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Topic: Planes of symmetry in triangular pyramid (Read 5861 times) |
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Altamira_64
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Planes of symmetry in triangular pyramid
« on: Sep 11th, 2012, 3:38pm » |
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How many planes of symmetry does a triangular pyramid have?
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Noke Lieu
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Re: Planes of symmetry in triangular pyramid
« Reply #1 on: Sep 11th, 2012, 7:01pm » |
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Without being too cautious, I think I could build one with none. I wonder how many isosceles triangle I could taunt that construction with?
essentially, irregular hexagons aren't always symmetrical
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| « Last Edit: Sep 11th, 2012, 7:18pm by Noke Lieu » |
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Grimbal
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Re: Planes of symmetry in triangular pyramid
« Reply #2 on: Sep 12th, 2012, 12:24am » |
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Assuming you are talking about a tetrahedron,
I count 6 planes, each plane contains one edge and the center.
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peoplepower
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Re: Planes of symmetry in triangular pyramid
« Reply #3 on: Sep 12th, 2012, 1:03am » |
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I suppose in general, we only need to look in
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.
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| « Last Edit: Sep 12th, 2012, 1:05am by peoplepower » |
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rmsgrey
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Re: Planes of symmetry in triangular pyramid
« Reply #4 on: Sep 12th, 2012, 5:04am » |
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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 
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Grimbal
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Re: Planes of symmetry in triangular pyramid
« Reply #5 on: Sep 12th, 2012, 7:01am » |
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I am not sure about the terminology. I assume by "plane symmetry" Altamira_64 was refering to the symmetry plane of a 3D reflection.
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peoplepower
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Re: Planes of symmetry in triangular pyramid
« Reply #6 on: Sep 12th, 2012, 4:55pm » |
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on Sep 12th, 2012, 7:01am, 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.
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Those happen to correspond to 2-cycles in S4, since they preserve one edge (two vertices) and flip the other two vertices.
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