Online since 2002. Over 3300 puzzles, 2600 worldwide members, and 270,000 messages.

TwistyPuzzles.com Forum
 It is currently Thu Jul 24, 2014 7:02 pm

 All times are UTC - 5 hours

 Page 1 of 1 [ 5 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: A question for polyhedronsPosted: Wed Mar 27, 2013 9:10 am

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.

Top

 Post subject: Re: A question for polyhedronsPosted: Wed Mar 27, 2013 9:25 am

Joined: Sat May 19, 2012 8:35 am
Location: Singapore
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?

Top

 Post subject: Re: A question for polyhedronsPosted: Wed Mar 27, 2013 9:56 am

Joined: Wed Mar 15, 2000 9:11 pm
Location: Delft, the Netherlands
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.

_________________
Jaap

Jaap's Puzzle Page:
http://www.jaapsch.net/puzzles/

Top

 Post subject: Re: A question for polyhedronsPosted: Wed Mar 27, 2013 11:16 am

Joined: Mon Mar 30, 2009 5:13 pm
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.

_________________
If you want something you’ve never had, you’ve got to do something you’ve never done - Thomas Jefferson

Top

 Post subject: Re: A question for polyhedronsPosted: Fri Mar 29, 2013 4:53 am

Joined: Sun Nov 23, 2008 2:18 am
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?

_________________
Just so you know, I am blind.

I pledge allegiance to the whole of humanity, and to the world in which we live: one people under the heavens, indivisible, with Liberty and Equality for all.

My Shapeways Shop

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 5 posts ]

 All times are UTC - 5 hours

#### Who is online

Users browsing this forum: Yahoo [Bot] and 8 guests

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Announcements General Puzzle Topics New Puzzles Puzzle Building and Modding Puzzle Collecting Solving Puzzles Marketplace Non-Twisty Puzzles Site Comments, Suggestions & Questions Content Moderators Off Topic

Forum powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group