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Thread: Question about Euclidean Space Perception for a TNT TV show

  1. #1 Question about Euclidean Space Perception for a TNT TV show 
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    Hello. My name is Jon and I'm a researcher for several TV shows including CSI, CSI: Miami, Bones and the TNT show RIZZOLI & ISLES. It's my job to make sure our scripts are as accurate as possible.

    I'm wondering if anyone here could help us better understand Euclidean Space Perception. How would you explain this to a regular person...or TV writer.

    Three Demensions of space and one for time? What else. What's it all mean?

    We really appreciate any help you can provide.

    Sincerely,

    Jon
    Researcher
    www.entertainmentresearchconsultants.com


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  3. #2  
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    is all around us. See this thread; http://www.thescienceforum.com/viewt...r=asc&start=15.


    I was some of the mud that got to sit up and look around.
    Lucky me. Lucky mud.
    -Kurt Vonnegut Jr.-
    Cat's Cradle.
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  4. #3  
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    Perhaps this may help with an overview of some of the related concepts involved here?

    http://www.geom.uiuc.edu/video/sos/materials/overview/ - The Shape of Space
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  5. #4 Re: Question about Euclidean Space Perception for a TNT TV s 
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    Quote Originally Posted by jonfromtv
    Hello. My name is Jon and I'm a researcher for several TV shows including CSI, CSI: Miami, Bones and the TNT show RIZZOLI & ISLES. It's my job to make sure our scripts are as accurate as possible.

    I'm wondering if anyone here could help us better understand Euclidean Space Perception. How would you explain this to a regular person...or TV writer.

    Three Demensions of space and one for time? What else. What's it all mean?

    We really appreciate any help you can provide.

    Sincerely,

    Jon
    Researcher
    www.entertainmentresearchconsultants.com
    1. Rizzoli & Isles is very good.

    2. The term "Euclidean Space Perception" is unusual. Euclidean space refers to ordinary space, of any dimension, in which the Pythagorean theorem is true.

    The geometry of the (flat) plane is Euclidean, as is what is usually thought of as 3-dimensional space.

    It is common in physics to add time as a fourth dimension and consider a thing called spacetime. In Newtonian physics spacetime is just Euclidean 3-space plus time. That might be reasonably called the "Euclidean Space Perception".

    That picture can be contrasted with the picture presented by Einstein's theory of relativity. In relativity, spacetime is not Euclidean. The non-Euclidean nature of relativistic spacetime gives rise to some effects that are well-proved by experiment but counter-intuitive. Such effects include length contraction, time dilation and the fact that observers in motion with respect to one another will disagree as to what events are simultaneous.

    One can also contrast Euclidean geometry with the geometry of the surface of a sphere. On a sphere the geodesics (straight lines or locally the shortest path between points)) are great circles. That is why airlines fly "great circle routes". So there are no parallel lines (since two great circles will intersect). The sphere is positively curved, so a triangle will have angles that sum to more than 180 degrees.

    Either of these distinctions might come up in the plot for one of your shows.
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  6. #5  
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    Gotcha DrR !!!

    Euclidian space or geometry is one in which Euclid's 5th postulate applies. You know, the one about a line and a point through which only one parallel line can be drawn. It was investigatuon of this postulate, and wether it could be eliminated as superfluous, that led Gauss, Bolyai and a russian ( I want to say Lobchevsky but I don't remember ) to investigate spaces or geometries where its not the case and triangles' angles add to more or less than 180 deg. Gauss' student Riemann finally gave us a more complete understanding of non-Euclidian geometries.

    Now of course I'm going to have to investigate wether the Pythagorean Theorem is due exlusively to Euclid's 5th or if its a result of the previous four.
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  7. #6  
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    Incidentally, I've seen CSI, CSI-Miami and even Bones but I've never even heard of Rizzoli and Isles. Is it any good? What's it deal with? How long has it been runnig?

    I want to say I'll download some torrents, if available, but I don't know if its a good idea with one of the show's consultants on this forum.
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  8. #7  
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    Quote Originally Posted by MigL
    Gotcha DrR !!!


    Now of course I'm going to have to investigate wether the Pythagorean Theorem is due exlusively to Euclid's 5th or if its a result of the previous four.
    When you finish you may need to reconsider your first sentence.

    The sum of the angles of a triangle is a test for flatness and it is what distinguishes Euclidean distance from other metrics..
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  9. #8 c^2 = a^2 + b^2 
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    c^2 = a^2 + b^2 is the only metric for which length is independent of the coordinate system

    Which, needless to say, makes euclidean space much simpler than any other
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  10. #9 Re: c^2 = a^2 + b^2 
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    Quote Originally Posted by granpa
    c^2 = a^2 + b^2 is the only metric for which length is independent of the coordinate system

    Which, needless to say, makes euclidean space much simpler than any other
    What in that supposed to mean ?

    I have held a PhD in mathematics for more than 30 years and I can find no way to interpret that statement so that it makes any sense.
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    well use the taxicab metric and see what you get

    c = a + b

    draw a circle of radius 1 using this metric. It looks like a diamond

    now turn your coordinate system 45 degrees

    draw the circle again. I end up with a square inside a diamond



    this shouldnt be too surprising since one way to derive the pythagorean theorem is to assume that the length is independent of coordinate system
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    Oh come on you guys, at least grant me that, hystorically, curved geometries arose because of investigation into Euclid's fifth postulate and not bby investigating geometries where the pythagorean theorem is not true.
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    Oh, and I meant the 'gotcha' in jest DrR. I hope you didn't take it the wrong way.
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    i dont know anything about curved geometries.

    i thought you just wanted to know about euclidean geometry.
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    What I was getting at granpa, is that strictly defined Euclidian geometry, is the one defined by Euclid 2000+ (?) yrs ago in his monumental work 'Elements'. It is defined by five postulates, the fifth of which states that only one other parallel line can be drawn to a line on a plane through any point.
    There was always thought that the fifth postulate was unnecessary and the first four were enough for a consistent geometry and various peole tried to show that the fifth could be obtained from the first four.
    Nobody succeded, but while investigating the results of a fifth that was untrue (either more than one parallel lines or no parallel lines ), in the 18th and 19th century, Gauss and later Reimann gave us curved geometries ( positive, triangle angle sum>180 deg. and negative, triangle angle sum<180 deg.)

    Both you and DrR are using the modern definition of Euclidian space, which while correct, is not the historical or for that matter, Euclid's definition.
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  16. #15  
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    Quote Originally Posted by granpa
    well use the taxicab metric and see what you get

    c = a + b

    draw a circle of radius 1 using this metric. It looks like a diamond

    now turn your coordinate system 45 degrees

    draw the circle again. I end up with a square inside a diamond



    this shouldnt be too surprising since one way to derive the pythagorean theorem is to assume that the length is independent of coordinate system
    That in no way shows anything like "the Euclidean metric is the only metric that is independent of the coordinate system". The Euclidean metric is in fact defined in terms of a coordinate system.

    What it shows is simply that rotations are isometries of the Euclidean metric.

    c=a+b is NOT a metric. The metric to which you refer comes from the norm This yields the "diamond" as the unit circle. This norm is often called the norm.

    The "square" unit circle is associated with the norm

    Note that these metrics are metrics in the sense of topological metric spaces, not in the sense of inner products. Any metric derivable from a positive-definite inner product on 2-space (or n-space) will be isomorphic to the usual Euclidean metric -- though you can define any two non-co-linear vectors to be orthogonal simply by using the "dot product" as your inner product. This construction demonstrates just how dependent on the coordinate system the Euclidean inner product really is.


    One interesting fact is that, while the metrics arising from norms are different, they result in exactly the same topologies on 2-space.
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    The "square" unit circle is actually a diamond because the coordinate system is rotated 45 degrees. Its still the taxicab metric
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  18. #17  
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    Quote Originally Posted by granpa
    The "square" unit circle is actually a diamond because the coordinate system is rotated 45 degrees. Its still the taxicab metric
    The taxicab metric is the metric derived from the norm.

    That is different from the norm.

    Those are not the same metrics. If the were they would define the same unit ball. They don't. The square and the diamond are not the same thing, even if related by a rotation. The Euclidean norm is rotation invariant. These other norms are not.

    http://en.wikipedia.org/wiki/Taxicab_geometry

    Edit. The two normed spaces are isometric is an isometry. But this is not a rotation, but rather a rotation and dilation.
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    I know that they arent the same thing thats why I responded.
    I dont know why you even mentioned the l^infinity norm.
    Its completely irrelevant.
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  20. #19  
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    Quote Originally Posted by granpa
    I know that they arent the same thing thats why I responded.
    I dont know why you even mentioned the l^infinity norm.
    Of course you don't.

    Quote Originally Posted by granpa
    Its completely irrelevant.
    No. It is not.

    In fact the transformation (x,y) --->(x-y,x+y) is an isometry taking the unit (square) in L1 to the unit ball (diamond) in L-infinity. I had missed that earlier.
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  21. #20  
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    Quote Originally Posted by MigL
    Oh come on you guys, at least grant me that, hystorically, curved geometries arose because of investigation into Euclid's fifth postulate and not bby investigating geometries where the pythagorean theorem is not true.
    They are the same thing. The parallel postulate and the Pythagorean theorem are equivalent.

    I thought that you were researching this for yourself.

    http://en.wikipedia.org/wiki/Pythagorean_theorem
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  22. #21  
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    DrRocket:

    With all due respect, it is obvious that you didnt do what I asked you to do.
    You didnt work through the exercise that I gave you.
    You obviously just glanced at it from your ivory tower and then spat out an response.
    If you work through the exercise you will see that L^infinity never occurs anywhere in the problem and is completely irrelevant. Obviously if one coordinate system give a diamond for a circle then using a coordinates system turned 45 degrees will give a square. Both are using the taxicab metric. Any 10 year old can see this.

    the important thing here is that the square will be INSIDE the diamond.
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  23. #22  
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    Quote Originally Posted by granpa
    the important thing here is that the square will be INSIDE the diamond.
    I worked out quite a bit. What you state is both true and utterly irrelevant to your assertion that the Euclidean metric is independent of coordinates. It is not.

    All that you have noticed is that the "circle" is a diamond with diagonal of lEuclidean length 2 hence sides of length and the "circle" is a square with sides of length 2 and diagonals of length .

    Euclidean distance is invariant under the action of the orthogonal group, but the inner product from which it is derived, the so-called dot product, is in fact based on a coordinate choice. One can make a dot product from any choice of basis vectors. The resulting geometries will be isomorphic, but not the same -- the identity map will in general not be an isometry, and neither will elements of the usual orthogonal group -- this is precisely the case with the and norms, although neither arise from an inner product.

    Ii think you are confusing "invariant under rotations by the orthogonal group" with "independent of choice of coordinates".
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  24. #23  
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    I did work it out for myself DrR. But the original question was for the definition of Euclidian geometry, so I gave Euclid's definition.
    My own modern definition is similiar to yours and grampa's.
    Given the right angled triangle with sides a, b and hypotenuse c then the relationship la^2+mb^2=nc^2 is valid in euclidian, flat geometry only if l=m=n=1 . For all other values of l,m,n the geometry is not euclidean or flat, but curved.
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  25. #24  
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    Quote Originally Posted by MigL
    I did work it out for myself DrR. But the original question was for the definition of Euclidian geometry, so I gave Euclid's definition.
    My own modern definition is similiar to yours and grampa's.
    Given the right angled triangle with sides a, b and hypotenuse c then the relationship la^2+mb^2=nc^2 is valid in euclidian, flat geometry only if l=m=n=1 . For all other values of l,m,n the geometry is not euclidean or flat, but curved.
    Did you notice the point that the Pythagorean theorem is equivalent to the parallel postulate ?
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  26. #25  
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    Euclid knew the Pythagorian Theorem since it was known in even earlier times Phoenicians maybe ??) yet he chose not to use it in his basic postulates. Does that mean he didn't think it was primitive and that the parallel line postulate is primitive and Pythagoras can be derived from it. I'm asking since I don't know and its beyond me to determine 'primitiveness'.
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  27. #26  
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    Quote Originally Posted by MigL
    Euclid knew the Pythagorian Theorem since it was known in even earlier times Phoenicians maybe ??) yet he chose not to use it in his basic postulates. Does that mean he didn't think it was primitive and that the parallel line postulate is primitive and Pythagoras can be derived from it. I'm asking since I don't know and its beyond me to determine 'primitiveness'.
    I don't know if Euclid Knew that the two were equivalent.

    In some sense the parallel postulate is more "intuitively obvious" and that makes it more attractive as a postulate.

    From a strictly logical perspective one can assume either and derive the other.

    "Primitiveness" is in the eye of the beholder.

    Euclid's choice seems reasonable and generated little criticism over a suitably long period, so I would opine that it is OK.
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