1
$\begingroup$

Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon. This is a conjecture I came across while trying to solve this problem. I was drawing many figures and three dimensional figures and pyramids were the first choice. And I observed this. I have no proof, or even ideas to a proof. But seems true. Any counter example would be nice. Thanks.

$\endgroup$
9
  • 2
    $\begingroup$ In my opinion, this is not suitable to MO. Sketch of proof: 1. a section which is a $(n+1)$-gon must meet all faces of the pyramid; 2. it is therefore not parallel to he base; 3. By an explicit computation, show that not edges of the section can have the same length. $\endgroup$ – Benoît Kloeckner Jun 23 '14 at 16:48
  • 1
    $\begingroup$ math.SE has a more general scope regarding math questions, you can try there if you do not solve the question by yourself. $\endgroup$ – Benoît Kloeckner Jun 23 '14 at 17:06
  • 1
    $\begingroup$ You might observe that "three edges of the n+1 gon have to fit in two edges of the n gon". $\endgroup$ – The Masked Avenger Jun 23 '14 at 17:54
  • 1
    $\begingroup$ The pyramid faces provide n edges of the cross section; the remaining edge has to be a chord of the base, and either lie on an edge of the base, or "clip off" a corner of the base. If both polygons are to be regular, you will have length issues with the sides as you are "trying to fit three edges in the space of two" while at the same time keeping the edges in the cross section of the same length. $\endgroup$ – The Masked Avenger Jun 24 '14 at 15:27
  • 1
    $\begingroup$ I suspect something more general is true, that you can't have an ngonal base, an (n+1)gonal cross section, and insist on some combination of equiangular or equilateral to apply to the polygons. $\endgroup$ – The Masked Avenger Jun 24 '14 at 15:31