[Mathematics] Sum of angles of triangle [Updated]
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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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((λ (x) `(,x ',x)) '(λ (x) `(,x ',x))) The Scheme Programming Language – Fourth Edition -
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.
I wanna be a eunuchs developer! Pass me a bread knife!
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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.
Ah. I should have read further before commenting.
I wanna be a eunuchs developer! Pass me a bread knife!
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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.
Rob Graham wrote:
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
more precisely, the long side is some multiple (the convenient length one chose) of the square root of 3/4 (one half root three) I believe the confusion that ripples through this discussion is between accurate drawing and precise measuring. As you note, accurate drawing is simple. Measuring the bisector is tougher, but the limit is in our tools, not our perceptions. We can calculate the length to any arbitrary precision, even though it is an irrational product. Of course the valid precision of the calculated value is limited be the precision associated with original line. An inconvenient proof? :)
there was only ever one purely original computer program, that was the first one...everything since is derivative to some degree - unknown
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no, it just means you can't accurately measure the sqrt(3) side.
Seriously? then prove that you can draw a line of length unit == 1 -there was a similar discussion in like the 1800's(date?) concerning infinity, where every time you subdivide a line (of any length) you can never get to any point on that line no matter how many divisions you make i.e if an arrow is shot, it both never leaves it's origin nor passes through a midpoint nor hits an end point. we know that these things do occur. when you are analyzing the meaning of a line your looking at the set [AB]... whether those points are real, rational, imaginary, or in any other coordinate system.. based on the axiom of a definition of a plane in that coordinate system. triangles are not defined as (AB) which doesn't include the "end point" also you place your own un-doing in the irrational you your self state.
I'd blame it on the Brain farts.. But let's be honest, it really is more like a Methane factory between my ears some days then it is anything else...
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now your changing the parameters of the problem. not fair.
No, actually I'm not. There aren't any ways to distinguish which type of "sphere" you are on without having knowledge of the space that it is embedded in.
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
-
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.
Reimann Geometry. However, has nothing to do with your example.
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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.
- Triangles in Euclidean space ALWAYS have the sum of the 3 internal angles = 180. 2) By your same thinking, you can never draw the diagonal across a square of side = 1 since its length is 2^1/2 (which is also irrational). 3) A right angled triangle having two sides length 1 and 2 will have a hypotenuse of 3^1/2. (This is Pythagoras' Theorem). 4) If you are to 'draw' the triangle in (3), you will ONLY be drawing an approximation of it. 5) To expound further on 4, the actual value of numbers and thier geometric representation: - Draw a line of any length and divide into ten equal parts - Starting from the left, mark each divide 0,1,2,3,4,5,6,7,8,9,10 - Now try to find, and mark, the following numbers - e = 2.718....... - pi = 3.14285..... - 3^1/2 = 1.713....... - 2^1/2 = 1.414....... - 1/3 = 0.333....... As hard as they are (or rather - impossible) to pin point exactly on this line, I think we can agree that these five numbers do exist. But the line is only an approximation. On the same reasoning: Think of the largest number you can think of. Now think of an even larger number. In this exercise we are 'measuring' the magnitude of infinity. We can reason that infinity is the largest number we know of but acknowledge that there is an even greater infinity. In a nutshell: perhaps your reasoning is sound on 'linguistic' level but mathematically flawed. Mathematical thinking is abstract. Also refer to: http://en.wikipedia.org/wiki/Number[^]
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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 think you've forgotten your pythagoras. Try again with sides of 1, 2, and 5^(0.5) and at least this basic point won't distract us. (1^2 + 2^2 = 1 + 4 = 5)
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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.
Wait a minute...I don't understand what you are saying:
d@nish wrote:
Since one cannot draw a line of length 3^1/2, this triangle is not possible
What's undrawable about 3^(1/2)? It's about 1.73205081 units (to a reasonable level of precision -unless your planning on drawing in a scale of light years or miles)? Yes it's irrational, but so is Pi. It easy to see how moving the triangle off a eucldean/cartesian plane will change everything. Imagine torquing (twisting) the triangle so that it does not "lay flat", and then measuring the angles which now lay in their own planes, unrelated to the planes of the others. It's now an angular measurement in 3 dimensions (solid angle). Interesting and important, but not the same rule as you learnt in school. To think of it anther way, draw your triangle on an orange. This is actually way cool stuff - that I haven't thought about in about 20 years...! :omg:
No matter where you go, there you are.
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- Triangles in Euclidean space ALWAYS have the sum of the 3 internal angles = 180. 2) By your same thinking, you can never draw the diagonal across a square of side = 1 since its length is 2^1/2 (which is also irrational). 3) A right angled triangle having two sides length 1 and 2 will have a hypotenuse of 3^1/2. (This is Pythagoras' Theorem). 4) If you are to 'draw' the triangle in (3), you will ONLY be drawing an approximation of it. 5) To expound further on 4, the actual value of numbers and thier geometric representation: - Draw a line of any length and divide into ten equal parts - Starting from the left, mark each divide 0,1,2,3,4,5,6,7,8,9,10 - Now try to find, and mark, the following numbers - e = 2.718....... - pi = 3.14285..... - 3^1/2 = 1.713....... - 2^1/2 = 1.414....... - 1/3 = 0.333....... As hard as they are (or rather - impossible) to pin point exactly on this line, I think we can agree that these five numbers do exist. But the line is only an approximation. On the same reasoning: Think of the largest number you can think of. Now think of an even larger number. In this exercise we are 'measuring' the magnitude of infinity. We can reason that infinity is the largest number we know of but acknowledge that there is an even greater infinity. In a nutshell: perhaps your reasoning is sound on 'linguistic' level but mathematically flawed. Mathematical thinking is abstract. Also refer to: http://en.wikipedia.org/wiki/Number[^]
I can't believe you guys keep tripping up on Pythagoras while simultaneously writing away as if you knew geometry. I don't know geometry and won't attempt to explain how a triangle might have anything but 180 degrees, but I do at least know that pythagoras' theorem states a^2 + b^2 = c^2 and *not* (a+b)^2 = c^2. Hence with a=1, b=2, c=5^0.5.