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Topic: Multiplication of a Rotation and a Reflection (Read 897 times) |
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Whiskey Tango Foxtrot
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Multiplication of a Rotation and a Reflection
« on: Nov 30th, 2006, 9:32am » |
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The multiplication of a rotation, R, about the z-axis and through the angle pi and a reflection in the xy-plane yields a group. Show this group to have four elements, then write it as a direct product.
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"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
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Icarus
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Re: Multiplication of a Rotation and a Reflection
« Reply #1 on: Nov 30th, 2006, 7:52pm » |
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The first, A, carries (x,y,z) to (-x, -y, z). The second, B, carries (x,y,z) to (x, y, -z). Clearly A2 = I, B2 = I, and AB = BA. Hence |<A>| = 2, |<B>| = 2, and the full group is <A> x <B>, and |<A> x <B>| = 4.
I suppose you could say that I didn't solve it, since I showed the direct product first, and not the 4 elements... 
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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Whiskey Tango Foxtrot
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Re: Multiplication of a Rotation and a Reflection
« Reply #2 on: Nov 30th, 2006, 10:38pm » |
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It's fine by me and I'm the one who posted the riddle. I guess that means you win. It's nice to see some other people are interested in Group Theory. Now that I've been studying it, I don't understand why I wasn't taught it earlier. It should be right up there with algebra with respect to its universal applications. At least that's how I see it.
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"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
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SMQ
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Re: Multiplication of a Rotation and a Reflection
« Reply #3 on: Dec 1st, 2006, 5:44am » |
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on Nov 30th, 2006, 10:38pm, Whiskey Tango Foxtrot wrote:| I guess that means you win. It's nice to see some other people are interested in Group Theory. |
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I'm interested, but despite my undergrad math minor I never learned any in school, and since it has exactly zero applicability to my actual for-pay job writing real estate software, I have to pick it up on my own time... any recommendations for good introductory material -- preferably online?
--SMQ
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--SMQ
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Whiskey Tango Foxtrot
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Re: Multiplication of a Rotation and a Reflection
« Reply #4 on: Dec 1st, 2006, 8:51am » |
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http://www.math.niu.edu/~rusin/known-math/index/20-XX.html
A decent introduction to the ideas behind Group Theory.
http://web.usna.navy.mil/~wdj/tonybook/gpthry/node1.html
Pretty deep and well thought out. The pages are a little too dense for me, but it is still manageable.
http://www.nbi.dk/GroupTheory/
Not too great but it's free and it covers some of the commonly used applications.
If you really want to understand Group Theory, pick up an old textbook. They're relatively cheap. If you have any kind of interest in physics, I can highly recommend "Symmetry in Physics" by Elliott and Dawber. If you still have access to a college library I'm sure they have several very good books.
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| « Last Edit: Dec 1st, 2006, 8:52am by Whiskey Tango Foxtrot » |
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"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
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Barukh
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Re: Multiplication of a Rotation and a Reflection
« Reply #5 on: Dec 1st, 2006, 9:01am » |
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on Dec 1st, 2006, 5:44am, SMQ wrote:| any recommendations for good introductory material -- preferably online?--SMQ |
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I would suggest you find the following books:
Herstein "Topcis on Algebra" (ch. 2)
Fraleigh "A First Course in Abstract Algebra" (ch. 1-2).
For the geometric interpretation of groups, Coxeter's "Regular Polytopes" is an excellent source.
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Whiskey Tango Foxtrot
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Re: Multiplication of a Rotation and a Reflection
« Reply #6 on: Dec 1st, 2006, 9:10am » |
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Herstein can be hard to find. My friend came across a used copy a while ago. I read portions of it, most of chapters 1 and 2 I think, and thoroughly enjoyed it.
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"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
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Barukh
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Re: Multiplication of a Rotation and a Reflection
« Reply #7 on: Dec 1st, 2006, 10:03am » |
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To the question of WTF: Consider the following degenerated polygon: it has 2 vertices, connected by two distinct (curved) sides (it is called 2-gon). Put it in the 3D-space so that its vertices lie on z-axis. Then, operation A interchanges only sides, operation B interchanges only vertices, and AB = BA interchange both vertices and sides. Thus, we get the full symmetry group of a 2-gon, which is of course dihedral group D2 = C2 x C2.
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Sameer
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Pie = pi * e
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Re: Multiplication of a Rotation and a Reflection
« Reply #8 on: Dec 9th, 2006, 2:04am » |
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on Dec 1st, 2006, 9:10am, Whiskey Tango Foxtrot wrote:| Herstein can be hard to find. My friend came across a used copy a while ago. I read portions of it, most of chapters 1 and 2 I think, and thoroughly enjoyed it. |
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You can find it on amazon ... btw i do remember this book because earlier I had asked the same question and Icarus suggested this book... I got it and it is pretty awesome.. even though mathematics is not my primary field (am an EE  ) i do really like math and this is one good book... Farleigh is expensive... i think I saw it for 150 bucks.. thats expensive for my pass reading 
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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Icarus
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Re: Multiplication of a Rotation and a Reflection
« Reply #9 on: Dec 9th, 2006, 7:03am » |
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on Dec 9th, 2006, 2:04am, Sameer wrote:| btw i do remember this book because earlier I had asked the same question and Icarus suggested this book... |
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Not me. I've never read it.
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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Sameer
Uberpuzzler
Pie = pi * e
Gender:
Posts: 1261
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Re: Multiplication of a Rotation and a Reflection
« Reply #10 on: Dec 9th, 2006, 12:11pm » |
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on Dec 9th, 2006, 7:03am, Icarus wrote:
Not me. I've never read it.
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Whoops it was a long time ago.. but somebody definitely suggested it.. so thank you whoever did!! 
Edit: Apparently it was Barukh the same ol' person who suggested it  .. i found the post where I asked the question and got the response
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_eas y;action=display;num=1078017611;start=
Icarus you did suggest the book on number theory though, which is indeed awesome...
Of course I read through the post and saw my newbie ramblings in there without contributing anything to the posted problem
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| « Last Edit: Dec 9th, 2006, 12:18pm by Sameer » |
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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ThudnBlunder
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Re: Multiplication of a Rotation and a Reflection
« Reply #11 on: Dec 9th, 2006, 12:22pm » |
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Barukh recommended it again (to SMQ) in this thread only 8 days ago.
I have quite a few Algebra ebooks that I am willing to share if anybody is interested.
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| « Last Edit: Dec 9th, 2006, 1:27pm by ThudnBlunder » |
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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Whiskey Tango Foxtrot
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Sorry Goose, it's time to buzz a tower.
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Re: Multiplication of a Rotation and a Reflection
« Reply #12 on: Dec 10th, 2006, 11:55am » |
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I'd love an algebra ebook.
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"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
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Sameer
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Pie = pi * e
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Re: Multiplication of a Rotation and a Reflection
« Reply #13 on: Dec 10th, 2006, 12:57pm » |
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Yes me too...
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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