Is there any pyramid like that?

[karosz]

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

 

[jedishrfu]

Welcome to PF!

Do you mean a real pyramid like the ones in Egypt or mathematically?

 

[karosz]

mathematically. Something tells me one can construct such a pyramid, but I can't give explicit example..

 

[BvU]

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 ?

 

[karosz]

Yes, only non horizontal ones. I tried the same but somehow its hard to show anything "mathematically"

 

[jedishrfu]

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.

 

[karosz]

Me too, I'll try it. But how it can be proven using elementary geometry?

 

[jedishrfu]

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...

 

[karosz]

Ive tried that already but it didnt lead me to anywhere

 

[jedishrfu]

Are you trying to proof that such a shape exists?

 

[karosz]

I firmly believe that it exists - I did an experiment with strings

But I can't guarantee it exists