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Is there any pyramid like that?

  1. Oct 9, 2015 #1
    I had geometry quite a while ago and I wonder if anyone has any idea how to tackle this problem:

    Is there any ABCDS pyramid (where ABCD is a rectangle) in which each 2 edges have different lengths and |AS|+|CS|=|BS|+|DS|

    Thanks
     
  2. jcsd
  3. Oct 9, 2015 #2

    jedishrfu

    Staff: Mentor

    Welcome to PF!

    Do you mean a real pyramid like the ones in Egypt or mathematically?
     
  4. Oct 9, 2015 #3
    :smile: mathematically. Something tells me one can construct such a pyramid, but I cant give explicit example..
     
  5. Oct 9, 2015 #4

    BvU

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    No Egyptian pyramids I know of have a rectangular floor plan....

    I assume the unequal edges are only the non-horizontal ones (because of this rectangular floor plan) ?

    In that case I look at a rectangle on the floor from above; then I imagine two equal length (length > diagonal of rectangle) wires, one from A to C and one from B to D. If I lift them with a hook or something, where can I end up with the top of the pyramid ? Can the projection of the tip on the floor be off both symmetry axes or only off one ?
     
  6. Oct 9, 2015 #5
    Yes, only non horizontal ones. I tried the same but somehow its hard to show anything "mathematically"
     
  7. Oct 9, 2015 #6

    jedishrfu

    Staff: Mentor

    You could draw the rectangular base and attach threads to each corner and then pull them up to a common point. This is how I'd visualize it.
     
  8. Oct 9, 2015 #7
    Me too, I'll try it. But how it can be proven using elementary geometry?
     
  9. Oct 9, 2015 #8

    jedishrfu

    Staff: Mentor

    I don't know how to prove it other than starting with the rectangular base and drawing a line segment of a any length perpendicular to the base such that one endpoint is in the rectangle then you can draw a line segment from any vertice of the rectangular base to the free line segment endpoint. Something like that...
     
  10. Oct 9, 2015 #9
    Ive tried that already but it didnt lead me to anywhere
     
  11. Oct 9, 2015 #10

    jedishrfu

    Staff: Mentor

    Are you trying to proof that such a shape exists?
     
  12. Oct 9, 2015 #11
    I firmly believe that it exists - I did an experiment with strings

    But I cant guarantee it exists
     
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