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 Post subject: A question for polyhedrons
PostPosted: Wed Mar 27, 2013 9:10 am 
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Joined: Sat May 19, 2012 8:35 am
Location: Singapore
Hello all. Randomly thought of a question today. Take a polyhedron, any polyhedron. Find its centre of gravity and mark this point. Take each face and find its centre of gravity as well and mark it out, and draw lines connecting the centre of gravity of the polyhedron to the centre of gravities of the faces. My question is this: How many polyhedrons satisfy the rule whereby all the faces are perpendicular to their respective "radial lines"? I only know somehow so far (by intuition) that the five platonic solids do.


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 Post subject: Re: A question for polyhedrons
PostPosted: Wed Mar 27, 2013 9:25 am 
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Hmm. After some thought, I guess all regular prisms will satisfy this rule. All bipyramids, trapezohedra and regular pyramids have a version which also satisfies this rule. What other classes do?


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 Post subject: Re: A question for polyhedrons
PostPosted: Wed Mar 27, 2013 9:56 am 
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Most of the Archimedean solids can easily be proven to follow this rule. There tends to be enough rotational symmetry around the non-square faces, and mirror symmetry through the square faces.
The exceptions are the snub-cube and snub-dodecahedron, and I don't know if they satisfy the condition.

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 Post subject: Re: A question for polyhedrons
PostPosted: Wed Mar 27, 2013 11:16 am 
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I believe any symmetric polyhedron should satisfy this rule, provided each of the faces are symmetric in at least 2 dimensions, including equilateral triangles, squares, rectangles, rhombic faces, regular pentagons, hexagons, etc. I'm not sure about isosceles triangles and deltoidal faces, which are symmetric in only 1 dimension (axis of reflection).

EDIT: Although, a stretched or squashed parallelepiped or octahedron wouldn't fit this behaviour, so I don't know. :?

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 Post subject: Re: A question for polyhedrons
PostPosted: Fri Mar 29, 2013 4:53 am 
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Considering how common this property is for highly symmetric polyhedra(it holds for all platonics, most, if not all, archimedian, at least 2 Catalan(Rhombic Dodeca and Rhombic Triconta), regular prisms, and several other infinite families under the right constraints), I wonder:
-What is the most asymmetrical polyhedron with this property?
-Can you construct a polyhedron with this property such that no face as rotational or reflectional symmetry and the polyhedron as a hold has no rotational or reflectional symmetry?

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