Results 1 to 8 of 8

Math Help - symmetry groups of the regular solids

  1. #1
    Junior Member
    Joined
    Oct 2007
    Posts
    32

    symmetry groups of the regular solids

    The two paragraphs just explain the problem. You are supposed to prove that

    The group of rotational symmetries of a tetrahedron is
    A(4).


    The full group of symmetries of a tetrahedron is
    S(4).

    The group of rotational symmetries of a cube or of an octahedron
    is
    S(4).

    The group of rotational symmetries of an icosahedron or of a dodecahedron is
    A(5).


    based on:


    A regular solid is a 3-
    dimensional polyhedron in which each face is a regular polygon. Any two
    faces as well as any two vertices can be matched by an isometry (a rigid
    motion) of the 3-dimensional space. It is convenient to use a symbol
    {p, q}

    for a regular solid whose faces are regular
    p-gons with q of them situated
    around each vertex. We have proved that there are FIVE regular solids:

    TETRAHEDRON {3, 3},

    CUBE {4, 3},

    OCTAHEDRON {3, 4},

    DODECAHEDRON {5, 3},

    ICOSAHEDRON {3, 5}.

    Any regular solid may be inscribed in a sphere, and then any symmetry of
    any regular solid will leave the center of the sphere fixed and will transform
    the surface of the sphere onto itself. We call a rotational symmetry of a
    regular solid any rotation of the sphere (with respect to an axis passing
    through its center) mapping the regular solid into inself. A symmetry of a
    regular solid is either a rotational symmetry or a reflection with respect to
    a plane (also passing through the center of the sphere), mapping the regular
    solid onto itself.

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Oct 2007
    Posts
    32
    ...................
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,614
    Thanks
    1581
    Awards
    1
    Quote Originally Posted by anncar View Post
    ...................
    Is that a BUMP?
    Whatever that is.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2007
    Posts
    32
    nope. i posted something and decided I wanted to delete it. so i just put dots in place of it.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by anncar View Post
    The group of rotational symmetries of a tetrahedron is
    A[FONT=CMR10][SIZE=3](4).

    [LEFT]The full group of symmetries of a tetrahedron is
    S[FONT=CMR10][SIZE=3](4).
    Look at the picture below and note all the ways you can rotate it. So you can have an identity rotation represented by the cycle (1) you can swap 1 and 4 leaving 2 and 3 unchanged that is (1,4). You can swap 2 and 3 leaving 1 and 4 unchanged that is (2,3). And so on ... You will get a group table.
    Attached Thumbnails Attached Thumbnails symmetry groups of the regular solids-picture.gif  
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    For the cube, look at the four long diagonals connecting a vertex to the opposite vertex on the opposite face. You can get any permutation of this set of four diagonals by a suitable rotation.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    I am too lazy to check this but is it in general true that the group of rotations on a solid is an alternating group?

    And if the above question is true is there any nice geometric signifigance to that, i.e. what happens when n\geq 5 and the group is not solvable? Is this any way related to the 5 Platonic solids?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by ThePerfectHacker View Post
    I am too lazy to check this but is it in general true that the group of rotations on a solid is an alternating group?
    This is not true for the cube or the octahedron, both of which have the full symmetric group S_4 as their symmetry group. There is a good explanation here.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: December 20th 2011, 05:37 PM
  2. Replies: 1
    Last Post: September 27th 2011, 02:42 PM
  3. Symmetry groups are indecomposible
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 12th 2009, 12:22 AM
  4. A subgroup of symmetry groups
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 9th 2008, 03:10 PM
  5. Abstract Algebra: Groups and Symmetry
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 11th 2008, 05:43 AM

Search Tags


/mathhelpforum @mathhelpforum