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science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Mon Jul 17, 2006 10:55 am    Post subject: Re: Rotations - why are they not vectors

<jw12jw12jw12@yahoo.com> wrote in message
 Quote: There is a nice, inexpensive little book by physicist Banesh Hoffman called "About Vectors" published by Dover (that I hope is still in print) which you might find interesting. jw

I have had a copy for a long time. I like it very much but it is too easy
for real mathematicians - most of whom as evidenced by my respondents don't
really know what a vector is.
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Mon Jul 17, 2006 11:05 am    Post subject: Re: Rotations - why are they not vectors

news:83Kug.6614\$tE5.874@news-server.bigpond.net.au...
 Quote: "ben" wrote in message news:1153124461.756415.119060@s13g2000cwa.googlegroups.com... Rotations (in Euclidean space) do not form a vector space over the reals because of the failure of the consistency axiom. Ben Thank you. I'll try to make sure I understand what you say. Can you provide an accessible reference text for the Consistency Axiom as basic as possible. I don't recall it being mentioned in any of the basic texts that I skim thru - I'll see if it is in Wikipedia.

Please note that I consider the Euclidean Space qualification to be invalid.

Am I correct in thinking that you have corrected your earlier view that 1-D
rotations ARE vectors ?
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Mon Jul 17, 2006 11:14 am    Post subject: Re: Rotations - why are they not vectors

news:9cKug.6626\$tE5.126@news-server.bigpond.net.au...
 Quote: Am I correct in thinking that you have corrected your earlier view that 1-D rotations ARE vectors ?

Please ignore this comment. My apolgies, being in my dotage I seem to have
confused your response with an earlier one from Jurgen.
mariano.suarezalvarez@gma

Joined: 28 Apr 2006
Posts: 58

Posted: Mon Jul 17, 2006 2:56 pm    Post subject: Re: Rotations - why are they not vectors

ben wrote:
 Quote: Rotations (in Euclidean space) do not form a vector space over the reals because of the failure of the consistency axiom. If r,s are real numbers and T is a rotation, then it is not necessarily the case that r(sT) = (rs) T For instance, let T denote rotation through one quarter-circle (90 degrees or pi/2 radians). Then 4T= Id, the identity rotation 1/4 ( 4T) = Id which is not the same as (1/4 * 4) T = T

What you call the "consistency axiom" is what most texts and most
mathematicans call the associativity.

-- m
Tim Golden
science forum Guru Wannabe

Joined: 12 May 2005
Posts: 176

Posted: Mon Jul 17, 2006 2:58 pm    Post subject: Re: Rotations - why are they not vectors

 Quote: I am bothered by the mathematics of rotations. It is I believe mathematically acceptable for any physical reality to be defined on an abstract axiomatic basis. Then anything that fulfills a given defining set of axioms for a type of mathematical object is a mathematically valid example of the defined mathematical object. Now consider simple (= 1-D) rotations of a spherical object about any given fixed axis. Superficially, to me (not a mathematician), such "angular displacements" meet all of the formal axioms for a Vector Space (as given in e.g. Halmos) as well as 1-D linear displacements do. Could someone point out to me in what way such 1-D rotations do NOT meet the axiomatic criteria for a Vector Space. If 1-D rotations are axiomatically vectors, why cannot they be axiomatically compounded into multi-dimensional vector spaces ? NB I am aware that 2-D rotations do-not-commute, but it seems to me that that has nothing to do with axiomatics or my questions. I am not suggesting that rotations ought to be physically vectors. I am just trying to get clarification of the math picture for vectors.

Neat question. I like rotation and think there is some underlying value
in its study.
I'm seeing 1D rotation as inversion. There is only a binary choice for
a 1D rotation.
A choice of -1 flips the handedness of a structure about that chosen
axis.
A choice of +1 preserves the structure identically.
Any change in magnitude is merely a scaling factor.
We don't generally worry about 1D rotation. I suppose the important
part of it is that if you do try the continuous rotation in 1D that
scaling becomes somewhat relevant. The image shrinks down to a point
before coming groeing out in it's inverted form. Raising this notion to
2D rotation allows for scaling within the concept which is not normally
considered.

Perhaps I am confusing your notion of rotation. Perhaps you are using
the standard 2D without scaling and so hence a 1D angle, but calling
that a 1D rotation I believe is a mistake.

-Tim
ben1
science forum beginner

Joined: 16 May 2006
Posts: 9

Posted: Tue Jul 18, 2006 5:48 am    Post subject: Re: Rotations - why are they not vectors

 Quote: ben wrote: Rotations (in Euclidean space) do not form a vector space over the reals because of the failure of the consistency axiom. If r,s are real numbers and T is a rotation, then it is not necessarily the case that r(sT) = (rs) T For instance, let T denote rotation through one quarter-circle (90 degrees or pi/2 radians). Then 4T= Id, the identity rotation 1/4 ( 4T) = Id which is not the same as (1/4 * 4) T = T What you call the "consistency axiom" is what most texts and most mathematicans call the associativity. -- m

I see wikipedia refers to it that way. On paper, it looks like
associativity, but I don't think it's called that properly. I heard the
name "consistency axiom" in a course on Modules & Rings. The reason I
don't want to refer to this as associativity is that there are two
operations in question. One is a "scalar multiplication" from
(Field)x(Set) -> Set and the other "field multiplication" from
(Field)x(Field) -> (Field), and the "associativity" axiom is really
asserting the consistency of the two operations.

None of Lang, Dummit & Foote or Lam's "First Course in Noncommutative
Rings" names the consistency/associativity axiom. I note, though, that
wikipedia's definition of associativity is at odds with the article on
vector spaces.

Ben
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Tue Jul 18, 2006 12:25 pm    Post subject: Re: Rotations - why are they not vectors

"ben" <benedict.williams@gmail.com> wrote in message
 Quote: Rotations (in Euclidean space) do not form a vector space over the reals because of the failure of the consistency axiom. If r,s are real numbers and T is a rotation, then it is not necessarily the case that r(sT) = (rs) T For instance, let T denote rotation through one quarter-circle (90 degrees or pi/2 radians). Then 4T= Id, the identity rotation 1/4 ( 4T) = Id which is not the same as (1/4 * 4) T = T Ben

Now that I've had some time to digest this, I have to tell you your argument
is wrong. I have read your later comment on the Wikipedia stuff but that
is irrelevant.

Also by referring to Id's you are sneaking in non-vector stuff. The special
vector is the Zero (or Null) vector - not the Id

Also for rotations there is no rational place for values scaled by "pi". I
complete rotation has the value = 1.

Here is why your argument is wrong;

1. The Zero vector is defined as the algebraic sum of any PAIR of equal but
opposite vectors i.e. 0 = +x + -x

2. For rotations every vector is multi-valued. If something is
multi-valued it does NOT mean that all the values are equal; it means all
the values are different.

3. So - I know this may come as a great shock after all your professors
have told you, so I hope you are sitting down with a strong restorative

One complete rotation, 1 Turn (value = 1) is NOT the same as no rotation
(value =0).

i.e. your statement that when T = 1/4 of 1 Turn, 4T = Id = 0 is incorrect.
So your argument from consistency or associavity or whatever fails.

4. Moreover after 1 complete Turn we do not have a PAIR of opposing values
so again we cannot by definition/axiom have 0, the Zero vector after 1 Turn.
Again the argument fails.

You may be aware of things called winding numbers. They are needed
precisely because of this situation; so that differences in the number of
Turns can be recognised. The differences are real; e.g. Midnight on
Wednesday is not the same as Midnight on Thursday.

So your argument - based on the axioms- fails. So we still don't now
whether rotations are vectors or not.
Tim Golden
science forum Guru Wannabe

Joined: 12 May 2005
Posts: 176

Posted: Tue Jul 18, 2006 5:48 pm    Post subject: Re: Rotations - why are they not vectors

These 'winding numbers' can be applied to polysign numbers:
http://bandtechnology.com/PolySigned/PolySigned.html
But I do not do anything with them. Instead they disappear via modulo
math.
I believe that the modulo math is correct, but when a product is taken
perhaps the net value could be accounted for. Is there a physical
meaning that you would assign? In terms of string you could have an
unwinding almost as an independent entity since the usual math just
throws it away. At a macro level we could look at the solar system and
ask if its winding is influential. Could magnetic winding and unwinding
explain the discrepancy of the planetary magnetic moments? They are
claimed to oscillate.
Perhaps electron spin is a more appropriate thing to worry about.
There is a funny 2pi/sign relation to do with electron spin.
There seems to be an important distinction between continuous and
discrete rotation.
If we rotate a globe on its axis half way around we cannot inherently
say which way it went unless the rotation was continuous. The very word
'rotation' implies continuous I think, but the polysign construction
allows discrete rotations. This stuff seems pretty suggestive.

-Tim
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Wed Jul 19, 2006 12:25 am    Post subject: Re: Rotations - why are they not vectors

"Timothy Golden BandTechnology.com" <tttpppggg@yahoo.com> wrote in message
 Quote: These 'winding numbers' can be applied to polysign numbers: http://bandtechnology.com/PolySigned/PolySigned.html But I do not do anything with them. Instead they disappear via modulo math. I believe that the modulo math is correct, but when a product is taken perhaps the net value could be accounted for. Is there a physical meaning that you would assign? In terms of string you could have an unwinding almost as an independent entity since the usual math just throws it away. At a macro level we could look at the solar system and ask if its winding is influential. Could magnetic winding and unwinding explain the discrepancy of the planetary magnetic moments? They are claimed to oscillate. Perhaps electron spin is a more appropriate thing to worry about. There is a funny 2pi/sign relation to do with electron spin. There seems to be an important distinction between continuous and discrete rotation. If we rotate a globe on its axis half way around we cannot inherently say which way it went unless the rotation was continuous. The very word 'rotation' implies continuous I think, but the polysign construction allows discrete rotations. This stuff seems pretty suggestive. -Tim

Most of what you write is correct - i.e. heading in the right direction -
except for that "Pi" stuff; but remember there are innocent mathematicians
reading this and we don't want to frighten them.
However if you attach the magic phrase "forgetful functors" to what you have
written they will mistake you for one of their senior wizards and let you
pass without attacking you - unlike what they try on me.

For now I am just trying to get them to think about vectors. It is not
easy. They are taught so much rubbish it is hard for them to think clearly.
You ask them a simple, but not easy, question about vector space axioms and
they - with one or two honourable exceptions -automatically talk about
groups, matrices, Rn, Euclidean space, Lie Algebras, "pi", and god knows
what; any old rubbish except considering and responding to the question ;
and of course they think they are really smart mathematicians - god help us
!
Tim Golden
science forum Guru Wannabe

Joined: 12 May 2005
Posts: 176

Posted: Wed Jul 19, 2006 2:34 am    Post subject: Re: Rotations - why are they not vectors

 Quote: Most of what you write is correct - i.e. heading in the right direction - except for that "Pi" stuff; but remember there are innocent mathematicians reading this and we don't want to frighten them. However if you attach the magic phrase "forgetful functors" to what you have written they will mistake you for one of their senior wizards and let you pass without attacking you - unlike what they try on me. For now I am just trying to get them to think about vectors. It is not easy. They are taught so much rubbish it is hard for them to think clearly. You ask them a simple, but not easy, question about vector space axioms and they - with one or two honourable exceptions -automatically talk about groups, matrices, Rn, Euclidean space, Lie Algebras, "pi", and god knows what; any old rubbish except considering and responding to the question ; and of course they think they are really smart mathematicians - god help us !

Rotation seems to be a product relationship. The complex numbers
perform geometrical rotation via product. Your winding concept is
pretty easily expressed in the polar coordinate system but what about
http://www.geom.uiuc.edu/~ross/webtex/webmmlSamples/node2.html
I see that it is mostly used in the context of paths. But should it
apply to the complex plane?

Here is one way to implement winding counters in them. So for example
rather than a+bi lets just look at just the a part.
a i i i i i = a i
would be the ordinary notation. But to implement counting we need to
add a component to the elemental structure like:
( 0, a ) i i i i i = 1, a i
or
( 2, 1.2 )( 3, 2.0 i ) = 5, 2.4 i
and
( 0 , -i )( 0 , -i ) = 1, i .
The question arises wether these are nondecreasing as more products are
taken, or can they unwind? I think they are nondecreasing; they are
coupled to the handedness of the coordinate system. So the discrete
product would produce a discontinuity on the positive real axis, where
a slight change toward -i from a slightly positive i value will cause
an abrupt change, though the usual math is undisturbed.

 Quote: From a physical point of view this may be meaningful since handedness plays a part in physics. If an iterative product raises the winding

count should an iterative division process bottom out? That would imply
disallowing negative winding numbers which might be a coherent choice.

It all seems contrived but might be worth keeping an awareness of these
options.

-Tim
Narcoleptic Insomniac
science forum Guru

Joined: 02 May 2005
Posts: 323

Posted: Wed Jul 19, 2006 3:26 am    Post subject: Re: Rotations - why are they not vectors

On Jul 18, 2006 9:34 PM CT, Timothy Golden wrote:

 Quote: Terry Padden wrote: Most of what you write is correct - i.e. heading in the right direction - except for that "Pi" stuff; but remember there are innocent mathematicians reading this and we don't want to frighten them. However if you attach the magic phrase "forgetful functors" to what you have written they will mistake you for one of their senior wizards and let you pass without attacking you - unlike what they try on me. For now I am just trying to get them to think about vectors. It is not easy. They are taught so much rubbish it is hard for them to think clearly. You ask them a simple, but not easy, question about vector space axioms and they - with one or two honourable exceptions -automatically talk about groups, matrices, Rn, Euclidean space, Lie Algebras, "pi", and god knows what; any old rubbish except considering and responding to the question ; and of course they think they are really smart mathematicians - god help us !

*bites tounge*

 Quote: Rotation seems to be a product relationship. The complex numbers perform geometrical rotation via product.

Indeed they do.

 Quote: Your winding concept is pretty easily expressed in the polar coordinate system but what about for a value a+bi? As I read around about winding number http://www.geom.uiuc.edu/~ross/webtex/webmmlSamples/node2.html I see that it is mostly used in the context of paths. But should it apply to the complex plane? Here is one way to implement winding counters in them. So for example rather than a+bi lets just look at just the a part. a i i i i i = a i would be the ordinary notation. But to implement counting we need to add a component to the elemental structure like: ( 0, a ) i i i i i = 1, a i or ( 2, 1.2 )( 3, 2.0 i ) = 5, 2.4 i and ( 0 , -i )( 0 , -i ) = 1, i .

It seems you are grasping at a way to determine what
branch you're working within after a given set of
rotations.

 Quote: The question arises wether these are nondecreasing as more products are taken, or can they unwind? I think they are nondecreasing; they are coupled to the handedness of the coordinate system. So the discrete product would produce a discontinuity on the positive real axis, where a slight change toward -i from a slightly positive i value will cause an abrupt change, though the usual math is undisturbed.

Again, it seems that you're thinking about moving from
one branch to the next.

 Quote: From a physical point of view this may be meaningful since handedness plays a part in physics. If an iterative product raises the winding count should an iterative division process bottom out? That would imply disallowing negative winding numbers which might be a coherent choice. It all seems contrived but might be worth keeping an awareness of these options. - Tim

Hmmmmm I dunno, one guy who tried to generalize this kind
of geometry of complex numbers ended up inventing the
quaternions...

...and we all know how horrible that turned out to be
(their imaginary parts can describe 3-D rotation and the
whole concept leaked into groups, matrices, and god knows
what).
Narasimham G.L.
science forum Guru Wannabe

Joined: 28 Apr 2005
Posts: 216

Posted: Wed Jul 19, 2006 5:42 am    Post subject: Re: Rotations - why are they not vectors

 Quote: mariano.suarezalvarez@gmail.com> wrote in message news:1153025545.663758.95150@75g2000cwc.googlegroups.com... Terry Padden wrote: I am bothered by the mathematics of rotations. It is I believe mathematically acceptable for any physical reality to be defined on an abstract axiomatic basis. Then anything that fulfills a given defining set of axioms for a type of mathematical object is a mathematically valid example of the defined mathematical object. A set does not make a vector space. You B***** Fool. Who mentioned sets !! ....... The obvious ones you moron. You can ADD rotations; and you can SCALE rotations by a suitable Field. Idiot !! Yes you fool - but that is in 1-D ; and I wrote do-not-commute in 2-D. Can't you even read a simple sentence ??? You may want to review Halmos' presentation of what a vector space is, if you think that commutativity is irrelevant, by the way... NO ! I want you to go AWAY !!!!!!

Your courtesy and manners seem to have undergone disorientation
somewhere along the line.It is never too late to rotate it backwards to
let others join in your quests in better harmony.
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Wed Jul 19, 2006 6:35 am    Post subject: Re: Rotations - why are they not vectors

"Timothy Golden BandTechnology.com" <tttpppggg@yahoo.com> wrote in message
 Quote: Rotation seems to be a product relationship. The complex numbers perform geometrical rotation via product. Your winding concept is pretty easily expressed in the polar coordinate system but what about for a value a+bi? As I read around about winding number http://www.geom.uiuc.edu/~ross/webtex/webmmlSamples/node2.html I see that it is mostly used in the context of paths. But should it apply to the complex plane?

I would reccommend that you discard conventional notions such as complex
numbers & polar co-ordinates as mathematical impositions if you wish to
continue making progress.
ben1
science forum beginner

Joined: 16 May 2006
Posts: 9

Posted: Wed Jul 19, 2006 9:06 am    Post subject: Re: Rotations - why are they not vectors

I'm quite happy with my understanding of rotations, which is that they
are orientation preserving linear isometries of a vector space. If you
want to use a different definition of rotation from me, go ahead.

I think it is very likely that rotations (in your sense of the word)
form a vector space. I would be greatly pleased if you could offer a
rigorous definition of what you understand by "a rotation", as this
might allow me to say for certain.

My guess is that a rotation, for you, is a measurement along with the
implicit understanding of what this rotation might do if you applied it
to something, soam guessing that your definition of rotation (again,
1-dimensional) might looks something like this:

Let f: R -> O(2) be the linear map taking t to the matrix [cos 2pi t,
-sin 2pi t; sin 2pi t, cos 2pi t]. Then a rotation is an element of R x
O(2) of the form (t, f(t)). Addition of rotations is given by
(a,f(a))+(b,f(b)) = (a+b,f(a+b)).

Under this definition, of course, rotations have all the properties you
ascribe to them, and also form a 1-dimensional vector space over the
reals.

A more subtle definition (the matrices are harder to write down) will
work for n-dimensions.

If one omits the map f, and says "a rotation is a measurement of an
angle", then one arrives at a theory of rotations that is essentially
useless, since there is no geometric content.

Ben
ben1
science forum beginner

Joined: 16 May 2006
Posts: 9

 Posted: Wed Jul 19, 2006 9:44 am    Post subject: Re: Rotations - why are they not vectors I said: "A more subtle definition (the matrices are harder to write down) will work for n-dimensions." This is completely false. Please disregard. Ben

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