1
$\begingroup$

It is easy to see that for isosceles tetrahedra (https://en.wikipedia.org/wiki/Disphenoid) all faces are equal acute triangles. If we consider regular tetrahedra and attach a regular triangular pyramid to each face, then we obtain a polyhedron with equal triangular faces; each face will be an isosceles triangle with obtuse angle $< 120^{\circ}$.

For which other triangles there is a polyhedron with all faces equal to this triangle?

The same question for quadrangles (see https://en.wikipedia.org/wiki/Trapezohedron for examples).

Improve this question
$\endgroup$
8
  • $\begingroup$ Take two identical pyramids with a regular polygonal base and stick the bases together. $\endgroup$
    – Gerry Myerson
    Commented Apr 4 at 21:56
  • $\begingroup$ But then we get a polyhedron, each face of which is an isosceles triangle with obtuse angle $<120^{\circ}$. We already have such an example. $\endgroup$
    – Fedor Nilov
    Commented Apr 5 at 6:06
  • $\begingroup$ Why would each face have an obtuse angle? The angles at the apex of a pyramid can be arbitrarily close to zero. $\endgroup$
    – Gerry Myerson
    Commented Apr 5 at 6:11
  • $\begingroup$ Yes, but for an arbitrary acute triangle we can construct an isosceles tetrahedra with faces equal to this triangle. The question is for which OTHER triangles we can construct a polyhedron with all faces equal to this triangle? $\endgroup$
    – Fedor Nilov
    Commented Apr 5 at 6:18
  • 1
    $\begingroup$ OK, but there's a lot more at that mathoverflow page than that one polyhedron. $\endgroup$
    – Gerry Myerson
    Commented Apr 5 at 8:16

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .