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 typo in section 16.2 
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Joined: 13 Mar 2008, 14:06
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Post typo in section 16.2
Where he's talking about magic circles, he says "every distance 1,2,3,...,q+1 can be uniquely represented as a sum of a cyclically successive collection of the a's"

- it should be 1,2,3,...,q(q+1)

Has this been fixed in the paperback version? I'm using the hardcover.
Laura


21 Mar 2008, 13:18
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Joined: 19 Mar 2008, 14:09
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Post Re: typo in section 16.2
fallingup wrote:
Where he's talking about magic circles, he says "every distance 1,2,3,...,q+1 can be uniquely represented as a sum of a cyclically successive collection of the a's"

- it should be 1,2,3,...,q(q+1)

Has this been fixed in the paperback version? I'm using the hardcover.
Laura


It's present in the paperback version.


24 Mar 2008, 18:58

Joined: 13 Mar 2008, 14:06
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Location: Ithaca NY
Post Re: typo in section 16.2
I got a nice note saying he'd looked at it and thanks for pointing out the typo. I suppose it'll appear on the roadsolutions website, and then it can be deleted from here - this forum being only needed as a kind of incubation area for corrections.
Larua


28 Mar 2008, 15:46

Joined: 19 Mar 2008, 13:44
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Location: Beirut, Lebanon
Post Re: typo in section 16.2
Do you think we need to report trivial typos as well, like spelling mistakes etc..?If so, there are some in the paperback edition...


29 Mar 2008, 06:26
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Post Re: typo in section 16.2
There shouldn't be any harm in reporting trivial errors such as spelling, but make sure its not already known about in the official list of errors.


29 Mar 2008, 11:08

Joined: 19 Mar 2008, 13:44
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Post Re: typo in section 16.2
Where can one find this "official list of errors"?


30 Mar 2008, 15:20
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Post Re: typo in section 16.2
Lemaitre wrote:
Where can one find this "official list of errors"?

Here:

http://www.roadsolutions.ox.ac.uk/


30 Mar 2008, 20:55

Joined: 19 Mar 2008, 13:44
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Location: Beirut, Lebanon
Post Re: typo in section 16.2
Thanks Kurdt!I have found some more typos in the paperback edition, and I will list them here shortly...
Now to come back to 16.2 and magic circles, can anyone(Laura?) tell me how to represent the distance 1, say, as a "sum of of the cyclically successive collection of the a's"? Unless you take its complement, i.e q(q+1)-1...in which case it is not "every distance" that could be represented in this manner.


31 Mar 2008, 05:04

Joined: 13 Mar 2008, 14:06
Posts: 42
Location: Ithaca NY
Post Re: typo in section 16.2
Lemaitre wrote:
Now to come back to 16.2 and magic circles, can anyone(Laura?) tell me how to represent the distance 1, say, as a "sum of of the cyclically successive collection of the a's"? Unless you take its complement, i.e q(q+1)-1...in which case it is not "every distance" that could be represented in this manner.

What he means by "cyclically successive" is that you start at one of the a's
and add one or more of the rest in sequence, going around the circle, but not all the way around. So in the case q=2, you have 1, 2, 1+2, 4, 4+1, 2+4.
An equivalent way of saying it is that you can choose the a's so that there are no repeated cyclically successive sums of a's. The number of possible strings of a's you can add together is q(q+1), so if there are no repeated cyclically successive sums, by the pigeonhole principle every number from 1 to q(q+1) is a cyclically successive sum. I bet the proof of the theorem uses the pigeonhole principle.
His language is kind of compressed, sometimes hard to understand.
I don't worry about minor typos, just things that interfere with understanding.
Laura


01 Apr 2008, 01:21

Joined: 19 Mar 2008, 13:44
Posts: 9
Location: Beirut, Lebanon
Post Re: typo in section 16.2
Thanks Laura!Your explanation is very helpful.
By the way, did you finish reading the book? I am still in Chapter 16, and I think this is a wonderful book, which contains all one needs to know to understand "Reality". However, many topics seem to be kind of a "bonus", i.e without any application to physics, like for example this story of the magic circles. On the other hand, other important issues are totally ignored, like, e.g., the CMB anisotropies and the study of the power spectrum...What do you think?


01 Apr 2008, 05:27

Joined: 13 Mar 2008, 14:06
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Post Re: typo in section 16.2
Lemaitre,
There are q+1 lines in P^2 (F_q). So what the "magic circles" show is that any two points in projective space define a unique line. Just like in Euclidean space.
I wonder if you could make a magic circle for RP^2. Or CP^2.
The "magic circle" defines a metric between the lines in the projective space. In P^2 (F_q) there are the same number of lines as points, that's why you can do this. The space of lines in P^2 (F_q) is the same thing as the Grassmanian of planes in F_q^3. The dimension of Gr_k(F^n), the Grassmanian of k-dimensional subspaces of F^n (F is any field), is k(n-k). So in this case the dimension of Gr_2(F^3) is 2, same dimension as FP^2. Gr_k(F^n) is naturally isomorphic to Gr_{n-k}(F^n), so Gr_2(F^3) is isomorphic to Gr_1(F^3) = FP^2.
If F is a finite field, it's of order q=p^a for some prime p. So F_q^3 has a field structure, if you think of it as the field of order p^{3a}. The nonzero elements of F_q^3 are a cyclic group. So P^2(F_q) also has a cyclic group structure, you would just factor out the group of non-0 elements of F_q^3 by the group of non-0 elements of F_q. Just find a generator of the cyclic group of elements of P^2(F_q), find the elements corresponding to lines going through a single point in P^2(F_q), and you would have the magic circle. So for finite fields you can squash the 2-dimensional P^2(F_q) onto a 1-dimensional circle.
There's a natural topology on Gr_2(R^3) and Gr_2(C^3). But that's a 2-dimensional topology. I don't know if you could squash Gr_2(R^3) onto a "magic circle". maybe by taking some kind of limit of what happens for finite fields. I don't think it could be a continuous map because of homology.
Time to go back to reading the book!
Laura


Last edited by fallingup on 03 Apr 2008, 00:51, edited 6 times in total.

01 Apr 2008, 18:40

Joined: 19 Mar 2008, 13:44
Posts: 9
Location: Beirut, Lebanon
Post Re: typo in section 16.2
Well, I will give it a try,Laura! But still, you did not answer my question: what is its application to "reality", i.e to physics??I am fairly well versed in GR, but I have never encountered "magic circles" there...Quantum field theory , maybe?


02 Apr 2008, 11:08
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