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Topic: convex polyhedron with almost all faces regular
Replies: 6   Last Post: Dec 6, 2012 5:43 PM

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quasi

Posts: 10,315
Registered: 7/15/05
Re: convex polyhedron with almost all faces regular
Posted: Dec 6, 2012 3:33 PM
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dy/dx <dydx-1@gmail.invalid> wrote:
>1treePetrifiedForestLane wrote:
>>
>> yes; the consequence of "one irregular facet" are to weird.
>>
>> "then all of the faces are regular polygons" as being true ...

>
>It's clear that if all but one of the polygons is *equilateral*,
>then they all must be.
>
>However, I exhibit a counterexample to the claim that if all
>but one are *regular* they all must be, if a dihedral angle of
>exactly 180 degrees is permitted:
>
>A rhombus with internal angles of 60 and 120 degrees, four
>square faces, and two equilateral triangular faces, forming a
>prism. The two triangular faces, combined, form a duplicate of
>the rhombic face.


Ok, but if I understand it correctly, your construction yields
a polyhedron with two adjacent coplanar faces, so those are not
acceptable as _distinct_ faces of a convex polyhedron.

quasi



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