Why can there not exist a pyramid with 29 corners?

I tried to find a pattern, it seems like I can draw a pyramids with corners, 3,4,5, and 6. 7 becomes tricky and 29 isn’t something I even can conceptualize. Ofcourse there is probably a pattern which one on the bases of would conclude that it is an impossibility.

Thanks for your assistance in advance.

  • 2
    $\begingroup$ How can you draw a pyramid with $3$ vertices? As far as I know, you cannot have less corners than a tetrahedron, which has 4 vertices. $\endgroup$
    – Toby Mak
    Sep 12, 2018 at 8:46
  • $\begingroup$ I thought of the numbers of vertices just in the base. I agree with your statement. $\endgroup$ Sep 12, 2018 at 8:55

1 Answer 1


Let the pyramids' base have $n$ vertices. There is also one more vertex on top, so there are in total $n+1$ vertices.

For a pyramid with $29$ vertices, the base must be a $28$-sided polygon. This might be hard to visualise because there are so many sides, but it is possible.

enter image description here

  • 3
    $\begingroup$ Added one such :-) $\endgroup$ Sep 12, 2018 at 10:49
  • $\begingroup$ @JyrkiLahtonen Thanks! How did you find that image? $\endgroup$
    – Toby Mak
    Sep 12, 2018 at 11:05
  • 1
    $\begingroup$ Being a tenured lecturer, I could afford to cough up the dough and purchase a personal license to Mathematica :-) It can be relatively easily coerced to produce such images. I first built a list of the coordinates and then defined a list of polygons forming this pyramid. Piece of cake, really. $\endgroup$ Sep 12, 2018 at 12:27

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