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Suppose that we have two sheets of paper $S,T$ and that each of $S,T$ is in the shape of a convex quadrilateral. Also, suppose that the length of the perimeter of $S$ equals that of $T$. (Note that $S$ and $T$ are not necessarily congruent.)

Then, here is my question.

Question : If we make a convex polyhedron with these sheets of paper in the following way, then can we make a convex polyhedron with $9$ faces?

(1) You can fold the paper along a line.

(2) You can make a convex polyhedron by pasting them at the edge.

Example : We can make a convex polyhedron with $10$ faces (see the figures below where $S$ and $T$ are congruent squares. Folded along red lines. Each red line crosses the midpoints of edges of the square.).

$\hspace1in$enter image description here

Let $N_f$ be the number of the faces of the convex polyhedron.

At math S.E, it's been proved that $N_f\le 12$ and that there are examples such that $N_f=4,5,6,7,8,10,11,12$. However, no example for $N_f=9$ can be found.

This page has a partial list of Enneahedra (polyhedra with 9 faces).

I have a conjecture that we can make the following polyhedron with $9$ faces. ($S$ and $T$ are not necessarily rectangles.) However, I can neither find any concrete example nor prove that it is impossible.

$\hspace1in$enter image description here

Update 1 : A polyhedron in the conjecture above given by $$A(a,b,0), B(a,c,0), C(-a,-b,0),D(-a,-c,0), $$$$E(d,e,f), F(g,h,i), G(-d,-e,f),H(-g,-h,i)$$ is almost what we desire (but it is not) where $$a=-2.71,b=0.931273,c=-0.963719,d=-2.681359,e=0.4689,$$$$f=4.28591,g=-2.66066,h=-1.368,i=3.74971.$$

Hence, the conjecture seems true, but I don't know how to get a concrete example strictly.

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    $\begingroup$ Just so I am clear: are you essentially asking which convex enneahedra can be cut in two along the edges of the polyhedra and unfolded into two convex quadrilaterals with the same perimeter? (Another possibility is that it be cut in two along "the middle of" faces instead.) Gerhard "Wrapping Head Around This Problem" Paseman, 2014.02.24 $\endgroup$ Feb 24, 2014 at 17:07
  • $\begingroup$ @GerhardPaseman: Yes. What I'm asking is also represented in the way you wrote. $\endgroup$
    – mathlove
    Feb 24, 2014 at 17:30

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