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        riddles >> medium >> Planes of symmetry in triangular pyramid
        
(Message started by: Altamira_64 on Sep 11^th, 2012, 3:38pm)

Title: Planes of symmetry in triangular pyramid
Post by Altamira_64 on Sep 11^th, 2012, 3:38pm

How many planes of symmetry does a triangular pyramid have?

Title: Re: Planes of symmetry in triangular pyramid
Post by Noke Lieu on Sep 11^th, 2012, 7:01pm

Without being too cautious, I think I could build one with none. I wonder how many isosceles triangle I could taunt that construction with?
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essentially, irregular hexagons aren't always symmetrical
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Title: Re: Planes of symmetry in triangular pyramid
Post by Grimbal on Sep 12^th, 2012, 12:24am

Assuming you are talking about a tetrahedron,
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I count 6 planes, each plane contains one edge and the center.
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Title: Re: Planes of symmetry in triangular pyramid
Post by peoplepower on Sep 12^th, 2012, 1:03am

I suppose in general, we only need to look in
[hide]2 dimensions: the base along with the projection of the apex; we look for lines of symmetry here which pass through the apex. I have a wishy-washy proof of this assumed one-to-one correspondence, but I would not be surprised if I was wrong.[/hide]

Title: Re: Planes of symmetry in triangular pyramid
Post by rmsgrey on Sep 12^th, 2012, 5:04am

A tetrahedron has 24 rigid symmetries, corresponding to the 24 permutations of the 4 vertices.

The even permutations (the identity, pairs of 2-cycles, and 3-cycles) are all rotations; odd permutations are either reflections or improper rotations. Counting and classifying them is left to the reader :)

Title: Re: Planes of symmetry in triangular pyramid
Post by Grimbal on Sep 12^th, 2012, 7:01am

I am not sure about the terminology. I assume by "plane symmetry" Altamira_64 was refering to the symmetry plane of a 3D reflection.

Title: Re: Planes of symmetry in triangular pyramid
Post by peoplepower on Sep 12^th, 2012, 4:55pm

on 09/12/12 at 07:01:50, Grimbal wrote:

I am not sure about the terminology. I assume by "plane symmetry" Altamira_64 was refering to the symmetry plane of a 3D reflection.

Those happen to correspond to 2-cycles in S₄, since they preserve one edge (two vertices) and flip the other two vertices.