this is rather elementary. assuming the mobius strip is infinitismally thin (has no z-dim thickness), spreading it out to a flat strip renders it 2 dimensional but by definition, this no longer is a mobius strip. so a mobius strip cannot be defined merely by its 2D surfaceness. the mobius strip is defined by a specific (spacific?) trajectory of a 2 dimensional surface within 3 dimensional space (and it should have its 3d/4d..etc equivalent within a 4d/5d space, no?) . now, mobius strip relative to itself, its own trajectory is 2 dimensional since by denominating it to itself, one can only view it while travelling along it (sort of like being on a rollercoaster). this also introduces the view of orthogonal space (the object matter of our conventional perception) from/denominated by the vantage gemeotry of the mobius strip. perhaps, if we build a huge box of a huge mobius rollercoaster and then get on board with a camera shooting interval shots and then attenmpt to combine those shots along a 3d mobius trajectory in hologram style (or 3d print and paint?) for the mobius representation of the mobius experience of orthographic space or...assemble the interval shots by overlapping as we do in panoramic shots to result in an orthographic representation of the mobius experience of orthographic space.
but still of course we define the mobius strip within an orthogonal space. can't have one without the other.
naturally, its nonsense to be objecting on the basis that a computer screen or a drawing are 2 dimensional anyways. to do so would be puerile, untintelligently nitpicky and constitutes nothing more an insult to everyone else's intelligence here. everyone knows that already and thats not the point.
a 2d object in 3d space is still 2d. i think that's why flatland was brought up. a single dimension point is a point, and probably can't conceive of the second dimension. a 2-d plane can't perceive the third dimension. if you put the 2d plane in a different environment, it still can't perceive the 3rd dimension. that's because a 2d plane bent in 3d space still only has 2-dimensions and thus should be referred to as 2d.
i read your post. i thought i was retorting therein.
you're saying that the mobius strip has to be considered a 3-d object because it exists within 3-d space. at least i think that's what you're saying.
but it's still a 2-d object. i'm suggesting that a 2-d object can exist in 3-d space. if you're describing a point on the 2-d mobius strip, you shouldn't need the 3rd identifier. pick an origin, and describe how many units away along the length and how many perpendicular to the length, and you will be able to find the point.
also, i think you're saying this:
if a 2d strip is spread flat, then it is 2d
if a 2d strip is spread flat, then it is not a mobius strip
conclusion: if it is 2d, it is not a mobius strip.
your premises might be correct, but [if a then b, if a then c, therefore if b then c] is flawed logic.
if we build a huge box of a huge mobius rollercoaster and then get on board with a camera shooting interval shots and then attenmpt to combine those shots along a 3d mobius trajectory
A 2D surface has no third dimension (Z, or height). A roller coaster moves through 3 dimensions. Therefore a 2D roller coaster is impossible.
for a moment, forget that a mobius strip does not have a thickness. by overlooking this, you are not contradicting the nature of the mobius strip, you are merely overlooking one of its aspects in order to bring out other facts about it.
now, look at the mobius strip not as a surface per se, but as a collection of points constituting this thing we call mobius strip. in other words, identify the mobius strip by all its coordinates.
these coordinates will have variable z-components
we know that in 2d x-y space, the z-component is invariable.
therefore, this set of points (that we used to call mobius strip) exist in three-dimensional disposition to each other
therefore, this set of points = mobius strip is geometrically transformed three dimensionally.
now, this property of three-dimensional transofrmation is instrinsic to a mobius strip; it is not intrinsic to a laid out flat strip with invariable z-component. it is thus not a mere adjective that renders it a specific subset of flat objects; it defines the nature of the mobius strip.
this is what i meant when i said : "so a mobius strip cannot be defined merely by its 2D surfaceness" & "we define the mobius strip within an (here i meant to say 3d orthogonal) orthogonal space. can't have one without the other"
both times, i used the word define because its not a description of a mobius strip to say that it occupies 3d space, it is within the nature of the mobius strip to exist and define itself within 3d space.
now, all 2d entities can exist in 2d space and in 3d space, that is true. and a 3d entity cannot exist in 2d but in 3d (lets not drag in more d's) but a mobius strip by virtue of the intrinsic requirement to transform itself 3 dimensionally, to distribute its points and parts three dimensionally, cannot solely exist in 2d space.
for the sake of just being more interesting, i also introduced the idea that if the mobius is taken as its own denominator and we flip over the dependency of the mobius strip on three-dimensional space so that orthogonal space is viewed from a mobius (mobial?) reference. the rollercoaster analogy. and if we do travel along a mobius strip (ie we imagine we are the mobius strip moving to generate itself), then yes, one can posit that it is a 2 dimensional space but then:
1. this is not how you define a mobius strip; its an interesting consequence of having first defined and thought of the mobus strip three dimensionally.
2. this negates an outer space altogether (let alone discussion of 2 or 3 dimensional space) and therefore any outer reference/observer. a mobius strip is only 2d dimensional within its own self awareness (and ours if we were imagining ourselves as this mobius strip generating itself). but this is not possible in our external appreciation and understanding within the isotropic space it exists in outside us.
allow me to continue this, because i know someone will jump on the wrong wagon just for the sake of it:
"and a 3d entity cannot exist in 2d but in 3d (lets not drag in more d's) but a mobius strip by virtue of the intrinsic requirement to transform itself 3 dimensionally, to distribute its points and parts three dimensionally, cannot solely exist in 2d space."
continues: now, because the mobius strip cannot exist in 2d space and can only exist in 3d space, it must therefore be a 3d entity (wherein the 3d-ness is instrinsic to its nature and not a divorcable property that can render it still a mobius strip in lack of said property- recall arguement above about necessary defining invariability of the z-component).
Miles, you are invariably wrong, smug and never actually interested in an exchange of ideas and you deem yourself above learning and above others. not a nice combo.
lately? no, his responses are typically that way.; he says he' snarky lately as someone would deliver a fashionable statement. and anyway i dont give a shit whether its lately or earlier. i see how he reacts to genuine interest in discussion and questions and thats enough for me. he esteems his personal attitude over and above the discussion and its relevance. i can do worse but this is enough; i don't want to get metaphorical.
i've already stated how one could define any point on a mobius strip with only 2 dimensions right? was that unlcear, or did i state it wrong?
you could set an origin wherever you want. it is certainly possible to set your origin such that a 3rd descriptor is required to describe the mobius strip. however, as i explained before, you can say all points are bound to the strip and because of that can be described with only 2 dimensions if you choose to see it that way.
if you lay your strip flat it is 2-dimensional. but, you can tilt the 2-d shape up, and then suddenly you need a 3rd dimension to describe all points. or, since the origin point is arbitrary, you keep the 2-d shape as-is and tilt your axis. then you would need a 3rd dimension to describe the points, even though nothing changed with the strip itself. it's not necessary in that case to add the 3rd dimension, but you can do so if you choose. same with a mobius strip. you don't need the 3rd dimension to describe any point on the strip, but if you want to complicate things you're free to do so.
you seem to be hung up on the idea that a mobius strip has to be 3d if it exists in 3d space, or something like that. didn't rem say "fuck context?" can't we apply that here?
some would say you do not exist in infinite dimension. some would say there are only 11 dimensions.
"didn't rem say "fuck context?" can't we apply that here?"
only if you yourself for yourself want to say that to yourself. i see that the context defines, to us, the mobius in the mobius strip. rewind repeat (or wind and pleat in this case).
tammuz, I've yet to be demonstrated wrong (so far in this thread at least), have indeed been accused of being smug (by those who can't prove me wrong) and would truly enjoy seeing you make a coherent statement about möbius strips and dimensionality that challenged my perceptions and opened my mind.
Instead of getting all huffy why don't you try to work through it constructively? Assume I'm an idiot (as it seems you already do) and explain it as simply as you can, because so far what you've said seems like utter nonsense to me.
the mobius strip is defined by a specific (spacific?) trajectory of a 2 dimensional surface within 3 dimensional space
False. A möbius strip is defined by having a single surface and a single boundary, and by being non-orientable. It can be defined as an object in 3D space, but that doesn't make it three dimensional any more than defining a point in 3D space makes the point 3D.
The surface of a sphere is two dimensional as well (and for some even harder to visualize as 2D), with the interesting property that traveling in any direction will eventually return you to your starting point. Some propose that the universe is curved and has this very same property. The möbius strip is interesting because when traveling in one direction you are as likely to return to your starting point as you are to hit the edge of the world (universe?).
no, you are smug because you are. i'm not huffy or puffy - that description you apply to me is evidnce of your smugness. your replies throughout archinect show what you are and thats all it takes. even this "(i) would truly enjoy seeing you make a coherent statement about möbius strips" is a belittling response.why do you think that this is about giving you joy; why do you assume such a superior position, a personable center that is to be pleased or dismayed? i'm not here to give you joy or to win yur admiration. either talk straight at me or don't talk to me at all but don't talk down to me.
i did not say you are an idiot. your other features background any requirement to make an evaluation on that front. i am also not saying that you cannot be idiotic.
you sayid prviously there is no z component. well, there is a formula that shows that there is. you don't even acknowledge this. why? because this is secondary to wanting to seem right. again, smug...and self righteous, wrongfully.
"A möbius strip is defined by having a single surface and a single boundary, and by being non-orientable" - this in no way contradicts the other defining charecteristic: "by a specific (spacific?) trajectory of a 2 dimensional surface within 3 dimensional space" which is the formula for the mobius strip defined by x, y and z. over and above all else, the most exacting definition is the formula. you will choose to contest the word define - i will tell you that your definition (ie from wikipedia) is insufficient. a cylinder suffices to fulfill the requisites you define the mobius strip with. the trajectory however is defined by the formula.
furthermore, you are choosing not to challenge my points concerning the imminence -and not causualness- of three dimentionality to the mobius geometry (curtkram did but in my opinion went into a non existent loop - this is my opinion with all due respect to curtkram's person) and therefore but chose to recycle your own conception. it is due to that imminence that i cannot perceive or qualify a mobius strip, let alone a sphere or cylinder as two dimensional surfaces. why? because of the variable third dimension component for one but more imoportantly becaucse we recognize these entities within a three dimensional coordinational space...we do not plot a mobius strip on its own surfaceness, on its own two dimensionality. of course i know you are incapable of seeing that this is - should be- an inherent quality to the nature of three dimensional surfaces (i.e. topology)where the third dimension is not the thickness of the surface but the behaviour of the surface three dimentionally.
what does the "non-existent loop" mean? are you specifically referring the mobius loop, and you're saying that the 2-dimensional mobius strip doesn't exist? or my argument doesn't exist, and also is a loop?
i'll loop one more time. if a point is bound to the surface of a mobius strip, it can be described with 2 numbers (x-axis, which might run along the length, and y-axis, which may be perpendicular). those 2 numbers are the 2 dimensions that exist in the world of that mobius strip. all points possible on that strip can be described with those 2 parameters. the third axis is non-variable. it is always 0 relative to the surface of the strip.
i understand you are defining a mobius strip as an object in 3-dimensial space. i understand you feel the need to define a third axis relative to a surface in that 3-dimensial space that is not the surface of the mobius strip. generally speaking, you think all coordinate planes have to be orthogonal, and you can't have an 'x-axis' that follows a surface of some sort.
but time-space is not orthogonal. that is what einstein's theory of relativity set out to prove, at least to my understanding of it. so to view space as something that can only be orthogonal based on your current point of observations does not, in my opinion, reflect all possible, rational, and realistic viewpoints. let's think outside of euclidean geometry and let the x-axis have some breathing room to move around a bit. here is where i step outside of my comfort zone and into things i know nothing about, but that is sort of what Riemannian geometry is about. unfortunately, Riemannian geometry is well outside my core competence.
you can have your 3d mobius strip. i'm not saying your wrong, given the context you defined. however, i don't think you've addressed how a 2-dimensional mobius strip is impossible based on the frame of reference i gave. if i understand you correctly, you simply dismissed it as non-existent.
So the nugget that requires chewing is: can a surface "behave" in a way that isn't two-dimensional? For the example with which I started this conversation, that of a 2D being that only understands 2D, it can't. But in my perception of it, as a being that experiences more than two dimensions, it can.
This whole conversation is definitely helping me feel more comfortable with the concept of multiple dimensions that we can't (yet?) perceive.
i think you're asking if you can bend a piece of paper, which of course you can.
does the paper cease to be what it is once bent? in my opinion, no. if something 2-dimensional lived on that piece of paper, would it notice it had been bent? i doubt it.
^ Exactly. And thus a 2D surface like a sphere, mobius strip, etc. remains two-dimensional however we in the third dimension perceive its shape. To the hypothetical inhabitants it would all be the same. I'll have to look, but I think that in Abbot's Flatland if you journeyed far enough you would return to your original position, which make Flatland the surface of a sphere (or at least gives it those properties).
I keep flashing on gravity wells and the distortion of space time and relativity but can't seem to make an analogy that fits. In our three dimensional world Euclidean geometry eventually fails yet we are only able to perceive that indirectly. Maybe that's what made Flatland so good although I'm still at a complete loss when it comes to tesseracts.
Quondam, I left off because of what I perceived as your inability to separate the concept of a 2D surface from a 3D shape. As for proving assertions, that is a two-way street. I backed some of mine with various references (linked), while yours are wholly unsupported. Thus the moon is green cheese.
tammmuz, you were so incensed about my rebutting your ideas that you copped a nasty attitude. Neither your inability to refute my points or the inability to make yours clear are justification for verbal abuse (smug, etc.). Now you're even more incensed about my pointed response to your rudeness. Go figure. Bottom line: your inability to make an argument that I can understand - or conversely my inability to understand whatever argument you are trying to make - is no reason to be a turd. Yes, that was slap. Now get over it and let's have some dimensional fun.
So there are all these posts about China and projects in China. What side of the street do they drive on? The left side, like the UK? Japan does as well, no?
I once was watching a program on Dubai and, from the aerials of their roadways, they drove on the right side of the street. I was stunned.
I don't do countries where they drive on the left side. I wouldn't rent a car. Actually, even if they drove on the right side, and they had a language I didn't speak or were corrupt and dangerous, I wouldn't rent a car there either. In some places you need to rent a car, because public transit offers poor frequencies, like in Sicily or on the island of Mallorca. In other places, you don't need to rent a car, like in Rio or B.A.
I'll jump back into the dimension mess later, but right now I wanted to share that I did in fact FAIL the stupid Illustrator certification test that I was forced to take, although it is NOT needed for me to do my job. I got a 55% and only needed a 66% to pass. I was SOO close! I figured I'd get maybe a 25. I'm impressed that I winged it so well, but now I'm annoyed that I was ELEVEN points away. Effing ELEVEN! Damn.
I was not rude to you Miles. I was responding to a build up of nasty one liners from you that obviously come from a reserve of disrepect you have towards other people's contributions here. If i call you distateful and smug, this is becauase I find you you distateful and smug, Miles Jaffe.You on the other hand go on to suggest that I'm a turd. Exactly who is being rude then? For the sake of not insulting your person, I myself did not reach out to such demaning metaphors.
Also, you did not rebut my arguments, you presented yours. You cannot say that you rebutted my arguments and at the same time state that "your inability to make an argument that I can understand - or conversely my inability to understand whatever argument you are trying to make". if you don't understand, then obviously you cannot have rebutted the content you lack comprehension of. ow can you be rusted with mathematics if you cannot be trusted with mere common sense? There is no shame in that. Simply admit your lack without trying to belittle others for what you lack.
Curved geometry is cool. It is amazing when the sum of angles of triangle is not 180. It's like the square root of -1. All bets are off. Not sure how a flat lander would feel if her home were warped into a moebius strip but the geometry must be defined by curved surfaces, no? What does Bucky say?
Unrelated but the new app for ikea that lets us look at the furniture in our own rooms in realtine and to scale is seriously cool shit.
"ll loop one more time. if a point is bound to the surface of a mobius strip, it can be described with 2 numbers (x-axis, which might run along the length, and y-axis, which may be perpendicular). those 2 numbers are the 2 dimensions that exist in the world of that mobius strip. all points possible on that strip can be described with those 2 parameters. the third axis is non-variable. it is always 0 relative to the surface of the strip."
I said previously almost exactly that: "now, mobius strip relative to itself, its own trajectory is 2 dimensional since by denominating it to itself"
however, this had been preceded by (my)statement: "so a mobius strip cannot be defined merely by its 2D surfaceness."
in a following post and digressing on that speciic thought: "for the sake of just being more interesting, i also introduced the idea that if the mobius is taken as its own denominator and we flip over the dependency of the mobius strip on three-dimensional space so that orthogonal space is viewed from a mobius (mobial?) reference. the rollercoaster analogy. and if we do travel along a mobius strip (ie we imagine we are the mobius strip moving to generate itself), then yes, one can posit that it is a 2 dimensional space but then:
1. this is not how you define a mobius strip; its an interesting consequence of having first defined and thought of the mobus strip three dimensionally.
2. this negates an outer space altogether (let alone discussion of 2 or 3 dimensional space) and therefore any outer reference/observer. a mobius strip is only 2d dimensional within its own self awareness (and ours if we were imagining ourselves as this mobius strip generating itself). but this is not possible in our external appreciation and understanding within the isotropic space it exists in outside us."
so, you see, this loop of a thought has aleady been entertained and so ironed out (on my part and for myself).
the latter half of your post (dragging einstein into this) is spinning of into the vacuous infinity of "huh? what the heck" :oP
i will also say that fundamentally, this disagreement is over semantical and descriptive accuracy - we all know to some extent what a mobius strip is and mathematicians know much more than we do. but on the issue of semantics: on your part, you see the surfaceness of the mobius strip, a warped 2D entity. i understand that - this is why i said a mobius, if related to its own self (its own surfaceness), can be ,seen as a 2D entity. but, for me this is insufficient semantics, insufficient qualifier of the mobius strip. it is methodically generated as a formula in three dimensions, or to put it nearer to your way of thinking its 2-dimensionality has been acted upon by a z-component factor. thus for me, the semantics cannot stop at describing its own condition of corporeality (ie 2 dimensionality) but also its condition of formal existence (3 dimensionality). so, its a choice to stop at describing it two dimensionally (and therefore there is no difference to a rectangular horizontal surface) or to be recognize its formal existence within a 3 dimensional world and therefore call it a 3 dimensional entity. if you look at its infinitismal thinness, then yes 2 d (a sort of partial tautological description). if you look that the coordinates are dispersed three dimensionally, the yes 3d. mathematically, the latter view is more relevant, in my estimation.
Assuming Flatland is a finite 2D surface as described by the surface of a sphere, a right angle is always < 90°. Thus the natural geometry is different from Euclidean space and the radius of the sphere would be a natural constant. An analogy would be a right angle in Euclidean space or the speed of light in a relativistic universe.
The topology of a mobius strip presents a different problem as an equivalent constant is seemingly non-existent, at least according to the the model we visualize that varies from concave to convex and appears to eliminate any obvious natural constant (so far as I can see). I won't even get into the single boundary, which makes me imagine a flat disk Earth with edges you can fall off of.
And now for something completely different.
"a build up of nasty one liners"
My two replies to tammuz's posts that justify his being a jerk:
1) A 2D surface has no third dimension (Z, or height). A roller coaster moves through 3 dimensions. Therefore a 2D roller coaster is impossible.
2) tammuz, That's exactly what you said, quoted verbatim.
Dude, you've got a problem. I'm not interested. Have a nice day.
Miles Jaffe, again you are wrong. these are not the replies i referred to. unfortunately, i come across you on other occasions and i know what sort of sordid online character i'm confronting here, one possessed by a smugness neither supported by characteristic intelligence nor by a wealth of knowledge .you are a bitter much older person who bickers with and tries to play cat and mouse with youngsters who ask quite innocent sounding questions; your ego and pickled attitude estimated above all else. you deride people with different valid opinions wanting only to prove yourself correct irrespective of the value of discussions and irrespective of how stunted your intellectual projection and empathy are and how circuitous your argument is whilst ignoring valid points raised apropos your non-argument. and i don't particularly care if you are interested or not. now off with your needlessly bloated head. and i don't care what sort of day you have.
1. i think it happened already; its a bit late, your request.
2. thread central is a topically open thread.
3. are you proprietor?
3. kindly ignore it if you don't like. i know i don't come here for pleasurable simulation within the usual thread central clique but i don't think its in your place to make someone not welcome, vado retro.
sorry to hear about your troubles jla. probably should keep in mind that, while the answer often does lie at the bottom of a bottle, the problems tend to come back the next day (with a headache). it's better to be busy than not.
and look at the bright side, you have many more weeks to live through! in a few years, this might not even be in the top 10 of bad weeks :)
Aww, I was enjoying - immensely, or should I say infinitely? - the discussion of Mobius strips until the namecalling started.
jla-x, can you share any of the troubles? I'm dealing with supreme embarrassment at how far behind I am on several projects. I just keep begging understanding of people, and try to figure out how to say NO to some things that really shouldn't be my responsibility.
jla, mediation works for me. If you can sit quietly for 20 minutes, eyes closed, doing nothing but breathing, a lot of the crap in your head will play itself out and leave you in a much better place. Good luck.
Re 2D: Interesting things about the surface of a sphere as a 2D finite universe:
If you travel halfway across the universe to the east, then halfway to the north, you will return to your original position upside down. "Hi honey, I'm home!"
Also the inside of the sphere makes just as good a model of a 2D universe as the outside.
I distrust all politicians for the most part and am neither a liberal nor a conservative, but this politician tops the list. And what a hypocrite - he wants Obamacare defunded while his adopted home state of MA has a similar program. And MA was one of the first to have same-sex marriage, as one would expect of liberal New England. That means that he knows what side his bread is buttered on, in terms of his constituency, and will deviate from what his religion and Republicanism tell him. However, on a grander scale, because the U.S. has a lot of "red" territory, he can be who he really is. I seriously never wanted to see this guy again, but his avarice gets in the way. I'm wondering if this next election is going to be Hillary versus Palin or Romney, in which case I will abstain. I've done that before. I don't know if talk that Christie of NJ would go Dem and jump into the race is nothing but mere b.s., but if he lost weight to make sure he remained in good health and ran as a Dem, I'd have to think that one through and he might get my vote. At any rate, here's someone, and a family, we don't need anywhere near the White House:
Thread Central
this is rather elementary. assuming the mobius strip is infinitismally thin (has no z-dim thickness), spreading it out to a flat strip renders it 2 dimensional but by definition, this no longer is a mobius strip. so a mobius strip cannot be defined merely by its 2D surfaceness. the mobius strip is defined by a specific (spacific?) trajectory of a 2 dimensional surface within 3 dimensional space (and it should have its 3d/4d..etc equivalent within a 4d/5d space, no?) . now, mobius strip relative to itself, its own trajectory is 2 dimensional since by denominating it to itself, one can only view it while travelling along it (sort of like being on a rollercoaster). this also introduces the view of orthogonal space (the object matter of our conventional perception) from/denominated by the vantage gemeotry of the mobius strip. perhaps, if we build a huge box of a huge mobius rollercoaster and then get on board with a camera shooting interval shots and then attenmpt to combine those shots along a 3d mobius trajectory in hologram style (or 3d print and paint?) for the mobius representation of the mobius experience of orthographic space or...assemble the interval shots by overlapping as we do in panoramic shots to result in an orthographic representation of the mobius experience of orthographic space.
but still of course we define the mobius strip within an orthogonal space. can't have one without the other.
naturally, its nonsense to be objecting on the basis that a computer screen or a drawing are 2 dimensional anyways. to do so would be puerile, untintelligently nitpicky and constitutes nothing more an insult to everyone else's intelligence here. everyone knows that already and thats not the point.
a 2d object in 3d space is still 2d. i think that's why flatland was brought up. a single dimension point is a point, and probably can't conceive of the second dimension. a 2-d plane can't perceive the third dimension. if you put the 2d plane in a different environment, it still can't perceive the 3rd dimension. that's because a 2d plane bent in 3d space still only has 2-dimensions and thus should be referred to as 2d.
read my post, retort therein.
i read your post. i thought i was retorting therein.
you're saying that the mobius strip has to be considered a 3-d object because it exists within 3-d space. at least i think that's what you're saying.
but it's still a 2-d object. i'm suggesting that a 2-d object can exist in 3-d space. if you're describing a point on the 2-d mobius strip, you shouldn't need the 3rd identifier. pick an origin, and describe how many units away along the length and how many perpendicular to the length, and you will be able to find the point.
also, i think you're saying this:
if a 2d strip is spread flat, then it is 2d
if a 2d strip is spread flat, then it is not a mobius strip
conclusion: if it is 2d, it is not a mobius strip.
your premises might be correct, but [if a then b, if a then c, therefore if b then c] is flawed logic.
if we build a huge box of a huge mobius rollercoaster and then get on board with a camera shooting interval shots and then attenmpt to combine those shots along a 3d mobius trajectory
A 2D surface has no third dimension (Z, or height). A roller coaster moves through 3 dimensions. Therefore a 2D roller coaster is impossible.
no, that not what i said. another way:
for a moment, forget that a mobius strip does not have a thickness. by overlooking this, you are not contradicting the nature of the mobius strip, you are merely overlooking one of its aspects in order to bring out other facts about it.
now, look at the mobius strip not as a surface per se, but as a collection of points constituting this thing we call mobius strip. in other words, identify the mobius strip by all its coordinates.
these coordinates will have variable z-components
we know that in 2d x-y space, the z-component is invariable.
therefore, this set of points (that we used to call mobius strip) exist in three-dimensional disposition to each other
therefore, this set of points = mobius strip is geometrically transformed three dimensionally.
now, this property of three-dimensional transofrmation is instrinsic to a mobius strip; it is not intrinsic to a laid out flat strip with invariable z-component. it is thus not a mere adjective that renders it a specific subset of flat objects; it defines the nature of the mobius strip.
this is what i meant when i said : "so a mobius strip cannot be defined merely by its 2D surfaceness" & "we define the mobius strip within an (here i meant to say 3d orthogonal) orthogonal space. can't have one without the other"
both times, i used the word define because its not a description of a mobius strip to say that it occupies 3d space, it is within the nature of the mobius strip to exist and define itself within 3d space.
now, all 2d entities can exist in 2d space and in 3d space, that is true. and a 3d entity cannot exist in 2d but in 3d (lets not drag in more d's) but a mobius strip by virtue of the intrinsic requirement to transform itself 3 dimensionally, to distribute its points and parts three dimensionally, cannot solely exist in 2d space.
for the sake of just being more interesting, i also introduced the idea that if the mobius is taken as its own denominator and we flip over the dependency of the mobius strip on three-dimensional space so that orthogonal space is viewed from a mobius (mobial?) reference. the rollercoaster analogy. and if we do travel along a mobius strip (ie we imagine we are the mobius strip moving to generate itself), then yes, one can posit that it is a 2 dimensional space but then:
1. this is not how you define a mobius strip; its an interesting consequence of having first defined and thought of the mobus strip three dimensionally.
2. this negates an outer space altogether (let alone discussion of 2 or 3 dimensional space) and therefore any outer reference/observer. a mobius strip is only 2d dimensional within its own self awareness (and ours if we were imagining ourselves as this mobius strip generating itself). but this is not possible in our external appreciation and understanding within the isotropic space it exists in outside us.
allow me to continue this, because i know someone will jump on the wrong wagon just for the sake of it:
"and a 3d entity cannot exist in 2d but in 3d (lets not drag in more d's) but a mobius strip by virtue of the intrinsic requirement to transform itself 3 dimensionally, to distribute its points and parts three dimensionally, cannot solely exist in 2d space."
continues: now, because the mobius strip cannot exist in 2d space and can only exist in 3d space, it must therefore be a 3d entity (wherein the 3d-ness is instrinsic to its nature and not a divorcable property that can render it still a mobius strip in lack of said property- recall arguement above about necessary defining invariability of the z-component).
tammuz, That's exactly what you said, quoted verbatim.
identify the mobius strip by all its coordinates.
these coordinates will have variable z-components
Every point on a mobius strip can be defined by 2 coordinates, X / Y. It has no Z coordinate. Thus a mobius strip is 2-dimensinoal.
You can't spread a mobius strip flat because it already is.
Miles, you are invariably wrong, smug and never actually interested in an exchange of ideas and you deem yourself above learning and above others. not a nice combo.
http://paulbourke.net/geometry/mobius/
I exist in infinite dimensions, but my sensory modalities don't register all of them.
I think Miles did mention that he's feeling particularly snarky lately, but I don't think he's averse to hearing/learning from others' opinions.
lately? no, his responses are typically that way.; he says he' snarky lately as someone would deliver a fashionable statement. and anyway i dont give a shit whether its lately or earlier. i see how he reacts to genuine interest in discussion and questions and thats enough for me. he esteems his personal attitude over and above the discussion and its relevance. i can do worse but this is enough; i don't want to get metaphorical.
i've already stated how one could define any point on a mobius strip with only 2 dimensions right? was that unlcear, or did i state it wrong?
you could set an origin wherever you want. it is certainly possible to set your origin such that a 3rd descriptor is required to describe the mobius strip. however, as i explained before, you can say all points are bound to the strip and because of that can be described with only 2 dimensions if you choose to see it that way.
if you lay your strip flat it is 2-dimensional. but, you can tilt the 2-d shape up, and then suddenly you need a 3rd dimension to describe all points. or, since the origin point is arbitrary, you keep the 2-d shape as-is and tilt your axis. then you would need a 3rd dimension to describe the points, even though nothing changed with the strip itself. it's not necessary in that case to add the 3rd dimension, but you can do so if you choose. same with a mobius strip. you don't need the 3rd dimension to describe any point on the strip, but if you want to complicate things you're free to do so.
you seem to be hung up on the idea that a mobius strip has to be 3d if it exists in 3d space, or something like that. didn't rem say "fuck context?" can't we apply that here?
some would say you do not exist in infinite dimension. some would say there are only 11 dimensions.
"didn't rem say "fuck context?" can't we apply that here?"
only if you yourself for yourself want to say that to yourself. i see that the context defines, to us, the mobius in the mobius strip. rewind repeat (or wind and pleat in this case).
tammuz, I've yet to be demonstrated wrong (so far in this thread at least), have indeed been accused of being smug (by those who can't prove me wrong) and would truly enjoy seeing you make a coherent statement about möbius strips and dimensionality that challenged my perceptions and opened my mind.
Instead of getting all huffy why don't you try to work through it constructively? Assume I'm an idiot (as it seems you already do) and explain it as simply as you can, because so far what you've said seems like utter nonsense to me.
the mobius strip is defined by a specific (spacific?) trajectory of a 2 dimensional surface within 3 dimensional space
False. A möbius strip is defined by having a single surface and a single boundary, and by being non-orientable. It can be defined as an object in 3D space, but that doesn't make it three dimensional any more than defining a point in 3D space makes the point 3D.
The surface of a sphere is two dimensional as well (and for some even harder to visualize as 2D), with the interesting property that traveling in any direction will eventually return you to your starting point. Some propose that the universe is curved and has this very same property. The möbius strip is interesting because when traveling in one direction you are as likely to return to your starting point as you are to hit the edge of the world (universe?).
no, you are smug because you are. i'm not huffy or puffy - that description you apply to me is evidnce of your smugness. your replies throughout archinect show what you are and thats all it takes. even this "(i) would truly enjoy seeing you make a coherent statement about möbius strips" is a belittling response.why do you think that this is about giving you joy; why do you assume such a superior position, a personable center that is to be pleased or dismayed? i'm not here to give you joy or to win yur admiration. either talk straight at me or don't talk to me at all but don't talk down to me.
i did not say you are an idiot. your other features background any requirement to make an evaluation on that front. i am also not saying that you cannot be idiotic.
you sayid prviously there is no z component. well, there is a formula that shows that there is. you don't even acknowledge this. why? because this is secondary to wanting to seem right. again, smug...and self righteous, wrongfully.
"A möbius strip is defined by having a single surface and a single boundary, and by being non-orientable" - this in no way contradicts the other defining charecteristic: "by a specific (spacific?) trajectory of a 2 dimensional surface within 3 dimensional space" which is the formula for the mobius strip defined by x, y and z. over and above all else, the most exacting definition is the formula. you will choose to contest the word define - i will tell you that your definition (ie from wikipedia) is insufficient. a cylinder suffices to fulfill the requisites you define the mobius strip with. the trajectory however is defined by the formula.
furthermore, you are choosing not to challenge my points concerning the imminence -and not causualness- of three dimentionality to the mobius geometry (curtkram did but in my opinion went into a non existent loop - this is my opinion with all due respect to curtkram's person) and therefore but chose to recycle your own conception. it is due to that imminence that i cannot perceive or qualify a mobius strip, let alone a sphere or cylinder as two dimensional surfaces. why? because of the variable third dimension component for one but more imoportantly becaucse we recognize these entities within a three dimensional coordinational space...we do not plot a mobius strip on its own surfaceness, on its own two dimensionality. of course i know you are incapable of seeing that this is - should be- an inherent quality to the nature of three dimensional surfaces (i.e. topology) where the third dimension is not the thickness of the surface but the behaviour of the surface three dimentionally.
i think thats about it.
...third dimension is not the thickness of the surface but the behaviour of the surface three dimensionally.
This is a nice nugget for me to chew on.
nuggets are not good for you, go watch Jamie Oliver x-naked chef. sigh...
Mods, please rename this thread "Math is Hard." TIA
what does the "non-existent loop" mean? are you specifically referring the mobius loop, and you're saying that the 2-dimensional mobius strip doesn't exist? or my argument doesn't exist, and also is a loop?
i'll loop one more time. if a point is bound to the surface of a mobius strip, it can be described with 2 numbers (x-axis, which might run along the length, and y-axis, which may be perpendicular). those 2 numbers are the 2 dimensions that exist in the world of that mobius strip. all points possible on that strip can be described with those 2 parameters. the third axis is non-variable. it is always 0 relative to the surface of the strip.
i understand you are defining a mobius strip as an object in 3-dimensial space. i understand you feel the need to define a third axis relative to a surface in that 3-dimensial space that is not the surface of the mobius strip. generally speaking, you think all coordinate planes have to be orthogonal, and you can't have an 'x-axis' that follows a surface of some sort.
but time-space is not orthogonal. that is what einstein's theory of relativity set out to prove, at least to my understanding of it. so to view space as something that can only be orthogonal based on your current point of observations does not, in my opinion, reflect all possible, rational, and realistic viewpoints. let's think outside of euclidean geometry and let the x-axis have some breathing room to move around a bit. here is where i step outside of my comfort zone and into things i know nothing about, but that is sort of what Riemannian geometry is about. unfortunately, Riemannian geometry is well outside my core competence.
you can have your 3d mobius strip. i'm not saying your wrong, given the context you defined. however, i don't think you've addressed how a 2-dimensional mobius strip is impossible based on the frame of reference i gave. if i understand you correctly, you simply dismissed it as non-existent.
correction: actually, i take back that a cylinder is nonorientable. it of course is. but otherwise, the logic stands as above.
So the nugget that requires chewing is: can a surface "behave" in a way that isn't two-dimensional? For the example with which I started this conversation, that of a 2D being that only understands 2D, it can't. But in my perception of it, as a being that experiences more than two dimensions, it can.
This whole conversation is definitely helping me feel more comfortable with the concept of multiple dimensions that we can't (yet?) perceive.
i think you're asking if you can bend a piece of paper, which of course you can.
does the paper cease to be what it is once bent? in my opinion, no. if something 2-dimensional lived on that piece of paper, would it notice it had been bent? i doubt it.
^ Exactly. And thus a 2D surface like a sphere, mobius strip, etc. remains two-dimensional however we in the third dimension perceive its shape. To the hypothetical inhabitants it would all be the same. I'll have to look, but I think that in Abbot's Flatland if you journeyed far enough you would return to your original position, which make Flatland the surface of a sphere (or at least gives it those properties).
I keep flashing on gravity wells and the distortion of space time and relativity but can't seem to make an analogy that fits. In our three dimensional world Euclidean geometry eventually fails yet we are only able to perceive that indirectly. Maybe that's what made Flatland so good although I'm still at a complete loss when it comes to tesseracts.
Quondam, I left off because of what I perceived as your inability to separate the concept of a 2D surface from a 3D shape. As for proving assertions, that is a two-way street. I backed some of mine with various references (linked), while yours are wholly unsupported. Thus the moon is green cheese.
tammmuz, you were so incensed about my rebutting your ideas that you copped a nasty attitude. Neither your inability to refute my points or the inability to make yours clear are justification for verbal abuse (smug, etc.). Now you're even more incensed about my pointed response to your rudeness. Go figure. Bottom line: your inability to make an argument that I can understand - or conversely my inability to understand whatever argument you are trying to make - is no reason to be a turd. Yes, that was slap. Now get over it and let's have some dimensional fun.
So there are all these posts about China and projects in China. What side of the street do they drive on? The left side, like the UK? Japan does as well, no?
I once was watching a program on Dubai and, from the aerials of their roadways, they drove on the right side of the street. I was stunned.
I don't do countries where they drive on the left side. I wouldn't rent a car. Actually, even if they drove on the right side, and they had a language I didn't speak or were corrupt and dangerous, I wouldn't rent a car there either. In some places you need to rent a car, because public transit offers poor frequencies, like in Sicily or on the island of Mallorca. In other places, you don't need to rent a car, like in Rio or B.A.
No bullshit, I can't do it. But it is a mathematical function known as spherical trigonometry that is commonly used in navigation.
Never attach your ego so closely to a position that when the position fails your ego goes with it.
I'll jump back into the dimension mess later, but right now I wanted to share that I did in fact FAIL the stupid Illustrator certification test that I was forced to take, although it is NOT needed for me to do my job. I got a 55% and only needed a 66% to pass. I was SOO close! I figured I'd get maybe a 25. I'm impressed that I winged it so well, but now I'm annoyed that I was ELEVEN points away. Effing ELEVEN! Damn.
Ok, continue...
^
So, when can you, or will you, take it again?
I was not rude to you Miles. I was responding to a build up of nasty one liners from you that obviously come from a reserve of disrepect you have towards other people's contributions here. If i call you distateful and smug, this is becauase I find you you distateful and smug, Miles Jaffe.You on the other hand go on to suggest that I'm a turd. Exactly who is being rude then? For the sake of not insulting your person, I myself did not reach out to such demaning metaphors.
Also, you did not rebut my arguments, you presented yours. You cannot say that you rebutted my arguments and at the same time state that "your inability to make an argument that I can understand - or conversely my inability to understand whatever argument you are trying to make". if you don't understand, then obviously you cannot have rebutted the content you lack comprehension of. ow can you be rusted with mathematics if you cannot be trusted with mere common sense? There is no shame in that. Simply admit your lack without trying to belittle others for what you lack.
Unrelated but the new app for ikea that lets us look at the furniture in our own rooms in realtine and to scale is seriously cool shit.
curkram, in one of your latest posts, you say:
"ll loop one more time. if a point is bound to the surface of a mobius strip, it can be described with 2 numbers (x-axis, which might run along the length, and y-axis, which may be perpendicular). those 2 numbers are the 2 dimensions that exist in the world of that mobius strip. all points possible on that strip can be described with those 2 parameters. the third axis is non-variable. it is always 0 relative to the surface of the strip."
I said previously almost exactly that: "now, mobius strip relative to itself, its own trajectory is 2 dimensional since by denominating it to itself"
however, this had been preceded by (my)statement: "so a mobius strip cannot be defined merely by its 2D surfaceness."
in a following post and digressing on that speciic thought: "for the sake of just being more interesting, i also introduced the idea that if the mobius is taken as its own denominator and we flip over the dependency of the mobius strip on three-dimensional space so that orthogonal space is viewed from a mobius (mobial?) reference. the rollercoaster analogy. and if we do travel along a mobius strip (ie we imagine we are the mobius strip moving to generate itself), then yes, one can posit that it is a 2 dimensional space but then:
1. this is not how you define a mobius strip; its an interesting consequence of having first defined and thought of the mobus strip three dimensionally.
2. this negates an outer space altogether (let alone discussion of 2 or 3 dimensional space) and therefore any outer reference/observer. a mobius strip is only 2d dimensional within its own self awareness (and ours if we were imagining ourselves as this mobius strip generating itself). but this is not possible in our external appreciation and understanding within the isotropic space it exists in outside us."
so, you see, this loop of a thought has aleady been entertained and so ironed out (on my part and for myself).
the latter half of your post (dragging einstein into this) is spinning of into the vacuous infinity of "huh? what the heck" :oP
i will also say that fundamentally, this disagreement is over semantical and descriptive accuracy - we all know to some extent what a mobius strip is and mathematicians know much more than we do. but on the issue of semantics: on your part, you see the surfaceness of the mobius strip, a warped 2D entity. i understand that - this is why i said a mobius, if related to its own self (its own surfaceness), can be ,seen as a 2D entity. but, for me this is insufficient semantics, insufficient qualifier of the mobius strip. it is methodically generated as a formula in three dimensions, or to put it nearer to your way of thinking its 2-dimensionality has been acted upon by a z-component factor. thus for me, the semantics cannot stop at describing its own condition of corporeality (ie 2 dimensionality) but also its condition of formal existence (3 dimensionality). so, its a choice to stop at describing it two dimensionally (and therefore there is no difference to a rectangular horizontal surface) or to be recognize its formal existence within a 3 dimensional world and therefore call it a 3 dimensional entity. if you look at its infinitismal thinness, then yes 2 d (a sort of partial tautological description). if you look that the coordinates are dispersed three dimensionally, the yes 3d. mathematically, the latter view is more relevant, in my estimation.
OK, yeah, that IKEA app is pretty cool.
Assuming Flatland is a finite 2D surface as described by the surface of a sphere, a right angle is always < 90°. Thus the natural geometry is different from Euclidean space and the radius of the sphere would be a natural constant. An analogy would be a right angle in Euclidean space or the speed of light in a relativistic universe.
The topology of a mobius strip presents a different problem as an equivalent constant is seemingly non-existent, at least according to the the model we visualize that varies from concave to convex and appears to eliminate any obvious natural constant (so far as I can see). I won't even get into the single boundary, which makes me imagine a flat disk Earth with edges you can fall off of.
And now for something completely different.
"a build up of nasty one liners"
My two replies to tammuz's posts that justify his being a jerk:
1) A 2D surface has no third dimension (Z, or height). A roller coaster moves through 3 dimensions. Therefore a 2D roller coaster is impossible.
2) tammuz, That's exactly what you said, quoted verbatim.
Dude, you've got a problem. I'm not interested. Have a nice day.
Miles Jaffe, again you are wrong. these are not the replies i referred to. unfortunately, i come across you on other occasions and i know what sort of sordid online character i'm confronting here, one possessed by a smugness neither supported by characteristic intelligence nor by a wealth of knowledge .you are a bitter much older person who bickers with and tries to play cat and mouse with youngsters who ask quite innocent sounding questions; your ego and pickled attitude estimated above all else. you deride people with different valid opinions wanting only to prove yourself correct irrespective of the value of discussions and irrespective of how stunted your intellectual projection and empathy are and how circuitous your argument is whilst ignoring valid points raised apropos your non-argument. and i don't particularly care if you are interested or not. now off with your needlessly bloated head. and i don't care what sort of day you have.
start a mobius strip thread and leave us out of this.
1. i think it happened already; its a bit late, your request.
2. thread central is a topically open thread.
3. are you proprietor?
3. kindly ignore it if you don't like. i know i don't come here for pleasurable simulation within the usual thread central clique but i don't think its in your place to make someone not welcome, vado retro.
consider yourself kindly ignored.
kewl
sorry to hear about your troubles jla. probably should keep in mind that, while the answer often does lie at the bottom of a bottle, the problems tend to come back the next day (with a headache). it's better to be busy than not.
and look at the bright side, you have many more weeks to live through! in a few years, this might not even be in the top 10 of bad weeks :)
jla-x sorry to hear about your troubles.
re: the 2d/3d discussion(s), while interesting what vado said...
yeah, get this shit out of here, or else i'll start posting photos of guys junk in mason jars; and you know i'll do it.
Aww, I was enjoying - immensely, or should I say infinitely? - the discussion of Mobius strips until the namecalling started.
jla-x, can you share any of the troubles? I'm dealing with supreme embarrassment at how far behind I am on several projects. I just keep begging understanding of people, and try to figure out how to say NO to some things that really shouldn't be my responsibility.
jla, mediation works for me. If you can sit quietly for 20 minutes, eyes closed, doing nothing but breathing, a lot of the crap in your head will play itself out and leave you in a much better place. Good luck.
Re 2D: Interesting things about the surface of a sphere as a 2D finite universe:
If you travel halfway across the universe to the east, then halfway to the north, you will return to your original position upside down. "Hi honey, I'm home!"
Also the inside of the sphere makes just as good a model of a 2D universe as the outside.
jla-x: Hang in there. That roller coaster called life often sucks.
@Miles I assume you meant meditation? though depending on the problem mediation might be what is needed....
So much for spell check ... but maybe you're right. Karma is funny that way.
see, if you sat down and meditated a bit it would have cleared the junk out of your head and you would have been better prepared to proofread.
Better prepared, maybe. Better able, I doubt it.
I've got lots of junk. I'm looking for the mental equivalent of a 40 yard dumpster.
I distrust all politicians for the most part and am neither a liberal nor a conservative, but this politician tops the list. And what a hypocrite - he wants Obamacare defunded while his adopted home state of MA has a similar program. And MA was one of the first to have same-sex marriage, as one would expect of liberal New England. That means that he knows what side his bread is buttered on, in terms of his constituency, and will deviate from what his religion and Republicanism tell him. However, on a grander scale, because the U.S. has a lot of "red" territory, he can be who he really is. I seriously never wanted to see this guy again, but his avarice gets in the way. I'm wondering if this next election is going to be Hillary versus Palin or Romney, in which case I will abstain. I've done that before. I don't know if talk that Christie of NJ would go Dem and jump into the race is nothing but mere b.s., but if he lost weight to make sure he remained in good health and ran as a Dem, I'd have to think that one through and he might get my vote. At any rate, here's someone, and a family, we don't need anywhere near the White House:
http://firstread.nbcnews.com/_news/2013/08/07/19914511-romney-re-enters-gop-fray?lite
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