrarily as 2x. Arrange them into a triangle.
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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.
It's a 2-dimensional curved manifold, not a 2 dimensional flat plane.
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
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You are incorrect: "number" and "infinity" are unrelated concepts. While the square root of three cannot be expressed precicely, it is possible to express it as an asymptotic convergence to a single value; that is to say, construct f(x) such that f(x) -> 3^(1/2) as x increases. That makes the number finite. The fact that there is no value of x such that f(x) = 3^(1/2) is what makes it irrational.
i didn't say anything about infinity. i talked about non-finite.
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It's a 2-dimensional curved manifold, not a 2 dimensional flat plane.
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
Then the curve is the third dimension.
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Then the curve is the third dimension.
That's only if it is embedded in 3 dimensions. If all you have is a coordinate atlas and distance function you have no way of knowing what shape it might embed into. There are plenty of objects isomorphic to a sphere that you would never recognize until you generated a map which proved it.
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
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That's only if it is embedded in 3 dimensions. If all you have is a coordinate atlas and distance function you have no way of knowing what shape it might embed into. There are plenty of objects isomorphic to a sphere that you would never recognize until you generated a map which proved it.
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
now your changing the parameters of the problem. not fair.
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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
Dan Neely wrote:
No. the surface of any 3d object is 2d.
true. however, we didn't talk about just one point, we talked about being "on" the surface of a sphere and since a person occupies more than one point, then it's no longer 2d.
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). 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. 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? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
You are being irrational!
xacc.ide
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). 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. 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? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
I did a bit of this at uni.. non-Euclidean geometries are based on non-flat surfaces Riemann and Lobachevsky both came up with kind of convex/concave non-euclidean geometries if you imagine a triangle on a curved surface, the angles are exaggerated and can add up to more or less than 180 Euclid's parallel postulate was one of the axioms that he used to prove his geometry assuming his axioms to be true, his proof was correct but if you do not assume it, the non-Euclidean geometries become consistent theories. people have tried and failed to prove the parallel postulate is true i think the whole curved surface thing ties in with Einstein.. :zzz:
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i didn't say anything about infinity. i talked about non-finite.
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We have learnt that sum of angles of a triangle is 180 degree. Now, 1, 3^1/2 and 2 form a triangle (based on the trigonometry). 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. 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? Edit: It is past midnight here. Time to sleep. Have a good time everyone.
d@nish wrote:
3^1/2
Just because there isn't a line for that on a plastic ruler doesn't mean that the length does not exist. If it exists, it can be drawn -- you just can't measure it accurately with a plastic ruler (meaning that you probably can't measure the angles accurately enough, either). I'd be intrigued to see this research involving three stars, if only to understand why it should work with three stars but not three points on a piece of paper.
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