Moebius Band is not homeomorphic with a Torus
Narasimham
it.
Mathematica formula and images of the unifying set of tori in toroidal
co-ordinate system I have indicated in:
http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg
I made a square section steel ring of m = 2 configuration by
cut/twist/weld and feel that the surface morphology of all these
surfaces has a unifying character by its fiber bundle.Of course the
standard Moebius Band is made by cut/twist/paste of a rectangular sheet
having two sides, small thickness of the sheet is conveniently
overlooked to result in "one side" only! There are actually four sides
in all and two tracks.
By varying m or Aspect Ratio of a rectangular section of a tube (or
solid section) of torus one can morph/animate images among all tori
including the Moebius Band.
I am not comfortable with the Moebius Band singled out topologically
either in terms of orientability or homeomorphability.
What exactly am I missing?
Regards,
Narasimham
wodoro...@gmail.com
Narasimham napisal(a):
There are many ways to show that MB is not homeomorphic with T.
For example: Torus is a surface, Moebius Band is not a surface.
Hox
Rupert
I'd call it a surface. What's your definition of surface?
One distinction is that a Moebius band has boundary, whereas a torus
doesn't.
> Hox
wodoro...@gmail.com
Surface is a 2-dimensional manifold.
In other words, for every point x of space Y, there must exist
a neighborhood (an open set, with point x) in Y,
homeomorphic with E^2 (Euclidian 2-dim space)
> One distinction is that a Moebius band has boundary, whereas a torus
> doesn't.
>
And so, for a point x from boundary of MB, there are
no neighborhood of x in MB,
that could satisfy above condition.
> > Hox
Jyrki Lahtonen
> And so, for a point x from boundary of MB, there are
> no neighborhood of x in MB,
> that could satisfy above condition.
Who said the points on the boundary are part of MB?
The *open* MB is a 2-manifold. It's not compact,
it's not orientable, so there are plenty of ways
of telling it's not homeomorphic (or diffeomorphic
if you prefer that) to the torus, but it is in this
sense a 2-manifold.
Cheers,
Jyrki
wodoro...@gmail.com
The author, Narasimham wrote:
> One distinction is that a Moebius band has boundary, whereas a torus
> doesn't.
I think that words "has boundary" means that it has boundary :-)
and we are talking about *close* MB.
Hox
Peter Webb
"Narasimham" <math...@hotmail.com> wrote in message
news:1161851477.0...@h48g2000cwc.googlegroups.com...
> Moebius Band is not homeomorphic with a Torus or even orientable like
> it.
>
> Mathematica formula and images of the unifying set of tori in toroidal
> co-ordinate system I have indicated in:
>
> http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg
>
> I made a square section steel ring of m = 2 configuration by
> cut/twist/weld and feel that the surface morphology of all these
> surfaces has a unifying character by its fiber bundle.Of course the
> standard Moebius Band is made by cut/twist/paste of a rectangular sheet
> having two sides, small thickness of the sheet is conveniently
> overlooked to result in "one side" only! There are actually four sides
> in all and two tracks.
>
No, a model made from steel has thickness; a Moebius Band has no width,
anymore than does a mathematical line.
Peter Webb
"Rupert" <rupertm...@yahoo.com> wrote in message
news:1161862130....@m73g2000cwd.googlegroups.com...
Not that I disagree with your argument, but can a Mobius Band have infinite
width and therefore be free of boundaries?
Stephen Montgomery-Smith
Let me summarize.
Person A: Is X true X?
Person B: No, X is not true because of Y.
Person C: Y is not true because of Z.
Person D: X is true because of U.
Person B: Person D is wrong because U is incompatible with Z.
Where does B's logic fail?
Hero
> A Moebius Band does not automatically become a 2D surface from a 3D
> solid body of finite thickness. It is self-intersecting after one
> revolution but has a differently directed oriented distinct normal
> vector.
A moebius band is never a solid or has thickness, as it seems from the
paper-strip, just like a line is not material 3D, also this seems to be
from the graphit of a pencil or from the chalk. It is 2D, part of a
plane deformed in 3D.
For a real example of a part of 2D surface, take an air pressure of
let's say 0.5 at. Now look at the air masses above your home place. As
the pressure is diminishing with height with continuity one can
determine this 2D-iso-pressure surface between the volume of air with
more, and the volume with less this pressure, as exact as one likes to.
A moebius band is also never the surface or part of a surface of a
solid, as such a kind of surface separates inner from outer space and
gives so orientation to it's surface: from it to the inside or to the
outside.
With friendly greetings
Hero
Narasimham
Is it so? Consider a simple case. A sheet of paper of say A4 size has
six faces. Four of them (two thickness X length and two thickness X
breadth areas) are so narrow to be negligible in area compared to two
main areas length X breadth of paper that we use for writing upon. But
that does not mean it has zero thickness. If thickness is zero, the
sheet vanishes entirely along with all its features.
In mathematical idealization an irrelevant feature need not be
incorporated, but can be invoked if a need arises. When a needed
feature cannot be invoked, the idealization itself is in jeopardy.
The surface Moebius Band is self-intersecting after one revolution but
has a differently directed oriented distinct surface normal vector.
With finite thickness Moebius Band retains a common homeomorphic
identity with the other members of the thick wall torus set as well as
a unique orientation. To demonstrate this, some radial separation of
tori into individual parts is necessary to be imparted, it is clearly
and separately shown:
http://img351.imageshack.us/img351/8537/moebiustrickds5.jpg
The cases m = 1 and 2 (Moebius Band and square tube) have suffered no
disorientation.
Looks to me as if the Moebius Trick is a grand success! ;-) ...
Narasimham
Proginoskes
Also:
(1) Euler characteristic (X(MB) = -1, X(T) = 0)
(2) Every graph embedded on the MB has a vertex of degree < 6; hence
any such graph can be colored with 6 vertices, no two adjacent vertices
getting the same color. This is not true for T; K7 embeds on the torus.
The MB is equivalent to the projective plane with a disk removed.
--- Christopher Heckman
victor_me...@yahoo.co.uk
> Moebius Band is not homeomorphic with a Torus or even orientable like
> it.
However you define a Mobius strip (with or without a boundary)
its fundamental group is cyclic and that of the torus isn't.
For each of these surfaces it's easy to write down explicitly
the universal cover and this gets the fundamental groups
immediately.
> I made a square section steel ring of m = 2 configuration by
> cut/twist/weld and feel that the surface morphology of all these
> surfaces has a unifying character by its fiber bundle.
Neither your steel constructions nor your "feelings"
for "surface morphology" have any mathematical force.
> What exactly am I missing?
Possibly the desire or the ability to think mathematically?
Victor Meldrew
Narasimham
tube to consider role of both sides of the surface necessary to fully
describe the surface.
The tube was filled with thermosetting adhesive. Air is gradually
evacuated from the inside of the tube, keeping inclination at four
quarter points to ground plane at 0, + 45, 90, + 45 degrees at quarter
points with local stretching as needed. The tube is heated and inside
surfaces allowed to stick together so that there is no more any
"inside" due to a monolithic rubber sheet formation of double
thickness.The sketches show how opposite points are brought together
(or just joined if you like) to make an MB out of a Torus. For MB
visibility only a half of the Torus is shown.
http://img79.imageshack.us/img79/8162/mbtorussl6.jpg
What results by this process is a Moebius strip that is supposedly
non-orientable when converted from the oriented Torus surface. The only
new requirement is that 720 degrees of polar rotation is needed instead
of 360 degrees to come back to the same point with same direction/sense
of normal vector without crossing the band edges.
Narasimham
Narasimham
> Narasimham wrote:
> > Moebius Band is not homeomorphic with a Torus or even orientable like it.
>
> However you define a Mobius strip (with or without a boundary)
> its fundamental group is cyclic and that of the torus isn't.
Even though as shown m = 6 type cases with their edges/boundaries
juxtaposed MBs together are making up a torus !!
> For each of these surfaces it's easy to write down explicitly
> the universal cover and this gets the fundamental groups
> immediately.
>
> > I made a square section steel ring of m = 2 configuration by
> > cut/twist/weld and feel that the surface morphology of all these
> > surfaces has a unifying character by its fiber bundle.
May be not in proper full context, but I was trying to follow Dave
Rusin's earlier post here:
http://www.math.niu.edu/~rusin/known-math/98/dimension
If the business of embedding is too dis-orient-ing (sorry) we can mimic
the Flatlander description of topology by discussing three-dimensional
manifolds. A thickened Moebius strip is an example of a 3-dimensional
manifold: ... this vector cannot be picked in a continuously-varying
way across the whole of the space... Now, topologists recognize that
the thickened Moebius strip is nothing but a solid torus... etc
Please refer to my previous images of MB and Torus where the
semi-circular meridian part of Torus shrinks by a factor 2/pi to become
MB...apparently with good homeomorphism .
> Neither your steel constructions nor your "feelings" for "surface morphology"
> have any mathematical force.
>
> > What exactly am I missing?
>
> Possibly the desire or the ability to think mathematically?
May be I should learn up more on concepts in manifolds and topology.
But,however unqualified I am to say so, but say it nevertheless
..Topolgy seems to hide behind concepts too involved that seem unable
to address simple geometrical situations with a matching simplicity.
> Victor Meldrew
Kindly help if possible with more/at least adequate/relevant arguments
or information before helping with judgments. I am not alone in such a
situation.Several of those who posted in this newsgroup expressed
dismay about absence of a satisfactory axiomatic base in topology.
Narasimham
Narasimham
Continued..
The source of my discomfiture with what to me always appeared as some
sort of a Moebius Trick is simple and I shall make an encore..
A half Torus between the main two points of its meridian(with tangents
parallel to the original uncut Torus axis) can be homeo - compressed to
a flat ring meridionally. When the fibers are twisted by a toroidal
co-ordinate rotation double that of the polar angle, the twisted torus
can likewise be homeo - compressed to a Moebius Band.
But the topology goes awry and way off !! I am unable to accept any
topological difference between a flat concentric ring and a its
twisted counterpart, the MB.The justification to the contrary I cannot
understand in simple terms. And I think it ought to be expressed in
simple terms starting from the postulates.
Best Regards
Narasimham
Hero
> A half Torus between the main two points of its meridian(with tangents
> parallel to the original uncut Torus axis) can be homeo - compressed to
> a flat ring meridionally. When the fibers are twisted by a toroidal
> co-ordinate rotation double that of the polar angle, the twisted torus
> can likewise be homeo - compressed to a Moebius Band.
>
> But the topology goes awry and way off !! I am unable to accept any
> topological difference between a flat concentric ring and a its
> twisted counterpart, the MB.The justification to the contrary I cannot
> understand in simple terms. And I think it ought to be expressed in
> simple terms starting from the postulates.
>
When You compress to a flat ring or a Moebius Band a triangle is no
longer on one side of these 2D- in 3D surfaces. As You state it, one
looks at the fibers. Now take a skew triangle with 30 - 80 - 70
degrees and translate it on, better in both surfaces. There is a
difference.
With friendly greetings
Hero
Narasimham
How so? What exactly do you mean to say? I understand what you say to
be: " When you compress a curved triangle to a flat ring or to a
Moebius Band, it is no longer on one side of these 2D- in 3D surfaces
.." OK ?
But, neither in the twisted Torus:
http://img79.imageshack.us/img79/8162/mbtorussl6.jpg
nor in the simple straight Torus whose image I just upload:
http://img176.imageshack.us/img176/2199/straighttorustb0.jpg
the compressed surface suffers any disorientation.
In mapping diffeomorphisms there is no translation or gliding on the
surface.Successive points are reposted to alternate locations.
Regards,
Narasimham
victor_me...@yahoo.co.uk
> > May be I should learn up more on concepts in manifolds and topology.
> > But,however unqualified I am to say so, but say it nevertheless
> > ..Topolgy seems to hide behind concepts too involved that seem unable
> > to address simple geometrical situations with a matching simplicity.
"seems"? "seems" to you perhaps.
> > Kindly help if possible with more/at least adequate/relevant arguments
> > or information before helping with judgments
Perhaps you might start by kindly posting only "adequate/relevant"
material to this group.
> >Several of those who posted in this newsgroup expressed
> > dismay about absence of a satisfactory axiomatic base in topology.
Where "several" means "none".
Topology is formalized in set theory, and there are various axiom
systems
for set theory.
> I am unable to accept any
> topological difference between a flat concentric ring and a its
> twisted counterpart, the MB.
This is a group for discusssion of mathematics, not your cognitive
deficits. The annulus and the Mobius strip are not homeomorphic
(they are "topologically different") whether or not you are
"able to accept" this or not.
I presume my comments about the fundamental groups of these
spaces went straight over your head.
Victor Meldrew
victor_me...@yahoo.co.uk
> Is it so? Consider a simple case. A sheet of paper of say A4 size has
> six faces. Four of them (two thickness X length and two thickness X
> breadth areas) are so narrow to be negligible in area compared to two
> main areas length X breadth of paper that we use for writing upon. But
> that does not mean it has zero thickness. If thickness is zero, the
> sheet vanishes entirely along with all its features.
You really are most resolute in your refusal to think mathematically.
Victor Meldrew
victor_me...@yahoo.co.uk
> The following experiment was performed on an automobile wheel rubber
> tube
Try sci.engr.
Victor Meldrew
Hero
victor_me...@yahoo.co.uk schrieb:
> Narasimham wrote:
>
> > > May be I should learn up more on concepts in manifolds and topology.
> > > But,however unqualified I am to say so, but say it nevertheless
> > > ..Topolgy seems to hide behind concepts too involved that seem unable
> > > to address simple geometrical situations with a matching simplicity.
>
> "seems"? "seems" to you perhaps.
>
> > > Kindly help if possible with more/at least adequate/relevant arguments
> > > or information before helping with judgments
>
> Perhaps you might start by kindly posting only "adequate/relevant"
> material to this group.
>
>
> > I am unable to accept any
> > topological difference between a flat concentric ring and a its
> > twisted counterpart, the MB.
>
> This is a group for discusssion of mathematics, not your cognitive
> deficits. The annulus and the Mobius strip are not homeomorphic
> (they are "topologically different") whether or not you are
> "able to accept" this or not.
>
> I presume my comments about the fundamental groups of these
> spaces went straight over your head.
>
So we have to look up to Victor from below.
But he is really afraid up there.
May be there is some mathematical material, which can shatter his
foundations?
Narasimham, keep fighting.
Read what galathaea wrote in
:: nonsense for earnest students of math and science ::
http://groups.google.com/group/sci.physics/browse_frm/thread/11a90d0461d84632/9b6865b73ce65a04?lnk=gst&q=galathaea&rnum=9#9b6865b73ce65a04
Friendly greetings
Hero
Hero
> Hero wrote:
> > Narasimham schrieb:
> >
>
> How so? What exactly do you mean to say?
Let me tell You, what i know, in more details.
A surface in math doesn't has to be the surface of a solid in space, a
plane is called surface too. Or to give it analytical form, in R3 with
cartesian coordinates, this is given with the points (x, y, z ) which
satisfy an algebraic equation F(x, y , z ) = 0, or a combination of
these,so f.e. a hyperboloid is a surface too and two connected sides of
a cube too.
A piece of paper is of course a solid, whereas a mathematical surface
has thickness zero.
Often we talk about the two sides of a surface, just like as if our
mathematical surfaces have two sides. But intrinsical a surface is only
one side. Not intrinsical - considering the surrounding space, there
are (often) two sides to a surface, not in it. At least locally one
can look at it from (!) one side (of space) onto it, or one can also
pass from one side through it to the other side, or return.
So we start drawing ink onto a mathematical moebius band in form of a
line, and we start where it is ,,glued together". When we finished
one rotation and we are back to the part, where it is ,,glued", the
line is in another part of the space. A direct connection of the
ink-point from the beginning to the ink-point of the finish has to pass
through the surface. So one can talk about orientation with taking the
surrounding space into account. But intrinsically? You are not allowed
to refer to the surrounding space. ,,Draw" a similar line in the
middle of the surface and it will return to itself after one rotation.
,,Draw" a skew triangle into a plane, by moving it around, by
scrolling ( translating) or rotation in the plane one will never get
the chiral opposite form of it. For this, one has to do a ,,line
reflection" or rotate it through space outside the plane. The same is
true on, no better in the surface of a solid. Now a twisted surface
does a rotation in space. So a skew triangle moved around the moebius
band does follow this twist. Or to put it in another way, a skew
traingle in a plane removed from it, rotated and returned, so it is
chiral opposite, will sweep out a surface of the moebius type or more
complicated types.
With friendly greetings
Hero
Narasimham
> Narasimham wrote:
> > The following experiment was performed on an automobile wheel rubber tube.
>
> Try sci.engr.
>
> Victor Meldrew
Posted in sci.engr.mech per your suggestion, thanks.
Narasimham
Narasimham
> Narasimham wrote:
>
> > > ..Topolgy seems to hide behind concepts too involved that seem unable
> > > to address simple geometrical situations with a matching simplicity.
> "seems"? "seems" to you perhaps.
> > > Kindly help if possible with more/at least adequate/relevant arguments
> > > or information before helping with judgments
>
> Perhaps you might start by kindly posting only "adequate/relevant"
> material to this group.
>
> > >Several of those who posted in this newsgroup expressed
> > > dismay about absence of a satisfactory axiomatic base in topology.
>
> Where "several" means "none".
> Topology is formalized in set theory, and there are various axiom
> systems for set theory.
No information, references or links ....
> > I am unable to accept any
> > topological difference between a flat concentric ring and a its
> > twisted counterpart, the MB.
>
> This is a group for discusssion of mathematics, not your cognitive deficits.
This is a group also for taking guidance, sharing and helping those in
need by discussions in mathematics,including to posters sincerely
wanting to learn a new subject by self help but whose cognitive
faculties of any sort may have been temporarily impaired.Read the
charter to catch its spirit.
> The annulus and the Mobius strip are not homeomorphic, they are "topologically
> different" whether or not you are "able to accept" this. (or not).
The straight torus is homeomorphic to the annnulus,and twisted torus to
the MB.As the torus is homeomorphable between twisted and straight ones
the annulus is homepomorph to the MB.That is one of the arguments that
you readily negate but go no further.
> Victor Meldrew
Below is given a half of non-classical MB consisting of positive, zero
and negative Gauss Curvatures cut out of a torus. Pray tell me what
topological provisions or requirements would pronounce this surface an
odd one out as regards orientation. Anyone can see, feel, intuit,
calculate and formulate its 3D parameterization to assert that this
half portion of the Clifford torus is predictably orientable in its
direction of its surface normal defined quite uniquely, clearly and
with analytical continuity at each and every point.
http://img288.imageshack.us/img288/9648/mbhalfje0.jpg
Narasimham
David Marcus
> > > dismay about absence of a satisfactory axiomatic base in topology.
> >
> > Where "several" means "none".
> > Topology is formalized in set theory, and there are various axiom
> > systems for set theory.
>
> No information, references or links ....
Are you asking for a book on topology?
> > The annulus and the Mobius strip are not homeomorphic, they are "topologically
> > different" whether or not you are "able to accept" this. (or not).
>
> The straight torus is homeomorphic to the annnulus,and twisted torus to
> the MB.
I know what an annulus and a Möbius strip are. What's a "straight
torus" and a "twisted torus"?
> As the torus is homeomorphable between twisted and straight ones
What does "homeomorphable" mean?
> the annulus is homepomorph to the MB.
Of course, an annulus is not homeomorphic to a Möbius strip.
> That is one of the arguments that you readily negate but go no further.
>
> > Victor Meldrew
>
> Below is given a half of non-classical MB consisting of positive, zero
> and negative Gauss Curvatures cut out of a torus. Pray tell me what
> topological provisions or requirements would pronounce this surface an
> odd one out as regards orientation. Anyone can see, feel, intuit,
> calculate and formulate its 3D parameterization to assert that this
> half portion of the Clifford torus is predictably orientable in its
> direction of its surface normal defined quite uniquely, clearly and
> with analytical continuity at each and every point.
>
> http://img288.imageshack.us/img288/9648/mbhalfje0.jpg
What's your point?
--
David Marcus
victor_me...@yahoo.co.uk
> > This is a group for discusssion of mathematics, not your cognitive deficits.
>
> This is a group also for taking guidance, sharing and helping those in
> need by discussions in mathematics,including to posters sincerely
> wanting to learn a new subject by self help but whose cognitive
> faculties of any sort may have been temporarily impaired.
I'm afraid I noticed no signs of sincerity in any of your
previous postings; neither did I see evidence that your
cognitive deficits were temporary.
> > The annulus and the Mobius strip are not homeomorphic, they are "topologically
> > different" whether or not you are "able to accept" this. (or not).
>
> The straight torus is homeomorphic to the annnulus,and twisted torus to
> the MB.As the torus is homeomorphable between twisted and straight ones
> the annulus is homepomorph to the MB.That is one of the arguments that
> you readily negate but go no further.
That is not an argument, but a succession of unsupported assertions
involving various undefined neologisms; for instance "straight torus"
and "twisted torus".
> Below is given a half of non-classical MB consisting of positive, zero
> and negative Gauss Curvatures cut out of a torus.
Not so, what is actually given below is a URL to a JPEG.
The JPEG does not appear to depict a Mobius strip.
Victor Meldrew
Narasimham
> Narasimham wrote:
------
> > Below is given a half of non-classical MB consisting of positive, zero
> > and negative Gauss Curvatures cut out of a torus.
>
> Not so, what is actually given below is a URL to a JPEG.
> The JPEG does not appear to depict a Mobius strip.
http://img288.imageshack.us/img288/9648/mbhalfje0.jpg
> Victor Meldrew
The following I consider to be important from viewpoint of the subject
of topology, in spite of the fact that I know so little of it.Yes,as
indicated already, the Band whose image you see after clicking on the
above URL is not that of the classical MB. It is a representation of a
half toroidal surface that I define here and now name it as a Half
Torus Helical Band (HTHB). Each filament or line on the Band is given
by 2 ph = th where ph and th are the latitude and longitude in a
toroidal co-ordinate system (similar to spherical co-ordinate system
except for the eccentrically shifted center of the sweeping circle away
from axis) respectively. The x,y,z parameterization I have already
given.I have undraped an MB here to reveal the HTHB by ripping off and
exposing the two hidden, separate but concealed layers of a
conventional (ramified?) MB, where the two hidden layers are positioned
back-to-back to exhibit a deceptive single entity appearance of a
two-sided surface. The HTHB has the main characteristics of an MB__
reversal of normal vector direction after 1rotation, coming back again
at the same point after 2 rotations with same orientation.The HTHB is
homeomorphable to the conventional MB clearly, except now the
indecisive state of orientation is removed and along with it (what to
me appears as) the false mantle/myth of its non-orientability.
Narasimham
> Narasimham wrote:
> > victor_me...@yahoo.co.uk wrote:
> > > Narasimham wrote:
> > > > Several of those who posted in this newsgroup expressed
> > > > dismay about absence of a satisfactory axiomatic base in topology.
> > >
> > > Where "several" means "none".
> > > Topology is formalized in set theory, and there are various axiom
> > > systems for set theory.
> >
> > No information, references or links ....
>
> Are you asking for a book on topology?
>
> > > The annulus and the Mobius strip are not homeomorphic, they are "topologically
> > > different" whether or not you are "able to accept" this. (or not).
> >
> > The straight torus is homeomorphic to the annnulus,and twisted torus to
> > the MB.
>
> I know what an annulus and a Möbius strip are. What's a "straight
> torus" and a "twisted torus"?
The straight torus is standard and has well known parameterization. The
twisted torus picture is already indicated.Its filaments are twisted as
per 2*ph = th. Please see about HTHB in comments to Victor's post.
David Marcus
> David Marcus wrote:
> > Narasimham wrote:
> > > victor_me...@yahoo.co.uk wrote:
> > > > Narasimham wrote:
> > > > > Several of those who posted in this newsgroup expressed
> > > > > dismay about absence of a satisfactory axiomatic base in topology.
> > > >
> > > > Where "several" means "none".
> > > > Topology is formalized in set theory, and there are various axiom
> > > > systems for set theory.
> > >
> > > No information, references or links ....
> >
> > Are you asking for a book on topology?
> I want to clarify this before reading as this doubt will linger on.
You mean you want to clarify something before reading a topology book?
What do you want to clarify, exactly?
> >
> > > > The annulus and the Mobius strip are not homeomorphic, they are "topologically
> > > > different" whether or not you are "able to accept" this. (or not).
> > >
> > > The straight torus is homeomorphic to the annnulus,and twisted torus to
> > > the MB.
> >
> > I know what an annulus and a M=F6bius strip are. What's a "straight
> > torus" and a "twisted torus"?
>
> The straight torus is standard and has well known parameterization. The
> twisted torus picture is already indicated.Its filaments are twisted as
> per 2*ph = th. Please see about HTHB in comments to Victor's post.
So, the twisted torus is a torus, but you use a different
parameterization. Yes?
> > > As the torus is homeomorphable between twisted and straight ones
> >
> > What does "homeomorphable" mean?
What does "homeomorphable" mean?
> > > the annulus is homepomorph to the MB.
> >
> > Of course, an annulus is not homeomorphic to a M=F6bius strip.
> >
> > > That is one of the arguments that you readily negate but go no further.
> > >
> > > > Victor Meldrew
> > >
> > > Below is given a half of non-classical MB consisting of positive, zero
> > > and negative Gauss Curvatures cut out of a torus. Pray tell me what
> > > topological provisions or requirements would pronounce this surface an
> > > odd one out as regards orientation. Anyone can see, feel, intuit,
> > > calculate and formulate its 3D parameterization to assert that this
> > > half portion of the Clifford torus is predictably orientable in its
> > > direction of its surface normal defined quite uniquely, clearly and
> > > with analytical continuity at each and every point.
> > >
> > > http://img288.imageshack.us/img288/9648/mbhalfje0.jpg
> >
> > What's your point?
It looks like you took a strip, gave it two twists, glued the ends
together, glued the edges together, and got a torus. So what?
--
David Marcus
victor_me...@yahoo.co.uk
> The following I consider to be important from viewpoint of the subject
> of topology, in spite of the fact that I know so little of it.
I, I, I,. me, me, me.
>The HTHB is
> homeomorphable to the conventional MB clearly,
Clearly it isn't.
> except now the
> indecisive state of orientation is removed and along with it (what to
> me appears as) the false mantle/myth of its non-orientability.
There is no such "myth/mantle".
Victor Meldrew
victor_me...@yahoo.co.uk
> It looks like you took a strip, gave it two twists, glued the ends
> together, glued the edges together, and got a torus. So what?
Except he/she didn't get a torus, but only a closed subset of
a torus homeomorphic to a closed annulus.
Quite underwhelming.
Victor Meldrew
David Marcus
> David Marcus wrote:
>
> > It looks like you took a strip, gave it two twists, glued the ends
> > together, glued the edges together, and got a torus. So what?
>
> Except he/she didn't get a torus, but only a closed subset of
> a torus homeomorphic to a closed annulus.
Oh.
> Quite underwhelming.
Indeed.
--
David Marcus
Narasimham
> >
> > The following I consider to be important from viewpoint of the subject
> > of topology, in spite of the fact that I know so little of it.
>
> I, I, I,. me, me, me.
>
> >The HTHB is
> > homeomorphable to the conventional MB clearly,
>
> Clearly it isn't.
A slight error on my part, what was implied or explicitly stated was
the join of separate parts of the HTHB.
> > except now the
> > indecisive state of orientation is removed and along with it (what to
> > me appears as) the false mantle/myth of its non-orientability.
>
> There is no such "myth/mantle".
>
> Victor Meldrew
Perhaps it was not adequately clear what was being talked about. It has
to be imagined as below.
http://img119.imageshack.us/img119/3520/mbaustoruslt9.jpg
In the above modified picture corresponding points are to be brought
together along directions indicated by red arrows continuously along
the long edges as well breadth contacting the inner areas together.
Such rubber sheet deformations are permitted. I feel (sorry there is no
way the first person can avoid a pronoun while referring to the first
person himself) that this double-thick coalesced band (it can be left
as a hollow lens shaped section as well)that would result by such a
joining is a Moebius Band of course sans orientation.
Also in the opening post of this thread, pictures of tori for n = 4,3,2
are orientable, but the fact of its description, that it is not any
more orientable just for m = 1 MB case....that is not appealing. The
normals have opposing directions even though one torus dimension tends
to zero. Both directions are valid but to say that it cannot be
oriented at all, that is not clear, it beats the first person.
David Marcus
> Perhaps it was not adequately clear what was being talked about. It has
> to be imagined as below.
>
> http://img119.imageshack.us/img119/3520/mbaustoruslt9.jpg
>
> In the above modified picture corresponding points are to be brought
> together along directions indicated by red arrows continuously along
> the long edges as well breadth contacting the inner areas together.
> Such rubber sheet deformations are permitted. I feel (sorry there is no
> way the first person can avoid a pronoun while referring to the first
> person himself) that this double-thick coalesced band (it can be left
> as a hollow lens shaped section as well)that would result by such a
> joining is a Moebius Band of course sans orientation.
Is the following right? You take a long strip of paper, give it two
twists and glue the short edges together. Then you take the long edges
and glue them together.
Doing this gives you a closed surface without boundary. It doesn't give
you a Moebius strip. Topologically, you get either a torus or a Klein
bottle, depending on how you glue the long edges together.
--
David Marcus
Phil Carmody
> http://img119.imageshack.us/img119/3520/mbaustoruslt9.jpg
You seem to have discovered that if you cut a MB down the
middle along its length, you get a single band with a double
twist, which we all knew aged 10, and then wrapped it up
with the obfuscation of reversing the process. Why?
Phil
--
"Home taping is killing big business profits. We left this side blank
so you can help." -- Dead Kennedys, written upon the B-side of tapes of
/In God We Trust, Inc./.
Narasimham
Absolutely so, without doubt.
However,the surface I refer to is not glued on the (thickness X long
edge) or (thickness X long edge ) sides of the strip as you
mentioned,but glued on entire areas indicated between red arrows as if
the two flat areas are brought together and cladded. It is formed by
joining parts of the same twisted torus HTHB I mentioned,somewhat like
Siamese twins conjoined back to back. I have no parameterization for
it now, it can be made out of China clay, for example.
Regards,
Narasimham
David Marcus
> David Marcus wrote:
> > Narasimham wrote:
> > > Perhaps it was not adequately clear what was being talked about. It has
> > > to be imagined as below.
> > >
> > > http://img119.imageshack.us/img119/3520/mbaustoruslt9.jpg
> > >
> > > In the above modified picture corresponding points are to be brought
> > > together along directions indicated by red arrows continuously along
> > > the long edges as well breadth contacting the inner areas together.
> > > Such rubber sheet deformations are permitted. I feel (sorry there is no
> > > way the first person can avoid a pronoun while referring to the first
> > > person himself) that this double-thick coalesced band (it can be left
> > > as a hollow lens shaped section as well)that would result by such a
> > > joining is a Moebius Band of course sans orientation.
> >
> > Is the following right? You take a long strip of paper, give it two
> > twists and glue the short edges together. Then you take the long edges
> > and glue them together.
> >
> > Doing this gives you a closed surface without boundary. It doesn't give
> > you a Moebius strip. Topologically, you get either a torus or a Klein
> > bottle, depending on how you glue the long edges together.
>
>
> However,the surface I refer to is not glued on the (thickness X long
> edge) or (thickness X long edge ) sides of the strip as you
> mentioned,but glued on entire areas indicated between red arrows as if
> the two flat areas are brought together and cladded. It is formed by
> joining parts of the same twisted torus HTHB I mentioned,somewhat like
> Siamese twins conjoined back to back. I have no parameterization for
> it now, it can be made out of China clay, for example.
You take a strip of paper, give it a full twist, glue the edges
together. Then you glue the surface of the band to itself. The result is
a Moebius band.
We can do it in the other order, and it won't look so mysterious. Take a
strip of paper, give it a half twist, but hold one edge to the middle.
Then wrap the other half around the strip. When you go all the way
around again, you can now glue the two edges together. Since you went
around the Mobius band twice, you have a full twist in your strip.
So, what's your point?
--
David Marcus
victor_me...@yahoo.co.uk
> I feel
Your feelings have no mathematical force. Perhaps
you should try thinking instead of feeling.
Victor Meldrew
Narasimham
> What does "homeomorphable" mean?
>
> David Marcus
By 'homeomorphable' I mean capable of being mapped (there are many
mappings, conformal, isometric etc among surfaces depending what is
held invariant between the differential surface elements (du,dv))
between patches of a surface generally according to sketch indicated:
http://img297.imageshack.us/img297/5171/surfacehomeomorphismqe0.jpg
It could be a small extension, compression or large surface
translations or even gliding 'parallel to' parameter lines (a
partial modification of my response to Hero) accompanied by extension
and compression. In all cases it is a simple change of parameters (u,v)
limits on a parameterized surface.
In my first post, please see that it is just parameter 'm' limits
that have changed for morphing a torus into a Square Tube and a Moebius
Band.
Narasimham
David Marcus
> David Marcus wrote:
> > Narasimham wrote:
> > > victor_me...@yahoo.co.uk wrote:
> > > > Narasimham wrote:
> --
> > What does "homeomorphable" mean?
> >
> > David Marcus
>
> By 'homeomorphable' I mean capable of being mapped (there are many
> mappings, conformal, isometric etc among surfaces depending what is
> held invariant between the differential surface elements (du,dv))
> between patches of a surface generally according to sketch indicated:
>
> http://img297.imageshack.us/img297/5171/surfacehomeomorphismqe0.jpg
> It could be a small extension, compression or large surface
> translations or even gliding 'parallel to' parameter lines (a
> partial modification of my response to Hero) accompanied by extension
> and compression. In all cases it is a simple change of parameters (u,v)
> limits on a parameterized surface.
Can't say that answers the question. Are you familiar with the standard
meanings of the words homeomorphism, homotopy, continuous map,
surjection, injection?
--
David Marcus
Narasimham
Yes, I understand the concepts now by seeing internet googled sources
e.g.,Wikipedia and Mathworld, .Hope an adequate and intuitive grasp is
there although I am not a full student of topology.You can see my past
posts in general,I have helped several students although I am not a
teacher in a university.
In continuum mechanics of materials I am very familiar with homotopy
althogh it does not go by that name.There geometrical strain/stress is
further subdivided into in-plane (tension, compression) and out- of
-plane (bending), dealt with in classical laminate theory but for small
deformations only. In topology large deformations are possible.Other
concepts are relatively easy.
Please indicate where exactly I am making an error in simple terms if
indeed I am. I appreciate improved objectivity,avoiding subjective
remarks without reasons stated.
The invalidating reasons should be plain,simple,easily understood, be
not so abtruse,recondite or recoverable only from the depths of a
special article in a recent journal or so.
Narasimham
David Marcus
> David Marcus wrote:
> > Narasimham wrote:
> > > > What does "homeomorphable" mean?
> > >
> > > By 'homeomorphable' I mean capable of being mapped (there are many
> > > mappings, conformal, isometric etc among surfaces depending what is
> > > held invariant between the differential surface elements (du,dv))
> > > between patches of a surface generally according to sketch indicated:
> > >
> > > http://img297.imageshack.us/img297/5171/surfacehomeomorphismqe0.jpg
>
> > > It could be a small extension, compression or large surface
> > > translations or even gliding 'parallel to' parameter lines (a
> > > partial modification of my response to Hero) accompanied by extension
> > > and compression. In all cases it is a simple change of parameters (u,v)
> > > limits on a parameterized surface.
> >
> > Can't say that answers the question. Are you familiar with the standard
> > meanings of the words homeomorphism, homotopy, continuous map,
> > surjection, injection?
>
> e.g.,Wikipedia and Mathworld, .Hope an adequate and intuitive grasp is
> there although I am not a full student of topology.You can see my past
> posts in general,I have helped several students although I am not a
> teacher in a university.
>
> In continuum mechanics of materials I am very familiar with homotopy
> althogh it does not go by that name.There geometrical strain/stress is
> further subdivided into in-plane (tension, compression) and out- of
> -plane (bending), dealt with in classical laminate theory but for small
> deformations only. In topology large deformations are possible.Other
> concepts are relatively easy.
>
> Please indicate where exactly I am making an error in simple terms if
> indeed I am. I appreciate improved objectivity,avoiding subjective
> remarks without reasons stated.
I don't think anyone here knows what point you are making. A Moebius
band is not homeomorphic to a torus (as these words are normally
defined). The picture you showed of gluing a strip back to back to
create a Moebius band is pretty, but so what?
You used the word "homeomorphable", but this does not have a standard
meaning and you didn't define what you meant. If you want to say
something, it will be easier if you use standard terminology, where it
exists, and define any new terminology you feel the need to invent.
> The invalidating reasons should be plain,simple,easily understood, be
> not so abtruse,recondite or recoverable only from the depths of a
> special article in a recent journal or so.
--
David Marcus
Narasimham
> > further subdivided into in-plane (tension, compression) and out- of
> > -plane (bending), dealt with in classical laminate theory but for small
> > deformations only. In topology large deformations are possible. Other
> > concepts are relatively easy.
> >
> > Please indicate where exactly I am making an error in simple terms if
> > indeed I am. I appreciate improved objectivity,avoiding subjective
> > remarks without reasons stated.
>
> I don't think anyone here knows what point you are making.
You may not think so. Respectfully I would say you cannot speak on
everyone's behalf without addressing the central question. Any one
going through this thread would suppose you and Victor constitute a
subset of topologists who cared to respond here.
God, OK ! if it is still not clear, I shall once again repeat it.
Is change of (u,v) parameter domain of 2D surface in 3-space (x,y,z)
of following parameterization in my first post a valid homeomorphism ?
b=3 ; a=1 ;
x = ( a Cos[th/2+ k Pi/ m ] + b ) Cos[th] ; y = ( a Cos[th/2+ k
Pi/m] + b ) Sin[th] ; z = a Sin[th/2+ k Pi/m] ;
Tori = ParametricPlot3D[{x,y,z},{th,0,4 Pi},{k,1,2 m },PlotPoints
->{101,2m},Boxed->False];
If so, you should accept m = 1 MB case as homeomorphic to the other
tori set members in my above mentioned first posting.If not, please
state why an exception has to be made in this case... What all axoims,
definitions, theorems, lemmas, tenets or whatever are brought into
fore for direct or implied mathematical/ topological logic to
invalidate an apparently simple and straight-forward embeddings of
these surfaces in 3D. It may be classical and so deeply ingrained. But
an enquiring student looks to a pointed answer while unlearning his
errors.
> A Moebius Band is not homeomorphic to a torus (as these words are normally defined).
I know it very well that this is (that MB and torus are not
homeomorphic) emphatically stated in all the textbooks. I myself quoted
from 'Geometry of Surfaces' in the very first post as Definition 5.
This appears to me (of course I have to ignore Victor's objections to
the reference to the first person as there is no other possibility) as
a contradiction based on my parameterization and that is why I at all
started this thread as OP. The point I ask here is how it deviates from
such a "normally" defined situation.
I suppose definitive words need not per se matter at all, really what
matters is what they should be connoting. New words can be always
coined to convey a concept at a later stage when it becomes an accepted
rule or tenet.
> The picture you showed of gluing a strip back to back to create a Moebius band is pretty.
Thanks, but pretty graphics credit is entirely due to Mathematica. A
great new kind of mathematical help. I used it to look at the m = 1
case "un-orientable" (Quotes until it is properly explained) MB
using RealTime 3D` with pressed mouse from all orientations. I shall
upload a few more pictures out of the same program, the MB and torus
sit neatly back to back obvoiusly a small distance that can be
compressed as a rubber membrane.Only the unifying parameterization in
above first post is due to me.
> but so what?
Your call to answer.
-----
> -----
> David Marcus
Narasimham
phil3...@yahoo.com
>
> The source of my discomfiture with what to me always appeared as some
> sort of a Moebius Trick is simple and I shall make an encore..
>
> A half Torus between the main two points of its meridian(with tangents
> parallel to the original uncut Torus axis) can be homeo - compressed to
> a flat ring meridionally. When the fibers are twisted by a toroidal
> co-ordinate rotation double that of the polar angle, the twisted torus
> can likewise be homeo - compressed to a Moebius Band.
>
> But the topology goes awry and way off !! I am unable to accept any
> topological difference between a flat concentric ring and a its
> understand in simple terms. And I think it ought to be expressed in
> simple terms starting from the postulates.
>
> Best Regards
>
> Narasimham
2 things...
1: The "width" of a Moebius band is about as thick as your desire to
understand its true existense. I do not mean this to be mean, but most
people who study this type of thing have devoted their entire lives to
understanding the subtle points which can't be just thought of without
training.
2: If the Moebius and and an annulus were the same things would be
VERY bad. Lets say for example you go jogging on track with 2 sides
divided by a center line. Now in the normal world this is an annulus
with a core circle drawn on it. You can run day in and day out and
never cross that line. However, if you were to do this on a Moebius
band (as you think it may be the same), on each lap you would return to
the "start" line but be on the OTHER side of the dividing line and thus
be crashing into oncoming traffic. A HA! you might say that you are on
the other side of the line but also upside down on the bottom. Now
notice that this ONLY fixes the dilemma if I can't now argue that you
were running "inside" the thickness of the band. This idea can be
cutely summed up in the classic trick: take a Moebius band, draw the
core circle, and cut along it. This will result in a longer band with 2
twists. Doing this with an annulus will obviously result in 2 annuli
being produced. If that isn't a good enough difference between the
Moebius band and an Annulus then I don't know what else you could want.
David Marcus
> God, OK ! if it is still not clear, I shall once again repeat it.
>
> Is change of (u,v) parameter domain of 2D surface in 3-space (x,y,z)
> of following parameterization in my first post a valid homeomorphism ?
>
> b=3 ; a=1 ;
> x = ( a Cos[th/2+ k Pi/ m ] + b ) Cos[th] ; y = ( a Cos[th/2+ k
> Pi/m] + b ) Sin[th] ; z = a Sin[th/2+ k Pi/m] ;
> Tori = ParametricPlot3D[{x,y,z},{th,0,4 Pi},{k,1,2 m },PlotPoints
> ->{101,2m},Boxed->False];
I don't use Mathematica, but tell me if the following is correct. Define
a function f:R^2 -> R^3 by
f(t,k) = ( ( cos(t/2 + k Pi/m) + 3 ) cos(t),
( cos(t/2 + k Pi/m) + 3 ) sin(t),
sin(t/2 + k Pi/m ) ).
You are asking whether if we restrict f to
{(t,k)| 0 < t < 4 Pi, 1 < k < 2m},
it is a homeomorphism. Is that your question?
--
David Marcus
Narasimham
No, my question is not about the character of functional dependence
f(t, k) but about the flexibility of the two parameter domain, the
rubber sheet area parameter domain, if you will. The question again:
For f (t, k) and for all arbitrary parameter domain sets (in your
notation) :
{(t, k) | tmin < t < tmax, kmin < k < kmax},
where t's are real and k's are integers, do we have a set of surfaces
with valid homeomorphism among them? Only in the present tori set we
have k interal,in general they also can be real.
f(t,k) is Monge's form of surface where we are varying the projected
window border limits of the surface whose topography is morphed in
homeomorphism as a rubber membrane.
Narasimham
Narasimham
Three Monge patches are shown,plan rectangles are cut out of the same
classic buckled surface z = sin(x)*sin(y).
Two patches are
overlapping/coalscing/occupying same 3D and 2D space/interferng in 2D
and 3D space.
Two patches are disjoint:
http://img58.imageshack.us/img58/1059/mongepatchhomeomorphismpu6.jpg
It is easy to see that between surfaces contained above any two plan
projections, membrane areas are inter-stretchable or homeomorphable.
Narasimham
David Marcus
> David Marcus wrote:
> > a function f:R^2 -> R^3 by
> >
> > f(t,k) = ( ( cos(t/2 + k Pi/m) + 3 ) cos(t),
> > ( cos(t/2 + k Pi/m) + 3 ) sin(t),
> > sin(t/2 + k Pi/m ) ).
> >
> > You are asking whether if we restrict f to
> >
> > {(t,k)| 0 < t < 4 Pi, 1 < k < 2m},
> >
> > it is a homeomorphism. Is that your question?
>
> f(t, k) but about the flexibility of the two parameter domain, the
> rubber sheet area parameter domain, if you will. The question again:
>
> For f (t, k) and for all arbitrary parameter domain sets (in your
> notation) :
>
> {(t, k) | tmin < t < tmax, kmin < k < kmax},
>
> where t's are real and k's are integers, do we have a set of surfaces
> with valid homeomorphism among them?
I don't understand what you are asking. I seem to be asking if something
is a homeomorphism. A homeomorphism is a map between two surfaces. So,
what are your two surfaces and what is the map? And, what does f have to
do with it?
> Only in the present tori set we
> have k interal,in general they also can be real.
--
David Marcus
David Marcus
> Continued...
>
> Three Monge patches are shown,plan rectangles are cut out of the
>
> 2D surfaces and two are disjoint:
>
> http://img58.imageshack.us/img58/1059/mongepatchhomeomorphismpu6.jpg
>
> It is easy to see that surfaces contained above any two plan
> projections areas are inter-stretchable or homeomorphable.
What does "homeomorphable" mean?
Not sure what point you are making.
--
David Marcus
Narasimham
I mean "homeomorphic".
Narasimham
The two surfaces are portions of the same surface f( k,m) =
(x,y,z).Mapping is possible between different m values.
Once again, the following is explained.
***********************************************************************
For 2D patches in 3D Space.
A patch is mapped onto another patch in a homeomorphism.
A patch of such a surface has 3 coordinate transformations for x,y, z
and 2 parameter domains for th and k. A set of tori considered here has
a full parametric representation as:
x=(a Cos[th/2+k Pi/m]+b) Cos[th]; y=(a Cos[th/2+k Pi/m]+b) Sin[th]; z=a
Sin[th/2+k Pi/m];
[{x,y,z},{th,0,4 Pi},{k,1,2 m}] or [f(th,k), {th,0,4 Pi},{k,1,2 m}]
Please see m =1,2 6 in first post imageshack jpeg. for b = 3, a = 1.
For a given coordinate transformation ANY change of parametric domain
represents a valid homeomorphism. Referred by the arrow between
parameter domains. I am referring to the two-way or bijective
homeomorphism of continuous functions, which in terms of a topological
space are essentially identical.
Homeomorphism exists between the above second domain changed sets for
some constant m = infinity,2,1.
There are infinity,4 and 2 orientations or directions of surface
normals to each parametric surface respectively.
The surfaces have infinity, 4,2 possibilities of orientation
respectively.
The surfaces are the torus, square tube and MB respectively. They are
essentially identical topological spaces.
The torus, square tube and MB are homeomorphic to each other.
To convert a torus to an MB, map from m = infinity to 1.To convert MB
to square tube, map from m = 1 to 2 . To convert an MB to a torus ,
map from m = 1 to infinity and so on. m = 6 is a smooth enough torus.
The torus, square tube and MB are orientable in infinity, 4,2 number of
ways.
The MB has 2 directions and is dually orientable, but not
un-orientable.
A narrow diangle line has 2 sides (but neither zero or indefinite!),
triangle three, rectangle four etc.
***************************************************************************
> --
> David Marcus
Hope I could make my point at least now.
Narasimham
Narasimham
**************************************************************************
Higher dimensional spaces of continuous bijective homeomorphsms
constitute an identical set as they are independent of at least one
dimension less independent arguments or independent parametric domains
defining the higher dimensional space. Directional properties of the
higher spaces like orientability are likewise independent of the
defining independent parametric domain.
*************************************************************************
You helped me express my above innate belief or generalization when I
totally rejected the statement of this thread title: "Moebius Band is
not homeomorphic with a Torus or even orientable like it". Posted here
even if it is no place to state beliefs or express related feelings,
but I know of no other forum this good.
Regards,
Narasimham
David Marcus
"f(k,m)=(x,y,z)" doesn't make much sense. What are k and m?
> Once again, the following is explained.
>
> ***********************************************************************
> For 2D patches in 3D Space.
>
> A patch is mapped onto another patch in a homeomorphism.
>
> A patch of such a surface has 3 coordinate transformations for x,y, z
> and 2 parameter domains for th and k.
What are th and k?
> A set of tori considered here has
> a full parametric representation as:
>
> x=(a Cos[th/2+k Pi/m]+b) Cos[th]; y=(a Cos[th/2+k Pi/m]+b) Sin[th]; z=a
> Sin[th/2+k Pi/m];
> [{x,y,z},{th,0,4 Pi},{k,1,2 m}] or [f(th,k), {th,0,4 Pi},{k,1,2 m}]
How do I get one torus from this set? What are the independent
variables?
Don't you think it would be better to switch to standard notation?
> Please see m =1,2 6 in first post imageshack jpeg. for b = 3, a = 1.
>
> For a given coordinate transformation ANY change of parametric domain
> represents a valid homeomorphism.
No idea what that means. What is a "change of parametric domain"?
> Referred by the arrow between
> parameter domains. I am referring to the two-way or bijective
> homeomorphism of continuous functions, which in terms of a topological
> space are essentially identical.
>
> Homeomorphism exists between the above second domain changed sets for
> some constant m = infinity,2,1.
What is m?
> There are infinity,4 and 2 orientations or directions of surface
> normals to each parametric surface respectively.
>
> The surfaces have infinity, 4,2 possibilities of orientation
> respectively.
>
> The surfaces are the torus, square tube and MB respectively. They are
> essentially identical topological spaces.
>
> The torus, square tube and MB are homeomorphic to each other.
>
> To convert a torus to an MB, map from m = infinity to 1.To convert MB
> to square tube, map from m = 1 to 2 . To convert an MB to a torus ,
> map from m = 1 to infinity and so on. m = 6 is a smooth enough torus.
>
> The torus, square tube and MB are orientable in infinity, 4,2 number of
> ways.
>
> The MB has 2 directions and is dually orientable, but not
> un-orientable.
>
> A narrow diangle line has 2 sides (but neither zero or indefinite!),
> triangle three, rectangle four etc.
>
> ***************************************************************************
>
> Hope I could make my point at least now.
Nope. Is the following what you are saying?
You have two surfaces that are subsets of three-dimensional space and
you have a map between these two surfaces that you say is a
homeomorphism. One of the surfaces is a torus and the other is a MB.
If that's correct, then what is the map?
--
David Marcus
David Marcus
> Higher dimensional spaces of continuous bijective homeomorphsms
Does "continuous bijective homeomorphism" just mean "homeomorphism"?
What is a "higher dimensional space of homeomorphisms"?
> constitute an identical set as they are independent of at least one
> dimension less independent arguments or independent parametric domains
> defining the higher dimensional space. Directional properties of the
> higher spaces like orientability are likewise independent of the
> defining independent parametric domain.
>
> You helped me express my above innate belief or generalization when I
> totally rejected the statement of this thread title: " Moebius Band is
> not homeomorphic with a Torus or even orientable like it ". Posted here
> even if it is no place to state beliefs or express related feelings,but
> I know of no other forum this good.
Considering it can be proved that a Moebius Band is not homeomorphic to
a torus, I don't see why you should "believe" otherwise. I suspect you
are confused as to what a homeomorphism is. The fact that you don't seem
to be able to clearly state what your "homeomorphism" is lends support
to my suspicion.
--
David Marcus
Narasimham
Your questions are addressed earlier. However, shall once again
clarify.
m -> Infinity gives torus
m = 1 gives Moebius Band
m = 2 gives Square tube
m = 6 gives 12-sided polygon tube
m -> Infinity gives torus
> What are the independent variables?
th and k are the independent parameters in a toroidal co-ordinate
system. Independent variables we call as parameters now.
th is the polar co-ordinate/paarmeter and k the latitude of the torus
(similar to spherical cordinates) coordinate/parameter.
> Don't you think it would be better to switch to standard notation?
This IS the standard notation, classical from earlier times. Whichever
format or symbol used, content/significance is same. Recently also used
by Alfred Gray etal.
> > Please see m =1,2 6 in first post imageshack jpeg. for b = 3, a = 1.
> >
> > For a given coordinate transformation ANY change of parametric domain
> > represents a valid homeomorphism.
>
> No idea what that means. What is a "change of parametric domain"?
My! attempting to explain again...
For a parabola on a plane( 1D immersed in 2D) y = x^2/4 can be
parametererized as x=2 t , y= t^2 bringing in parameter t. One can say
(x(t),y(t)) = f(t). ( "t" has significance as slope of tangent , but
does not matter for usage). Similarly (x(t),y(t),z(t)) = g(t) is a 1D
line immersed in 3D.
For 2D immersion in 3D examples are
( Independent parameters u and v, dependent variables x,y and z. )
z = sin(u) cos(v), x=u, y = v ; A wavy surface ; In Mathematica Plot3D
command is used.
x = u+v , y = u - v , z = u*v ; hyperboloid surface ; In Mathematica
ParametricPlot3D command is used.
By bending and stretching a parabolic arc, it can be made into another
straight line or parabola, a surface into another surface or same
surface from one patch to another patch or domain. Shifting domain
parameter interval limits can establish a dependent function into a
new range which can be attained by topological rubber deformation or
homeomorphism.
The independent parameters th and k each have an upper limit, a lower
limit. By parameter domain is meant for ( thmin < th < thmax) th
parameter domain and for (kmin < k < kmax) k parameter domains.
> > Referred by the arrow between
> > parameter domains. I am referring to the two-way or bijective
> > homeomorphism of continuous functions, which in terms of a topological
> > space are essentially identical.
> >
> > Homeomorphism exists between the above second domain changed sets for
> > some constant m = infinity,2,1.
>
> What is m?
The number of plot points taken to divide a line. If 2 points are used
to join two equi-spaced points on a circle we get a diangle, 3 a
triangle, 4 a rectangle, 20 an icosagon, infinity a circle etc.
Toplogically they are all identical, choice of number of plot points is
immaterial.
>
> > There are infinity,4 and 2 orientations or directions of surface
> > normals to each parametric surface respectively.
> >
> > The surfaces have infinity, 4,2 possibilities of orientation
> > respectively.
> >
> > The surfaces are the torus, square tube and MB respectively. They are
> > essentially identical topological spaces.
> >
> > The torus, square tube and MB are homeomorphic to each other.
> >
> > To convert a torus to an MB, map from m = infinity to 1.To convert MB
> > to square tube, map from m = 1 to 2 . To convert an MB to a torus ,
> > map from m = 1 to infinity and so on. m = 6 is a smooth enough torus.
> >
> > The torus, square tube and MB are orientable in infinity, 4,2 number of
> > ways.
> >
> > The MB has 2 directions and is dually orientable, but not
> > un-orientable.
> >
> > A narrow diangle line has 2 sides (but neither zero or indefinite!),
> > triangle three, rectangle four etc.
> >
> > ***************************************************************************
> >
> > Hope I could make my point at least now.
>
> Nope. Is the following what you are saying?
>
> You have two surfaces that are subsets of three-dimensional space and
> you have a map between these two surfaces that you say is a
> homeomorphism. One of the surfaces is a torus and the other is a MB.
>
> If that's correct, then what is the map?
If after you are through with what was replied in the above, my part
should be obvious, If not, we shall address it once again,to go
again..But then you may have to explain any inconsistency of the same
MB coming out from tori set I am indicating,also appearing by
consideriation of mapping.I shall not justify the map procedure.My
mapping is rubber sheet homeomorphism only. If you bring in a map or
any other advanced topological consideration and the foregone
conclusion along with it, it means you did not agree to consider or
partake in my argument. In such a situation we may agree to close the
thread perhaps inconclusively.
> --
> David Marcus
Narasimham
Narasimham
> Narasimham wrote:
> > David Marcus wrote:
> > > Narasimham wrote:
> > > > David Marcus wrote:
> > > > > I don't use Mathematica, but tell me if the following is correct. Define
> > > > > a function f:R^2 -> R^3 by
> > > > > f(t,k) = ( ( cos(t/2 + k Pi/m) + 3 ) cos(t),
> > > > > ( cos(t/2 + k Pi/m) + 3 ) sin(t),
> > > > > sin(t/2 + k Pi/m ) ).
(included in next post).
> "f(k,m)=(x,y,z)" doesn't make much sense. What are k and m?
Makes a full good sense. k and m are 2 parameters or independent
variables if you will, for surfaces from 2 parameters into 3D or R^3.
It is better to write ( x(k,m), y(k,m) , z(k,m) ). Some books avoid
paranthetical repetition.It is known as parametererization, included in
all 3D software. Seen it written that way in texts before.I chose k and
m as they are integer variables.
Narasimham
Narasimham
> Narasimham wrote:
> > Higher dimensional spaces of continuous bijective homeomorphsms
>
> Does "continuous bijective homeomorphism" just mean "homeomorphism"?
Continuous functions mapping both ways sort of homeomorphism, homotopy
or whatever... that is not too interesting to me so long as that is a
"valid" homeomorphism.
> What is a "higher dimensional space of homeomorphisms"?
Since no headway is being made with basics, can we agree to let this
one pass for the time being.
> > constitute an identical set as they are independent of at least one
> > dimension less independent arguments or independent parametric domains
> > defining the higher dimensional space. Directional properties of the
> > higher spaces like orientability are likewise independent of the
> > defining independent parametric domain.
> >
> > You helped me express my above innate belief or generalization when I
> > totally rejected the statement of this thread title: " Moebius Band is
> > not homeomorphic with a Torus or even orientable like it ". Posted here
> > even if it is no place to state beliefs or express related feelings,but
> > I know of no other forum this good.
>
> Considering it can be proved that a Moebius Band is not homeomorphic to
> a torus, I don't see why you should "believe" otherwise.
If you do not agree to examine parameter domain independence in some
valid homeomorphisms among a common torus family as basis and stick to
the classical line then status quo continues.
> I suspect you are confused as to what a homeomorphism is. The fact that you don't
> seem to be able to clearly state what your "homeomorphism" is lends support
> to my suspicion.
I have shown the images below and the parameterization / formula (but
held back nothing by way of stating something if at all I could ! )
Frankly, I do not attach any significance to what type of (my ?!)
homeomorphism it is, as long as it is a valid one.The surface is just
twisting, bending and stretching of a membrane to a slightly altered
position.Just like twisting a long quadrilateral to new helical
position. If had known I would mention.I first thought it may be
bijective homeomorphism described by two continuous surface functions
for each topography or homotopy of sorts...It may not be so correct a
description with my scant familiarity with topology and skepticism ...
http://img79.imageshack.us/img79/8162/mbtorussl6.jpg
Do you mean the two states of deformation do not belong to any sort of
homeomorphism, like one-to-one etc.?
I expected you to describe what the above is giving the correct name or
label to it.By seeing the above MB to torus deformation, you should be
able to say that, I myself dont know and thats why I posted the
picture. About what homeomorphism is depicted in the (pretty)picture
above, you remained so far without comment, expecting me to describe
it? It is not for me to say either way due to my non-familiarity and
because of which I am basically posting.
> David Marcus
With Regards
Narasimham
Narasimham
http://www.geocities.com/glnarasimham/MoebiusTorusMorph/TorusMoebiusMorph.htm
Narasimham
David Marcus
> I have shown the images below and the parameterization / formula (but
> held back nothing by way of stating something if at all I could ! )
> Frankly, I do not attach any significance to what type of (my ?!)
> homeomorphism it is, as long as it is a valid one.The surface is just
> twisting, bending and stretching of a membrane to a slightly altered
> position.Just like twisting a long quadrilateral to new helical
> position. If had known I would mention.I first thought it may be
> bijective homeomorphism described by two continuous surface functions
> for each topography or homotopy of sorts...It may not be so correct a
> description with my scant familiarity with topology and skepticism ...
>
> http://img79.imageshack.us/img79/8162/mbtorussl6.jpg
The top-left is a MB. The top-right is another surface (with two corners
touching). The bottom-left has the two together. The bottom-right looks
like the same as the bottom-left from another angle. So, what does this
show?
--
David Marcus
David Marcus
Notation changes over the years. Modern notation is clearer. And,
Mathematica notation is not the same as mathematical notation.
Here is what I think you are saying:
g: {(t,k,m)| 0 <= t <= 4 Pi, 1 <= k <= 2m, 1 <= m <= oo} -> R^3
g(t,k,m) := ( ( cos(t/2 + k Pi/m) + 3 ) cos(t),
( cos(t/2 + k Pi/m) + 3 ) sin(t),
sin(t/2 + k Pi/m ) ),
f_m(t,k) := g(t,k,m).
Then for any m, 1 <= m <= oo, the image of f_m is a surface. And, f_1
is a MB, and f_oo is a torus.
This is not a homeomorphism between a MB and a torus. If we restrict m
to be in some finite interval, then we get a homotopy. For example, g
gives a homotopy between the maps f_1 and f_2. However, two surfaces
can be homotopic without being homeomorphic. And, you can't let m be
oo if you are constructing a homotopy.
A homeomorphism is different. Let M be a subset of R^3 that is a MB.
Let T be a subset of R^3 that is a torus. Then a homeomorphism is a
map h from M to T such that h is bijective, continuous, and its
inverse is continuous.
--
David Marcus
Narasimham
That is fine, only now we are beginning to talk for the first time on
addressable terms.
> This is not a homeomorphism between a MB and a torus. If we restrict m
> to be in some finite interval, then we get a homotopy. For example, g
> gives a homotopy between the maps f_1 and f_2.
OK, there is homotopy between an MB and that I now call the square
tube S.
> However, two surfaces can be homotopic without being homeomorphic.
You have to give some more examples (of course other than the present
one) and its logic, if there is something that can be brought in here.
> And, you can't let m be oo if you are constructing a homotopy.
Most certainly we can.The function is bounded, finite intervals in its
range and domain m. The points occupy a slightly different position in
R^3, that is all. We can let m be 6, no? I am satisfied with m=6 or f_6
homotopy with torus m=oo or f_6, you too will surely agree to this. It
is like accepting a situation in which the sum of external angles of a
hexagon (m=6) and circle (m=oo) is the same total angle 2 pi, we are
quite indifferent to polygon smoothness in topology or the difference
between a hexagon and circle. M values 6 -> 5 -> 4 -> 3 -> 2 between
and among the maps f_2, f_3, f_4, f_5,and f_6 gives a homotopy. Then
why do you bar holds suddenly in front of 2 -> 1 map between f_2 and
f_1?
It makes no difference with inclusion of the third argument of
parameter domain, hope you can see it in Mathematica animation for the
entire torus set:
Do[ ParametricPlot3D[{x[th,k,m],y[th,k,m],z[th,k,m] },{th,0,4
Pi},{k,1,2 m},PlotPoints ->{101,2m},
Boxedï‚®False] ,{m,1,6,1}]:
Do[ ParametricPlot3D[{x[th,k,m],y[th,k,m],z[th,k,m] },{th,0,4
Pi},{k,1,2 m},PlotPoints ->{101,2m} ]
> A homeomorphism is different.
Is different how?
> Let M be a subset of R^3 that is a MB.
> Let T be a subset of R^3 that is a torus.
Alright, in one statement, (M,T) are subsets of R^3 as separate
embedments.
> Then a homeomorphism is a map h from M to T such that h is bijective, continuous, and its inverse is continuous.
Yes this holds, T -> S -> M homeomorphism.
If you consider torus f_oo is too long drawn out, consider only the
square tube f_2 of S.
Why is not a map s a homeomorphism from S to M such that s is
bijective, continuous, and its inverse is continuous? Between f_2 and
f_1?
Narasimham
> --
> David Marcus
David Marcus
> David Marcus wrote:
> > Here is what I think you are saying:
> > g: {(t,k,m)| 0 <=3D t <=3D 4 Pi, 1 <=3D k <=3D 2m, 1 <=3D m <=3D oo} -> R=
> ^3
> > g(t,k,m) :=3D ( ( cos(t/2 + k Pi/m) + 3 ) cos(t), ( cos(t/2 + k Pi/m) + 3=
> ) sin(t),
> > sin(t/2 + k Pi/m ) ), f_m(t,k) :=3D g(t,k,m).
>
> > Then for any m, 1 <= m <= oo, the image of f_m is a surface. And, f_1
> > is a MB, and f_oo is a torus.
>
> That is fine, only now we are beginning to talk for the first time on
> addressable terms.
>
> > This is not a homeomorphism between a MB and a torus. If we restrict m
> > to be in some finite interval, then we get a homotopy. For example, g
> > gives a homotopy between the maps f_1 and f_2.
>
> OK, there is homotopy between an MB and that I now call the square
> tube S.
>
> > However, two surfaces can be homotopic without being homeomorphic.
>
> You have to give some more examples (of course other than the present
> one) and its logic, if there is something that can be brought in here.
Do you mean you want an example of surfaces that are homotopic, but not
homeomorphic? I'll give a simpler example, which should give the idea.
Consider
f: {(x,t)| 0 <= x <= 2 pi, 0 <= t <= 1} -> R^2,
f(x,t) := (t cos(x), t sin(x) ),
g_t(x) := f(x,t).
Then the image of g_0 is a point and the image of g_1 is a circle. f is
a homotopy between a point and a circle. However a point and a circle
are not homeomorphic, since a circle has more than one point.
> > And, you can't let m be oo if you are constructing a homotopy.
>
> Most certainly we can.
It doesn't satisfy the definition of "homotopy". If you want to
construct a homotopy, you need to use a finite interval. In your case,
you can perhaps rewrite your formulas to use a finite interval. I didn't
check.
> The function is bounded, finite intervals in its
> range and domain m. The points occupy a slightly different position in
> R^3, that is all. We can let m be 6, no? I am satisfied with m=6 or f_6
> homotopy with torus m=oo or f_6, you too will surely agree to this. It
> is like accepting a situation in which the sum of external angles of a
> hexagon (m=6) and circle (m=oo) is the same total angle 2 pi, we are
> quite indifferent to polygon smoothness in topology or the difference
> between a hexagon and circle. M values 6 -> 5 -> 4 -> 3 -> 2 between
> and among the maps f_2, f_3, f_4, f_5,and f_6 gives a homotopy. Then
> why do you bar holds suddenly in front of 2 -> 1 map between f_2 and
> f_1?
>
> It makes no difference with inclusion of the third argument of
> parameter domain, hope you can see it in Mathematica animation for the
> entire torus set:
>
> Do[ ParametricPlot3D[{x[th,k,m],y[th,k,m],z[th,k,m] },{th,0,4
> Pi},{k,1,2 m},PlotPoints ->{101,2m},
> Boxed=EF=82=AEFalse] ,{m,1,6,1}]:
> Do[ ParametricPlot3D[{x[th,k,m],y[th,k,m],z[th,k,m] },{th,0,4
> Pi},{k,1,2 m},PlotPoints ->{101,2m} ]
>
> > A homeomorphism is different.
>
> Is different how?
Formally, the definitions are different. Conceptually, two things that
are homeomorphic have identical topological properties. Two things that
are homotopic can be continuously deformed into each other, but can have
very different topological properties.
> > Let M be a subset of R^3 that is a MB.
> > Let T be a subset of R^3 that is a torus.
>
> Alright, in one statement, (M,T) are subsets of R^3 as separate
> embedments.
>
> > Then a homeomorphism is a map h from M to T such that h is bijective,
> continuous, and its inverse is continuous.
>
> Yes this holds, T -> S -> M homeomorphism.
What holds? I said we need a map h: M -> T. You wrote "T -> S -> M". I
don't know what you mean.
> If you consider torus f_oo is too long drawn out, consider only the
> square tube f_2 of S.
>
> Why is not a map s a homeomorphism from S to M such that s is
> bijective, continuous, and its inverse is continuous?
You can't demonstrate two things are homeomorphic by writing down a
homotopy. What is your map between the two surfaces? Please write the
formulas. It should be a map from (a subset of) R^3 to (a subset of)
R^3.
> Between f_2 and f_1?
You can have a homotopy between f_2 and f_1, but the homeomorphism has
to be between the *image* of f_2 and the *image* of f_1. A homotopy is
between maps. A homeomorphism is between sets.
--
David Marcus
Lee Rudolph
>Narasimham wrote:
...
>> David Marcus wrote:
...
>> > However, two surfaces can be homotopic without being homeomorphic.
>>
>> You have to give some more examples (of course other than the present
>> one) and its logic, if there is something that can be brought in here.
>
>Do you mean you want an example of surfaces that are homotopic, but not
>homeomorphic?
A nice example, which ought to (but probably won't) clear up N.'s confusion,
is the following. Take that inner tube he had a while ago, and rip a
patch off it. This physical object, which many people would be content
to abstract to S, a torus with an open disc removed--i.e., an orientable
surface of genus 1 with a single boundary component--*does* have thickness,
and so it would appear that N. would like to abstract it differently,
as the "square tube" (if I understand his term correctly) of that
surface, in other words, as a trivial I-bundle over S, which is a
3-dimensional solid with piecewise smooth boundary comprising three
parts--a (very narrow) annulus (which is the restriction of the I-bundle
to the boundary of S) and two copies of S (the "inside" and "outside"
surfaces of the inner tube). Of course homeomorphisms don't care
about corners, so topologically this solid is a "handlebody of genus 2",
bounded by a surface F of genus 2 (with empty boundary); the residuum
of the corners is a decomposition (what Gabai calls a "suturing") of
F into an annulus and the two (as it happens, homeomorphic) components
of its complement.
Now start again, this time from a flat disk from the interior of which
two smaller open disks with disjoint closures have been removed. A
physical counterpart of this might be a mask with two eye-holes (and
no mouth or nostrils), or a pair of underpants (with no fly). Again,
many of us would be content to abstract the mask or pants to S',
the original orientable surface of genus 0 with three boundary components;
but N., presumably, would once again take the flat tube. This time
we *again* get a 3-dimensinal handlebody of genus 2, but now the
sutures are different: there are 3 of them.
The process of replacing S or S' with a tribial I-bundle over it does not
change the homotopy type, and indeed creates two homeomorphic total spaces;
yet (of course!) S is not homeomorphic to S'.
Put differently, cancellation doesn't work for homeomorphism types:
you can have A x X = B x X with X non-empty (even, like I, contractible)
and yet not have A = B.
Lee Rudolph
Narasimham
The square tube S of case f_2 has two identical tracks of length L = 4
pi b and smaller width pi a/2,the two tracks are sutured together along
the long edges of length L = 4 pi b to form a closed torus.
> Now start again, this time from a flat disk from the interior of which
> two smaller open disks with disjoint closures have been removed. A
> physical counterpart of this might be a mask with two eye-holes (and
> no mouth or nostrils), or a pair of underpants (with no fly). Again,
> many of us would be content to abstract the mask or pants to S',
> the original orientable surface of genus 0 with three boundary components;
> but N., presumably, would once again take the flat tube. This time
> we *again* get a 3-dimensinal handlebody of genus 2, but now the
> sutures are different: there are 3 of them.
>
> The process of replacing S or S' with a tribial I-bundle over it does not
> change the homotopy type, and indeed creates two homeomorphic total spaces;
> yet (of course!) S is not homeomorphic to S'.
>
> Put differently, cancellation doesn't work for homeomorphism types:
> you can have A x X = B x X with X non-empty (even, like I, contractible)
> and yet not have A = B.
>
> Lee Rudolph
What (to me) appears simple is however associated with something from
the dark unfathomed depths of the deep. Anyway,may I repeat four simple
questions governing mapping from domain/image to surface:
Is f_6 to f_2 homotopy ok? If so,why so and if not,why not? Is f_6 to
f_2 homeomorphism ok? If so,why so and if not,why not? Is f_2 to f_1
homotopy ok? If so,why so and if not,why not? Is f_2 to f_1
homeomorphism ok? If so,why so and if not,why not?
Regards,
Narasimham
Narasimham
> Narasimham wrote:
> > David Marcus wrote:
>
> > > Here is what I think you are saying:
> > > g: {(t,k,m)| 0 <=3D t <=3D 4 Pi, 1 <=3D k <=3D 2m, 1 <=3D m <=3D oo} -> R=
> > ^3
> > > g(t,k,m) :=3D ( ( cos(t/2 + k Pi/m) + 3 ) cos(t), ( cos(t/2 + k Pi/m) + 3=
> > ) sin(t),
> >
> > > Then for any m, 1 <= m <= oo, the image of f_m is a surface. And, f_1
> > > is a MB, and f_oo is a torus.
> >
> > That is fine, only now we are beginning to talk for the first time on
> > addressable terms.
> >
> > > This is not a homeomorphism between a MB and a torus. If we restrict m
> > > to be in some finite interval, then we get a homotopy. For example, g
> > > gives a homotopy between the maps f_1 and f_2.
> >
> > OK, there is homotopy between an MB and that I now call the square
> > tube S.
> >
> > > However, two surfaces can be homotopic without being homeomorphic.
> >
> > You have to give some more examples (of course other than the present
> > one) and its logic, if there is something that can be brought in here.
>
> Do you mean you want an example of surfaces that are homotopic, but not
> homeomorphic? I'll give a simpler example, which should give the idea.
> Consider
>
> f: {(x,t)| 0 <= x <= 2 pi, 0 <= t <= 1} -> R^2,
> f(x,t) := (t cos(x), t sin(x) ),
> g_t(x) := f(x,t).
>
> Then the image of g_0 is a point and the image of g_1 is a circle. f is
> a homotopy between a point and a circle. However a point and a circle
> are not homeomorphic, since a circle has more than one point.
In this example, there is a singularity at origin, in which size
shrinking is associated with annihilation of direction. When radius t
shrinks to zero at pole, normal vector direction of (x-t) field is
indeterminate. Likewise, exactly at the north/south poles the direction
of (lat, long) lines is lost when using spherical coordinates.
Quaternions are employed to have a definitive direction at all points.
In the helicoidal 3D surface ( t cos(x), t sin(x), x ) has a definite
direction of surface normal along z-axis but its projection considered
as 2D disc (t cos(x), t sin(x)) the center would not have any
determinate direction. Does not orientation suffer here while
considering homeomorphism? Could we consider such an example while
considering homemorphism?
> > > And, you can't let m be oo if you are constructing a homotopy.
> >
> > Most certainly we can.
>
> It doesn't satisfy the definition of "homotopy". If you want to
> construct a homotopy, you need to use a finite interval. In your case,
> you can perhaps rewrite your formulas to use a finite interval. I didn't
> check.
For all topological purposes we can ignore difference between f_6,
which is a ridged torus and the smooth torus f_oo, so there is no need
to rewrite the given formula serving this same purpose. (m values 1-6
or 2-6 are finite intervals).
That states the present position again; with no new inputs I cannot
understand why it should be so, appears still to me as not so
consequentially established facts stated a priori.
> > > Let M be a subset of R^3 that is a MB.
> > > Let T be a subset of R^3 that is a torus.
> >
> > Alright, in one statement, (M,T) are subsets of R^3 as separate
> > embedments.
> >
> > > Then a homeomorphism is a map h from M to T such that h is bijective,
> > continuous, and its inverse is continuous.
> >
> > Yes this holds, T -> S -> M homeomorphism.
>
> What holds? I said we need a map h: M -> T. You wrote "T -> S -> M". I
> don't know what you mean.
>
S is the square tube. It is beneficial to recognize its good vantage
topological position standing between MB and torus, close to MB. The
square tube S is not disoriented, has the same orientation as MB. As
the thickness disappears so the orientation disappears, that is all.
The map T -> S -> M or the reversed M -> S -> T homotopy is allowed
all the way down as a deformation. The part of it S -> T is also
homotopically OK, but S -> T is not being a homeomorphism seems to be
the present stumbling block, its nature as yet to me incomprehensible.
> > If you consider torus f_oo is too long drawn out, consider only the
> > square tube f_2 of S.
> >
> > Why is not a map s a homeomorphism from S to M such that s is
> > bijective, continuous, and its inverse is continuous?
>
> You can't demonstrate two things are homeomorphic by writing down a
> homotopy. What is your map between the two surfaces? Please write the
> formulas. It should be a map from (a subset of) R^3 to (a subset of)
> R^3.
>
The formula is already mentioned in (***) above, map between f_2 to f_1
is S -> T. The images are mapped as they belong to the same general
parametric domain.
m=1 for image of f_1 (MB) -> m=2 for image of f_2 (S)
((th,0,4 Pi ), (k,1,2)) -> ((th,0,4 Pi ), (k,1,4))
> > Between f_2 and f_1?
>
> You can have a homotopy between f_2 and f_1, but the homeomorphism has
> to be between the *image* of f_2 and the *image* of f_1. A homotopy is
> between maps. A homeomorphism is between sets.
>
In the example above, if we take the range of variation of deformed
points in 3D as homotopy, and the generator parameter space or the
generating image space as above, then, is not homeomorphism sufficient
and complete with definitions of identified domain as image and range
put and considered together?
I am going through the Staford University lecture notes. But much seems
to be hidden. I would like to see many more concrete examples on:
(homotopic, not homeomorphic).
Narasimham
> --
> David Marcus
Narasimham
Proginoskes wrote:
> wodoro...@gmail.com wrote:
> > Narasimham napisal(a):
> > > Moebius Band is not homeomorphic with a Torus or even orientable like
> > > it.
> > >
> > > co-ordinate system I have indicated in:
> > >
> > > http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg
> > >
> > > I made a square section steel ring of m = 2 configuration by
> > > cut/twist/weld and feel that the surface morphology of all these
> > > surfaces has a unifying character by its fiber bundle.Of course the
> > > standard Moebius Band is made by cut/twist/paste of a rectangular sheet
> > > having two sides, small thickness of the sheet is conveniently
> > > overlooked to result in "one side" only! There are actually four sides
> > > in all and two tracks.
> > >
> > > By varying m or Aspect Ratio of a rectangular section of a tube (or
> > > solid section) of torus one can morph/animate images among all tori
> > > including the Moebius Band.
> > >
> > > I am not comfortable with the Moebius Band singled out topologically
> > > either in terms of orientability or homeomorphability.
> > >
> > > What exactly am I missing?
> > >
> > > Regards,
> > > Narasimham
> >
> >
> > There are many ways to show that MB is not homeomorphic with T.
>
> Also:
>
> (1) Euler characteristic (X(MB) = -1, X(T) = 0)
But X(MB) = 0 !
> (2) Every graph embedded on the MB has a vertex of degree < 6; hence
> any such graph can be colored with 6 vertices, no two adjacent vertices
> getting the same color. This is not true for T; K7 embeds on the torus.
>
> The MB is equivalent to the projective plane with a disk removed.
>
> --- Christopher Heckman
>
> > For example: Torus is a surface, Moebius Band is not a surface.
> >
> > Hox
Narasimham
> Moebius Band is not homeomorphic with a Torus or even orientable like
> it.
>
> Mathematica formula and images of the unifying set of tori in toroidal
> co-ordinate system I have indicated in:
>
> http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg
>
> I made a square section steel ring of m = 2 configuration by
> cut/twist/weld and feel that the surface morphology of all these
> surfaces has a unifying character by its fiber bundle.Of course the
> standard Moebius Band is made by cut/twist/paste of a rectangular sheet
> having two sides, small thickness of the sheet is conveniently
> overlooked to result in "one side" only! There are actually four sides
> in all and two tracks.
>
> By varying m or Aspect Ratio of a rectangular section of a tube (or
> solid section) of torus one can morph/animate images among all tori
> including the Moebius Band.
>
> I am not comfortable with the Moebius Band singled out topologically
> either in terms of orientability or homeomorphability.
>
> What exactly am I missing?
>
> Regards,
> Narasimham
Time to close the thread.I thank David Marcus,Lee Rudolph and others
who participated.
To be considered for acceptance are:
1) If a torus T fully encloses/is enclosed by an MB, corresponding
points being identified by a common tori set formula given, there is
homotopy but no homeomorphism between an MB and T.
2) If a square tube S fully encloses /is enclosed by an MB,-Ditto-.
3) The above is valid, not withstanding the fact that the topological
invariant, Euler Characteristic X for MB, T and even the KB are the
same, equal to zero.
4) In a square tube there are two tracks. If one track ( homeomorphic
to a disc) is removed, the other track is not homeomorphic to an MB.
When googling this topic, I was very surprised to see a large Moebius
sculpure in stone like the square tube S.
" Keizo Ushimo starts with a massive granite ring having a hole width
equal to the thickness of the ring. He then drills into the granite to
slice it longitudinally, not the way you would normally slice a bagel
to get two halves, but with a cut that makes a 180-degree twist during
its travel around the ring.In effect, such a cut creates a space that
can be considered a Möbius strip ".
It is topologically same as what I had made in steel and rubber. In the
following link from which the above is quoted,in effect we should see
no (topological) difference between the square tube, Möbius strip and
the torus.
http://www.sciencenews.org/articles/20031220/mathtrek.asp
Regards,
Narasimham
David Marcus
> Is f_6 to f_2 homotopy ok?
Yes.
> If so,why so and if not,why not?
It satisfies the definition of "homotopy".
> Is f_6 to f_2 homeomorphism ok?
That's not a homeomorphism.
> If so,why so and if not,why not?
Because a homotopy is not a homeomorphism.
> Is f_2 to f_1
> homotopy ok? If so,why so and if not,why not? Is f_2 to f_1
> homeomorphism ok? If so,why so and if not,why not?
--
David Marcus
David Marcus
So? It is still a homotopy.
> In the helicoidal 3D surface ( t cos(x), t sin(x), x ) has a definite
> direction of surface normal along z-axis but its projection considered
> as 2D disc (t cos(x), t sin(x)) the center would not have any
> determinate direction. Does not orientation suffer here while
> considering homeomorphism? Could we consider such an example while
> considering homemorphism?
A homotopy is not a homeomorphism. If you want to show two things are
homeomorphic, you should exhibit the homeomorphism.
What are your definitions of the words "homotopy" and "homeomorphism"?
That's not what a homeomorphism is. If you insist on confusing two
different things, then naturally you will be confused.
> m=1 for image of f_1 (MB) -> m=2 for image of f_2 (S)
>
> ((th,0,4 Pi ), (k,1,2)) -> ((th,0,4 Pi ), (k,1,4))
>
> > > Between f_2 and f_1?
> >
> > You can have a homotopy between f_2 and f_1, but the homeomorphism has
> > to be between the *image* of f_2 and the *image* of f_1. A homotopy is
> > between maps. A homeomorphism is between sets.
> >
>
> In the example above, if we take the range of variation of deformed
> points in 3D as homotopy, and the generator parameter space or the
> generating image space as above, then, is not homeomorphism sufficient
> and complete with definitions of identified domain as image and range
> put and considered together?
No.
> I am going through the Staford University lecture notes. But much seems
> to be hidden. I would like to see many more concrete examples on:
> (homotopic, not homeomorphic).
I gave an example to show a circle is homotopic to a point. But, a
circle is not homeomorphic to a point. Doesn't this convince you that
the two are different?
--
David Marcus
Narasimham
David Marcus wrote:
> Narasimham wrote:
> > Is f_6 to f_2 homotopy ok?
>
> Yes.
>
> > If so,why so and if not,why not?
>
> It satisfies the definition of "homotopy".
>
> > Is f_6 to f_2 homeomorphism ok?
>
> That's not a homeomorphism.
As far as I can understand, toplogy recognizes no corners, either in
homotopy or homeomorphism. A ridged torus,square tube or smooth torus
are all visibly and intuitively the same.It is the straining effect by
pressurization in a ridged rubber tube turning it smooth that topology
ignores.
Narasimham
That point is singular. Forgive,I am not so much well read here,I
wanted to pick up something using only simple intuitive and basic
cognitive skills.To me it sounds too crude. Even David Hilbert's
geometrical axioms of point,line and surface had been refined later on,
serving only as good starting models.When we talk about orientability
and direction of surfaces [For example knowing the origin as projection
of center points of radial lines through the origin of a rotating screw
generator projection in helicoid parametric lines from 3D to 2D as
having no determinate direction..], when the direction we are
attempting to look at is itself vanishing in the final scene, that
sounds to me as an inappropriate choice for an illustrative example.
>
> --
> David Marcus
David Marcus
> David Marcus wrote:
> > Narasimham wrote:
> > > Is f_6 to f_2 homotopy ok?
> >
> > Yes.
> >
> > > If so,why so and if not,why not?
> >
> > It satisfies the definition of "homotopy".
> >
> > > Is f_6 to f_2 homeomorphism ok?
> >
> > That's not a homeomorphism.
>
> As far as I can understand, toplogy recognizes no corners,
This is an intuitive statement.
> either in homotopy or homeomorphism.
These are rigorously defined.
> A ridged torus,square tube or smooth torus
> are all visibly and intuitively the same.It is the straining effect by
> pressurization in a ridged rubber tube turning it smooth that topology
> ignores.
That only says that intuitively the surfaces are homeomorphic. You can't
prove they are homeomorphic by exhibiting a homotopy.
Suppose X and Y are topological spaces and f: X -> Y, g: X -> Y are
continuous maps. Then f and g can be homotopic even though f(X) and g(X)
are not homeomorphic. And, f(X) and g(X) can be homeomorphic even though
f and g are not homotopic.
So, the fact that you showed us a homotopy between f_6 and f_2 doesn't
by itself tell us anything about whether the images of f_6 and f_2 are
homeomorphic.
--
David Marcus
David Marcus
In the plane, a circle is homotopic to a figure eight, but the two are
not homeomorphic.
--
David Marcus
David Marcus
> In the plane, a circle is homotopic to a figure eight, but the two are
> not homeomorphic.
In fact, if f: X -> R^n and g: X -> R^n are continuous, then they are
homotopic. Simply define F: X x [0,1] -> R^n by
F(x,t) = (1-t) f(x) + t g(x).
--
David Marcus
Narasimham
How so?
By plotting the set of cardioids:
(x,y) = (cos(u)+v)*(cos(u),sin(u)), 0 < u < 2 pi, -2 < v < 1, (v > 0
curves are self-intersecting), we see the entire set homotoping to the
central case of the circle through origin (v = 0). Let us reflect
portions (assuming the procedure is valid for homotopy) of all
cardioids to the other side of the minimum x tangent lines so we get
more convex ovals for v < - 1 cases, (smoother than the cusp of v = -1
classical central cardoid case) and figures of eight for v > 0 cases
which all form a homotopic set. I don't readily see why this does not
also form a homeomorphic set.
Narasimham
David Marcus
> David Marcus wrote:
> > Narasimham wrote:
> > > David Marcus wrote:
> > > > Narasimham wrote:
> > > > > I am going through the Staford University lecture notes. But much seems
> > > > > to be hidden. I would like to see many more concrete examples on:
> > > > > (homotopic, not homeomorphic).
> > > >
> > > > I gave an example to show a circle is homotopic to a point. But, a
> > > > circle is not homeomorphic to a point. Doesn't this convince you that
> > > > the two are different?
> > >
> > > That point is singular. Forgive,I am not so much well read here, I
> > > wanted to pick up something using only simple intuitive and basic
> > > cognitive skills.To me it sounds too crude. Even David Hilbert's
> > > geometrical axioms of point,line and surface had been refined later on,
> > > serving only as good starting models.When we talk about orientability
> > > and direction of surfaces [For example knowing the origin as projection
> > > of center points of radial lines through the origin of a rotating screw
> > > generator projection in helicoid parametric lines from 3D to 2D as
> > > having no determinate direction..], when the direction we are
> > > attempting to look at is itself vanishing in the final scene, that
> > > sounds to me as an inappropriate choice for an illustrative example.
> >
> > In the plane, a circle is homotopic to a figure eight, but the two are
> > not homeomorphic.
>
>
> By plotting the set of cardioids:
> (x,y) = (cos(u)+v)*(cos(u),sin(u)), 0 < u < 2 pi, -2 < v < 1, (v > 0
> curves are self-intersecting), we see the entire set homotoping to the
> central case of the circle through origin (v = 0). Let us reflect
> portions (assuming the procedure is valid for homotopy) of all
> cardioids to the other side of the minimum x tangent lines so we get
> more convex ovals for v < - 1 cases, (smoother than the cusp of v = -1
> classical central cardoid case) and figures of eight for v > 0 cases
> which all form a homotopic set. I don't readily see why this does not
> also form a homeomorphic set.
What does the word "homeomorphism" mean? Please define it and give an
example.
--
David Marcus
Narasimham
David Marcus
> Narasimham wrote:
> > David Marcus wrote:
> > > In the plane, a circle is homotopic to a figure eight, but the two are
> > > not homeomorphic.
> >
> >
> > By plotting the set of cardioids:
> > (x,y) = (cos(u)+v)*(cos(u),sin(u)), 0 < u < 2 pi, -2 < v < 1, (v > 0
> > curves are self-intersecting), we see the entire set homotoping to the
> > central case of the circle through origin (v = 0). Let us reflect
> > portions (assuming the procedure is valid for homotopy) of all
> > cardioids to the other side of the minimum x tangent lines so we get
> > more convex ovals for v < - 1 cases, (smoother than the cusp of v = -1
> > classical central cardoid case) and figures of eight for v > 0 cases
> > which all form a homotopic set. I don't readily see why this does not
> > also form a homeomorphic set.
> >
> > Narasimham
>
> A v > 0 cardoid case is at
>
> http://img205.imageshack.us/img205/1647/cardoidscr6.jpg
Not sure what that is supposed to show.
Narasimham
A portion of a v > 0 cardoid has been reflected to result in a figure
eight.
> What does the word "homeomorphism" mean? Please define it and give an
> example.
>
> --
> David Marcus
Actually it is the learning process that is supposed to be going on
right now where I had expected some guidance from you and all.Just
purchased the Topology second edition by James R.Munkres, MIT.Came
across (Martin Fuchs Theorem) in chapter on Fundamental Groups:Two
spaces X and Y have same homotopy type iff they are homeomorphic to
deformation retracts of a single space Z. Must read up a lot more.
Narasimham
David Marcus
> David Marcus wrote:
> > What does the word "homeomorphism" mean? Please define it and give an
> > example.
>
> right now where I had expected some guidance from you and all.
Part of the process of learning is being able to give definitions and
examples. This also helps crystallize the ideas in your mind.
> Just
> purchased the Topology second edition by James R.Munkres, MIT.
I have the first edition. It is an excellent book. It was the textbook
in the first topology course that I took.
> Came
> across (Martin Fuchs Theorem) in chapter on Fundamental Groups:Two
> spaces X and Y have same homotopy type iff they are homeomorphic to
> deformation retracts of a single space Z. Must read up a lot more.
I suggest you start closer to the beginning of the book rather than
skipping to the last chapter. Try reading chapter 2.
--
David Marcus
Narasimham
Thanks,will do that slowly.You may have already guessed my apprehension
:) .. unwittingly accepting an innocuous text-book definition whose
consequence conflicts with intuition.
Narasimham
Aluminium Holocene Holodeck Zoroaster
wait'l I recall it!
> > http://img139.imageshack.us/img139/3489/mbnoorientci0.jpg
> >
> > I made a square section steel ring of m = 2 configuration by
> > cut/twist/weld and feel that the surface morphology of all these
> > surfaces has a unifying character by its fiber bundle.Of course the
> > standard Moebius Band is made by cut/twist/paste of a rectangular sheet
> > having two sides, small thickness of the sheet is conveniently
> > overlooked to result in "one side" only! There are actually four sides
> > in all and two tracks.
> >
> > By varying m or Aspect Ratio of a rectangular section of a tube (or
> > solid section) of torus one can morph/animate images among all tori
> > including the Moebius Band.
> http://www.sciencenews.org/articles/20031220/mathtrek.asp
thus:
no, it was CSPAN,
where I said we were actually trying to land
in the pentagonal courtyard;
I really got all scratched up!
> Didn't I watch you on CNN?
thus:
I'd start with the similarity of ABC with AFB.
> > >> ( http://myhome.hanafos.com/~freeasbird/usenet4.jpg )
--nerfmania!
David Marcus
> David Marcus wrote:
> > > Came
> > > across (Martin Fuchs Theorem) in chapter on Fundamental Groups:Two
> > > spaces X and Y have same homotopy type iff they are homeomorphic to
> > > deformation retracts of a single space Z. Must read up a lot more.
> >
> > I suggest you start closer to the beginning of the book rather than
> > skipping to the last chapter. Try reading chapter 2.
>
> :) .. unwittingly accepting an innocuous text-book definition whose
> consequence conflicts with intuition.
Definitions can't conflict with intuition. Theorems can, but that's why
we prove them.
--
David Marcus
Narasimham
> Narasimham wrote:
> > Narasimham wrote:
> > > David Marcus wrote:
> > > > In the plane, a circle is homotopic to a figure eight, but the two are
> > > > not homeomorphic.
> > >
> > > How so?
> > >
> > > By plotting the set of cardioids:
> > > (x,y) = (cos(u)+v)*(cos(u),sin(u)), 0 < u < 2 pi, -2 < v < 1, (v > 0
> > > curves are self-intersecting), we see the entire set homotoping to the
> > > central case of the circle through origin (v = 0). Let us reflect
> > > portions (assuming the procedure is valid for homotopy) of all
> > > cardioids to the other side of the minimum x tangent lines so we get
> > > more convex ovals for v < - 1 cases, (smoother than the cusp of v = -1
> > > classical central cardoid case) and figures of eight for v > 0 cases
> > > which all form a homotopic set. I don't readily see why this does not
> > > also form a homeomorphic set.
> > >
> > > Narasimham
> >
> What does the word "homeomorphism" mean? Please define it and give an
> example.
> --
> David Marcus
I repeat again, the normally accepted definition.
One to one mappings that stretch and bend their domains into their
ranges without tearing are called homeomorphisms.
There are essentially three components for a homeomorphism for 2-
domain parameter (u,v) function mapping into a 3-space range
parameters (x,y,z).
1) Parameter domain space, two elements (u,v)
2) Parameter range space,three elements (x,y,z)
3) Functional dependences, three relations x(u,v) = fx(u,v), y(u,v) =
fy(u,v), z(u,v) = fz(u,v).
[ u is short for u1 <= u <= u2 etc.]. plot3d,ParametricPlot3D surfaces
of Maple or Mathematica are standard examples of homeomorphisms.
I had believed topolgy is indifferent to number of plot points m in a
curve. It is__not__ a relevant parameter.OK?
Even if m is a relevant parameter _____
Embedding of (u,v,m) maps into R^3 or (x,y,z) should (IMHO) should
yield a torus set as I indicated homeomorphic between Moebius Band and
other tori agreeably in a somewhat constricted sense, but that should
be applicable to ALL elements of the torus set, not just to the m = 1
MB case alone. Is it not?
However, there should be no disagreement about that:
Embedding of (u,v,m) maps into R^4 or (w,x,y,z) yields a free torus
superset homeomorphism between Moebius Band and torus. As sufficient
(one extra degree of freedom) accomodation is available.
What is the confusion here? Why is not the central point (Topic: When
is a homotopy not a homeomorphism?) not simply stated and discussed ?
Narasimham
Narasimham
> Narasimham wrote:
> > Narasimham wrote:
> > > David Marcus wrote:
> > > > In the plane, a circle is homotopic to a figure eight, but the two are
> > > > not homeomorphic.
> > >
> > > How so?
> > >
> > > By plotting the set of cardioids:
> > > (x,y) = (cos(u)+v)*(cos(u),sin(u)), 0 < u < 2 pi, -2 < v < 1, (v > 0
> > > curves are self-intersecting), we see the entire set homotoping to the
> > > central case of the circle through origin (v = 0). Let us reflect
> > > portions (assuming the procedure is valid for homotopy) of all
> > > cardioids to the other side of the minimum x tangent lines so we get
> > > more convex ovals for v < - 1 cases, (smoother than the cusp of v = -1
> > > classical central cardoid case) and figures of eight for v > 0 cases
> > > which all form a homotopic set. I don't readily see why this does not
> > > also form a homeomorphic set.
> > >
> > > Narasimham
> >
> What does the word "homeomorphism" mean? Please define it and give an
> example.
> --
> David Marcus
I repeat again, the normally accepted definition.
Aluminium Holocene Holodeck Zoroaster
at this time; I'd say the main problem is that, although
it's a neat idea, your halftube doesn't actually stick-
together at the ends; it's really just an ordinary strip,
albeit curved like a gutter, with a full twist
in to loops, or what ever.
the "open" MB is also *occaissonally* known
as a "twisted cylinder," though I have seen
only a purely topographical treatment (in Stillwell,
I think his book on 3D manifolds).
thus:
so, what is the estimated date of greatcircling,
or what ever?
it's pretty obvious that peak oil was launched
by the second invasion of Iraq --
that wascawy Harry Potter!
the USA should clearly have a tarrif on oil, LNG etc.,
if not on electricity, itself, although
that would go against Dubya's policy of *caveat emptor*,
Suckers, and the Protocol of the Elders
of Kyoto [sik] -- which he loveth.
we have very little time,
with such enormous financial leverages,
to actually do anything about it.
>This 6-minute video is about the proposed global energy grid which was Fuller's highest priority project.
thus:
IEEE 754 was an article in *Computer*, an issue in 1980;
it's variety of implimentation is quite fractal,
as proven by Mandelbrot, when he gave his talk
at Royce Hall!
there was a later update, like 854 or so. in fact,
Flops are inherently chaotic.
thus:
I meant, I knew taht Clooney was an activist actor, but
not what he looked like. recalled the other seeming namesake,
was either "Bettman" or Beckman, the fictionalized Nazi
rocket-scientist for whom "Brandt" worked. so,
I'd never heard of anything like that, about the Caltech guy;
was he saved from Germany?
>I didn't know who he was). there were two interesting name-sakes
>in the movie, the other of which I forgot, but
>"the good German" actually was supposed to be "Emille Brandt,"
>which is almost a perfect anagram of the postwar President's
>-- and what an a-hole, he actually was!
>(I thought it was an exact anagram, but
>I'd "synethesiad" the M into a Dubya ... oops;
>there's two Is and one E in Willie, as well, but
>it's *still* not just a conincidence .-)
--nerfman!
Narasimham
> couldn't deal with the algebraic geometry-speak,
> at this time; I'd say the main problem is that, although
> it's a neat idea, your halftube doesn't actually stick-
> together at the ends; it's really just an ordinary strip,
> albeit curved like a gutter, with a full twist
> in to loops, or what ever.
The other pictures show a full tube when 0 < th < 4 pi. For example
when 0 < th < 3 pi there is plenty of common seam length of the twisted
tube for suturing along the helical partition line.
> the "open" MB is also *occassionally* known
> as a "twisted cylinder," though I have seen
> only a purely topographical treatment (in Stillwell,
> I think his book on 3D manifolds).
Thanks,first time hearing this confirmation.Request you for a page
reference in op cit.
--- non-maths snipped ---