Results 1 to 4 of 4

Math Help - symmetries of an octahedron

  1. #1
    Member
    Joined
    Apr 2008
    Posts
    204

    symmetries of an octahedron

    hi was just wondering how do u calculate the symmetries of an octahedron? i need to work out an enumeration problem, but i need to know how many distinct octahedra there are possible (all of the same size). they have 6 vertices, 12 edges and 8 triangular faces. thankssss
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by wik_chick88 View Post
    hi was just wondering how do u calculate the symmetries of an octahedron? i need to work out an enumeration problem, but i need to know how many distinct octahedra there are possible (all of the same size). they have 6 vertices, 12 edges and 8 triangular faces. thankssss
    Label the vertices of the octahedron from 1 to 6 in "order". What does it it mean in order? It mean we have 1,2,3,4 all along the same square with 5,6 in opposite sides - not sure if that makes sense. Now we need to replace the octahedron in such a way so that "order" is preserved. There are six ways to place the vertex 1 and then four possible places to choose for 2. After that is doen the "order" must be contained so along that square 3 and 4 are determined. This leaves us with 5,6 for which there are two locations on opposite sides. Multiply out the numbers and get that the symmetry group has order 48. You can find it here
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by wik_chick88 View Post
    hi was just wondering how do u calculate the symmetries of an octahedron? i need to work out an enumeration problem, but i need to know how many distinct octahedra there are possible (all of the same size). they have 6 vertices, 12 edges and 8 triangular faces.
    It depends what sort of transformations you are allowing. As explained by ThePerfectHacker, the full symmetry group of a regular octahedron has order 48. But only 24 of these can be physically realised by rotations. The other 24 are mirror images of the original octahedron.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Apr 2008
    Posts
    204
    ok well i need to figure out how many distinct octahedron i get if i can colour each face either red, blue or green. this is what i worked out. you said that there are 24 PHYSICAL rotations, not including mirror images. there are (8 + 3 - 1) choose 3 = 10 choose 3 = 120 different combinations of 3 colours on the eight faces. does this mean that there are 120/24 = 5 distinct octahedra with coloured faces? that cant be right, its not nearly enough. im very confused...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Symmetries question
    Posted in the Geometry Forum
    Replies: 2
    Last Post: July 9th 2011, 04:05 AM
  2. Vertices and Symmetries
    Posted in the Geometry Forum
    Replies: 2
    Last Post: June 16th 2011, 09:48 AM
  3. Symmetries
    Posted in the Geometry Forum
    Replies: 8
    Last Post: June 4th 2010, 11:14 AM
  4. Symmetries of cube
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 22nd 2009, 11:50 PM
  5. coloured faces of an octahedron
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: September 18th 2008, 05:51 PM

Search Tags


/mathhelpforum @mathhelpforum