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Possible error found in Keith Devlin's book The Math Gene Help?

  1. Apr 21, 2013 #1
    Possible error found in Keith Devlin's book "The Math Gene" Help?

    1. The problem statement, all variables and given/known data
    In the section "What is mathematics?" of Keith Devlin's book The Math Gene, he explains to the layman the purpose of abstractions. Devlin uses an example from group theory of symmetry to explain a point. In the example, there's an equilateral triangle with vertices Z,X,and Y from left to right respectively.
    This equilateral triangle can be transformed in 6 different ways and still maintain its symmetry.
    v = by being rotated 120°
    w = by being rotated 240°
    I = no change made to equilateral triangle
    x = by being reflected on line x, a 90° line*
    y = by being reflected on line y, a 330° line*
    z = by being reflected on line z, a 210° line*
    * I'm referring to angles on a cartesian plane (so where up is 90°, left 180°, down 270°, right 360°/0°)

    Devlin then states that you can get to one of the 6 transformations by "combining" two or more of the transformations.
    The first equation he writes explaining this concept is x ○ v = y
    2. Relevant equations
    "G1. For all x,y,z in G, (x * y) * z = x * (y * z)
    G2. There is an element e in G such that x * e = e * x = x for all x in G. (e is called an identity element.)
    G3. For each element x in G there is an element y in G such that x * y = y * x = e, where e is as in condition G2."
    All relevant equations and the copy of the page in the book "Math Gene" can be found here:
    http://bit.ly/13pzUHa
    Start with the paragraph that states "Consider, for example, a symmetry group..." on pg.108 and continue to pg.109

    3. The attempt at a solution
    The problem is that this equation doesn't make since to me after I checked it with the operations. I don't know if Devlin made a typo or if I'm doing something wrong.

    So I start with the equilateral triangle (z, x, y).
    Then I perform the 'x' transformation (reflection on line x) which switches the y and z getting (y, x, z)
    Then I perform the 'v' transformation (120° rotation) which moves the vertices down one getting (z, y, x).
    Lastly I check to see if this equals a 'y' transformation (reflection on line y) of the equilateral triangle which is (x, z, y).
    Well clearly (x,z,y) and (z,y,x) don't match up, so either I'm doing it wrong or he made a typo.

    Just to make sure I was doing the right thing I tried a different equation.
    x ○ y = v
    I take the equilateral triangle (z,x,y)
    I apply a x transformation (reflection on line x) (y,x,z)
    Next I apply a y transformation (reflection on line y) to the preceding transformation (y,z,x)
    I check to see if this matches a v transformation (rotating triangle 120°) (y,z,x)
    and it does.
    Here's a picture showing my work and the manipulations.
    UKNnlcB.jpg
    So now I'm confused. I'm trying to follow along with him logically, but I don't understand what happened here. (Maybe incorrect interpretation of the symbol '○' ?) If this is a mistake on his part, I'll begin to lose his respect. It could possibly be an error of my own. In any case I'm trying to LEARN.
    So does someone mind checking the validity of aforementioned equation?
    BTW: As for my math background, I'm a high school senior and I'm working (struggling...) through AP Calc BC this year--good grasp of the concepts, poor arithmetic skills :P I blame my crappy Pre-Cal teacher.
     
  2. jcsd
  3. Apr 21, 2013 #2
    Does the book say the rotations should be clockwise? It's not unusual for rotations to be specified counter clockwise in math and science. I found both x ○ v = y and x ○ y = v true assuming v and w are counter-clockwise rotations.
     
  4. Apr 21, 2013 #3
    Thanks for commenting.
    But Devlin states that v and w are indeed clockwise rotations of 120° and 240° respectively since rotating an equilateral triangle counterclockwise 120° is the same as rotating it clockwise 240° and vice versa. By the way, how did you solve x ○ y = v assuming that the rotation transformations are counterclockwise?
     
  5. Apr 21, 2013 #4
    I think I'm beginning to see your point. If you perform the second operation first then solve and use counterclockwise rotations they work. I wish he'd made that clearer.
     
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