Is there any pyramid like that?

In summary, the conversation discusses the possibility of constructing a pyramid with unequal edges in a rectangular base, where the lengths of the non-horizontal edges satisfy a certain equation. Various methods are suggested to visualize and potentially prove the existence of such a pyramid, but no definitive proof is given. The conversation ends with the belief that the pyramid likely exists, but cannot be guaranteed.
  • #1
karosz
6
0
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
 
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  • #2
Welcome to PF!

Do you mean a real pyramid like the ones in Egypt or mathematically?
 
  • #3
:smile: mathematically. Something tells me one can construct such a pyramid, but I can't give explicit example..
 
  • #4
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 ?
 
  • #5
Yes, only non horizontal ones. I tried the same but somehow its hard to show anything "mathematically"
 
  • #6
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.
 
  • #7
Me too, I'll try it. But how it can be proven using elementary geometry?
 
  • #8
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...
 
  • #9
Ive tried that already but it didnt lead me to anywhere
 
  • #11
I firmly believe that it exists - I did an experiment with strings

But I can't guarantee it exists
 

1. Is there any pyramid like that?

Yes, there are many pyramids located around the world. The most famous and well-known are the Egyptian pyramids, but there are also pyramids in other countries such as Mexico, Peru, and Sudan.

2. How were pyramids built?

The exact methods used to build pyramids varies depending on the location and time period. However, most pyramids were built using large blocks of stone or mud bricks that were stacked on top of each other to create a triangular shape. Some pyramids also had complex internal structures and chambers.

3. Why were pyramids built?

The purpose of pyramids also varies depending on the location. Some were built as tombs for pharaohs and rulers, while others served as ceremonial or religious sites. Pyramids were often seen as symbols of power and status in ancient societies.

4. How old are the pyramids?

The age of pyramids also varies, with some being thousands of years old. The oldest known pyramid is the Step Pyramid of Djoser in Egypt, which dates back to around 2630 BCE. The most recent pyramid was built in the 19th century in Sudan.

5. Are there any pyramids that have not been discovered yet?

It is possible that there are still undiscovered pyramids around the world. In recent years, new pyramids have been discovered using satellite imaging and ground-penetrating radar technology. It is also believed that there are many smaller and less well-known pyramids that have yet to be uncovered by archaeologists.

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