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Rotations - why are they not vectors
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Tim Golden
science forum Guru Wannabe


Joined: 12 May 2005
Posts: 176

PostPosted: Tue Jul 18, 2006 5:48 pm    Post subject: Re: Rotations - why are they not vectors Reply with quote

Terry Padden wrote:
Quote:
"ben" <benedict.williams@gmail.com> 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. 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
readily available -

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.

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
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Tue Jul 18, 2006 12:25 pm    Post subject: Re: Rotations - why are they not vectors Reply with quote

"ben" <benedict.williams@gmail.com> wrote in message
news:1153124461.756415.119060@s13g2000cwa.googlegroups.com...
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
readily available -

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.
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ben1
science forum beginner


Joined: 16 May 2006
Posts: 9

PostPosted: Tue Jul 18, 2006 5:48 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

mariano.suarezalvarez@gmail.com wrote:
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
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Tim Golden
science forum Guru Wannabe


Joined: 12 May 2005
Posts: 176

PostPosted: Mon Jul 17, 2006 2:58 pm    Post subject: Re: Rotations - why are they not vectors Reply with quote

Terry Padden wrote:
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
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mariano.suarezalvarez@gma
science forum addict


Joined: 28 Apr 2006
Posts: 58

PostPosted: Mon Jul 17, 2006 2:56 pm    Post subject: Re: Rotations - why are they not vectors Reply with quote

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
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 11:14 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"Terry Padden" <TPadden@bigpond.net.au> wrote in message
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.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 11:05 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"Terry Padden" <TPadden@bigpond.net.au> wrote in message
news:83Kug.6614$tE5.874@news-server.bigpond.net.au...
Quote:

"ben" <benedict.williams@gmail.com> 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 ?
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 10:55 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

<jw12jw12jw12@yahoo.com> wrote in message
news:1153110681.231177.132530@m73g2000cwd.googlegroups.com...
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.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 10:55 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"ben" <benedict.williams@gmail.com> wrote in message
news:1153124952.700118.157220@s13g2000cwa.googlegroups.com...
..
Quote:

I thought so; but hm ? So 2 x 1-D rotations is a vector space and 1-D
rotations are commutative. Where then from the axioms does the
non-commutaivity of 2-D rotations come from ? Puzzled I am.

I have tidied up your top-post


Quote:
The set of rotations of the plane form a group (although not a
vector-space), which we can denote O(1). Pairs of these form a group
O(1)xO(1). This group is not isomorphic to O(2); which is a fancy way
of stating a pair of 1D rotations is not the same as a 2D rotation. The
easiest way to see this is the case, however, is to note that one is a
commutative group and the other is not.

From my perspective this is cheating. We are not supposed to be discussing
groups - or any kind of Euclidean space.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 10:55 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"Stephen Montgomery-Smith" <stephen@math.missouri.edu> wrote in message
news:CaCug.1084014$xm3.732932@attbi_s21...
Quote:
I am reminded of a story told to me by my driving instructor - blather

My reply to you was quite civil - but corrected your errors. It seems you
wish to change the subject to ad hominem junk. I understand the psychology
of your response but so what. You may not approve of my methods of
improving the average mathematical competence of this newsgroup but it
works. When this thread is complete you will all have a clearer
understanding of vectors & rotations - thanks to me. No need for gratitude,
I see it as a work of charity.

One person seems to be having no problem providing reasonable responses to a
reasonable question. I am truly grateful, but not a changed person.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 10:54 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"ben" <benedict.williams@gmail.com> wrote in message
news:1153124461.756415.119060@s13g2000cwa.googlegroups.com...
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


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.
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ben1
science forum beginner


Joined: 16 May 2006
Posts: 9

PostPosted: Mon Jul 17, 2006 8:29 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

The set of rotations of the plane form a group (although not a
vector-space), which we can denote O(1). Pairs of these form a group
O(1)xO(1). This group is not isomorphic to O(2); which is a fancy way
of stating a pair of 1D rotations is not the same as a 2D rotation. The
easiest way to see this is the case, however, is to note that one is a
commutative group and the other is not.

Quote:
I thought so; but hm ? So 2 x 1-D rotations is a vector space and 1-D
rotations are commutative. Where then from the axioms does the
non-commutaivity of 2-D rotations come from ? Puzzled I am.
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ben1
science forum beginner


Joined: 16 May 2006
Posts: 9

PostPosted: Mon Jul 17, 2006 8:21 am    Post subject: Re: Rotations - why are they not vectors Reply with 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
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mariano.suarezalvarez@gma
science forum addict


Joined: 28 Apr 2006
Posts: 58

PostPosted: Mon Jul 17, 2006 5:34 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

jw12jw12jw12@yahoo.com wrote:
Quote:
Given the previous posts I'm hesitant about adding anything, but here
goes:

Heh.

Quote:
1-D rotations ARE a vector space...this space is isomorphic to R^1.For
example, one of the axioms of a vector space is: u+v=v+u , and in
the case of 1-D rotations a rotation of (for example) 30 degrees
followed by a rotation of 40 degrees is equivalent to a rotation of 40
degrees follwed by a rotation of 30 degrees. I would agree that this
is the obvious interpretyation of addition in this this case (although
others are possible). All other axioms woukd be satisfied in this
interpretation.

As I said before, making sense of a statement such as
"1-D rotations ARE a vector space" requires that one at the
very least decide what are "1-D rotations". In particular, one
needs to be able to determine when two "1-D rotations" are
the same "1-D rotation".

Are the "1-D rotations" of 360 degrees and of 0 degrees the
same rotation? If yes, then your attempt at regarding the set
of 1-D rotations as a vector space is doomed to fail. If no,
well, then the notion of 1-D rotation would be more usefully
called "oriented angle", and yes, oriented angles can be seen
as a vector space in a natural way.

-- m
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jw12jw12jw12@yahoo.com
science forum beginner


Joined: 28 Oct 2005
Posts: 16

PostPosted: Mon Jul 17, 2006 4:31 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

Terry Padden wrote:
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.

Given the previous posts I'm hesitant about adding anything, but here
goes:

1-D rotations ARE a vector space...this space is isomorphic to R^1.For
example, one of the axioms of a vector space is: u+v=v+u , and in
the case of 1-D rotations a rotation of (for example) 30 degrees
followed by a rotation of 40 degrees is equivalent to a rotation of 40
degrees follwed by a rotation of 30 degrees. I would agree that this
is the obvious interpretyation of addition in this this case (although
others are possible). All other axioms woukd be satisfied in this
interpretation.

As you point out u+v=v+u is not valid for general rotations in R^3 and
this is a reason why rotations in R^3 do not form a vector space.

I think the concept of a vector is a bit trickier than many people
realize. I've met math teachers who were unable to answer the following
questions, or incorrectly.
1. Let AB be the trip from Albany to Buffalo, let BC be the trip from
Buffalo to Chicago so AB+BC=AC in the sense that you have a trip from
Albany to Chicago. Is AB a vector?
Explain.

2. Are forces free vectors or a bound vectors?

3. Angular displacements in R^3 (i.e. rotations) are not vectors, so
why are angular veclocities in R^3 vectors?

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
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