i would have clicked 'most important thread topic ever' but that isn't an option, so i suppose this goes in 'random.'
it's a 2d object in a 3d environment. if you choose to describe a point on a mobius strip with 3 dimensions instead of the requisite 2, you can do that. you could probably explain it's existence in 4-dimensional space too if you're clever enough. use as many dimensions you want, it's still 2d.
I've seen numerous examples of mobius strip topology being used in envelope and overall building form, but does anyone have other examples of how it's used in building components, program, etc..?
There's the mobius house from UNStudio, but from what I understand it's mostly symbolic.
i agree trffl. quondam, as you can see, thinks that a mobius strip can only be thought of as a 3-dimensional entity. he says "one way to represent the mobius strip..." and concludes "mobius strip requires 3 dimensions." not an exact quote, but you can refer to above.
toaster, i don't think this has anything to do with architecture. i could be wrong though.
Well, if you embed in 3d then you can describe it with three coordinates. Quondam is just pasting the wikipedia explanation of how you do that. If we want to get specific here we should be using real math terminology.
Now that I see the discussion you guys are having in thread central... The problem is you are all not using the agreed-upon definitions for things and abusing terminology, that is why you are disagreeing about this stuff (or making very general statements that don't really mean anything).
i would say it could be either. i also think, if it can be either, my viewpoint of referring to a mobius strip as a 2d object is an accurate representation and not "wrong." i'm pretty sure quondam and tammuz are saying i'm "wrong."
All three factors in Quondam's first formula are constructed out of two data points: u and v. The parameter u runs around the strip while v moves from one edge to the other.
Polar coordinates are also a two coordinate system: radial (r) and angular (θ).
Thus a surface that can be described by either system can also be described as topologically two-dimensional. Which does not prevent you from mathematically describing such a construct in more dimensions, which does not make the object described more dimensional.
Einstein was very careful to construct his thought experiments from a clearly defined point of view, which is why it is called relativity. Everything is relative to the time/space location of the observer. Thus we return to Flatland.
More seriously, you guys are arguing about this in a really bizarre way. While it is true that, if you disregard everything except the object itself and treat it in isolation as a non-Euclidian manifold, a Mobius strip can be thought of as a two-dimensional object, this is akin to the old anecdote about economists and spherical cows. It's a trivial result that illuminates nothing of importance. Similarly, re-framing the description of the shape using UV coordinates simply formalizes the same error for the sake of expedience: ignoring everything outside the object itself. It's a handy way to manipulate objects abstractly, and essential for manipulating them at the computational abstraction level, but it doesn't tell you anything about the objects as they relate to everything else.
Back here in reality, a Mobius strip, while it has some fun properties due to its peculiar topology and construction, is still a three-dimensional object (esp. if you actually cut a strip of paper and make one). Arguing about whether or not this is true is an example of the common error of mistaking abstractions for reality. When you draw a building elevation, it's a highly abstracted representation in two dimensions of a three-dimensional object, not the object itself. Abstractions are great for focusing attention on important characteristics, describing complex things in simple terms, and cognitive efficiency, but they aren't real. Continuing with the building elevation example, it is a distressingly common conceptual failing among architects that we get so absorbed in thinking about building design via two-dimensional symbolic abstractions (plan, elevation, section, detail, etc.) that many of us become completely separated from the reality of three-dimensional building and space. I am convinced this is a big part of the reason most modern architecture, as produced by rank and file architects, is shit. It's all abstraction and very little reality.
You can mathematically describe a Mobius strip as a continuous, one-faced, two-dimensional warped planar manifold, but that doesn't mean it's really a two-dimensional shape. The math is still just an abstraction: a map, not the territory. Same goes for "fractal" geometry too. You can fill space with one- and two-dimensional objects. By doing so, you make the conjunction of those shapes three-dimensional even though the resulting configuration is not solid. Abstracting the geometric relationships and describing them in terms of fractional dimensionality helps us consider some of their properties differentially (structural open-ness, recursive structure, etc.), but it doesn't make the space-filling result anything less than three-dimensional.
tl;dr ... this thread is now about psychologists in wet t-shirts. Does anyone have those spring break pics of Sigmund Freud handy?
i don't think your abstract to concrete fallacy applies here. it's not a logic jump.
first of all, abstract thought is a good thing for our profession. being able to see beyond things such as a 2d elevation to the actual building it represents is a good thing, and practicing this sort of abstract thinking is a good thing. ignoring the obvious because your narrow point of observation is limited to certain objects is not good.
to your logic fail, you seem to be suggesting that since thinking of a mobius strip as 2-dimensional is abstract, that makes it not real. an example of the abstract to concrete fallacy is that a map is not the topography it represents. the map is an abstract representation of the topography. if i talk about a map, that doesn't mean the topography doesn't exist.
to simplify what you're saying g, the mobius strip is a non-euclidean manifold, thus it is 2-dimensional. however, in your current frame of reference, the world around you operates in 3-dimensions, and with euclidean coordinates. therefore, since your point of observation is in 3-dimensions, everything else must be in 3-dimensions, including mobius strips. those crazy physicists who talk about extra-dimensional space are just morons who can't understand physics as well as an architect, who's physics education likely stopped at statically determinate beams.
from wikipedia (where you got your previous math quondam) - this is under 'topology' around half way down:
"The Möbius strip is a two-dimensional compact manifold (i.e. a surface) with boundary. It is a standard example of a surface which is not orientable. In fact, the Möbius strip is the epitome of the topological phenomenon of nonorientability. This is because 1) two-dimensional shapes (surfaces) are the lowest-dimensional shapes for which nonorientability is possible, and 2) the Möbius strip is the only surface that is topologically a subspace of every non-orientable surface. As a result, any surface is non-orientable if and only if it contains a Möbius band as a subspace."
but that's not the 'statically determinate beam' sort of math we learn in architecture schools. also, there are what they call 3-manifolds, which is a manifold that operates in 3-dimensions.
so anyway, if you're looking for 'math' that creates a 2-dimensional mobius strip in a 3-dimensional euclidean coordinate system, i think you're generally asking the wrong question.
curtkram, you say you're not engaging in the reification fallacy, but then you go right ahead and try to reify your abstraction of a mobius strip as really being two-dimensional when in fact it is not. You can describe it that way by abstracting away its third dimension, but that doesn't mean it doesn't actually have a third dimension. Non-euclidian geometry is simply a convention for thinking about topological problems using a set of mathematical conventions. Just like Euclidian geometry is also. But anything that fills space possesses three dimensionality. Now, that space (or coordinate system) may itself have a shape (be non-isometric, which is the fundamental assumption of all non-Euclidian analysis), but that doesn't really change anything. You're still just manipulating abstractions.
And none of that has anything to do with whether or not there are other dimensional axes within which the object exists. If we take change through time as fourth dimension, then a mobius strip is also a four-dimensional object. It did not exist before I tape the ends of the twisted strip of paper together, and the form of its existence may change or be bounded in the future (I could burn it because the sophistry being displayed in this thread enrages me so, move (translate) it around by carrying it somewhere, or maybe change its form and topology by crumpling it up and using it as the design model for a Gehry-esque monstrosity). It may have other n-dimensional properties as well. So what? None of that changes because I tell you that, no matter how much you try to reify your abstraction of a mobius strip as a two-dimensional object it still possesses three-dimensionality.
Anyway, that's not what this thread is about anymore.
still not reification. are you suggesting all abstract thought is a logical fallacy? you're using words to describe the thoughts you're trying to convey. those words are essentially an abstract representation of your thoughts. in that sense, all communication is a reification fallacy. that wouldn't make sense.
i'm not saying a mobius strip is 2-d, then abstracting away a third dimension. i'm simply saying it's 2-d.
time is not the 4th dimension. that's kind of ridiculous. if i have a dot that is constrained to a single axis (1-dimensional) and move that dot, it moves in time. time and space are both contained in all dimensions. the 4th dimension would include both time and space, as all other dimensions do. unless the 4th dimension behaves different for some reason. i can't see the 4th dimension, what with being constrained to a 3-d environment and all (or, if i can see the 4th dimension, i'm only seeing a shadow of sorts).
you can't build a 2-dimensional shape with studs and drwall. not even ghery could do that. not even if he had a metal skin instead of drywall. it doesn't work that way. my guess is that your assumption of what 2-d space is is incorrect, and that's why your trying to place a logical fallacy on an argument that was never a logic construct to begin with.
You guys keep jumping between topology and how you'd represent the thing in "real life" (sometimes within the same post). You should pick one or the other to talk about as they are rather different.
If you are indeed trying to talk about this from a "mathy" standpoint, it would also help if people stopped using vague analogies. Get formal.
In reality a mobius strip is not a three dimensional object: it is an imaginary object with properties that define it two-dimensionally. It is represented in three dimensional space by a twisted paper model. At any given point the model has two sides - a mobius strip has only one. Some seem to be taking the model literally instead of as an abstraction. This is real the reification fallacy: treating an abstraction as if it were a real thing.
The point about architects falling in live with their abstractions and losing sight of what they represent is entirely correct, and It's so prevalent that I'm assuming that this is what they are trained to do in school.
Quondam, you're trying too hard to win a battle that only you are fighting, and now you're reduced the discussion to using semantics to create contradictions where none exist.
But if that's where you want to go:
Conscious thought is imagination, which has no reality. Therefore you do not exist.
Imagination does not mean lacking physical reality, it means lacking factual reality. In reality (the world or the state of things as they actually exist) there are things that exist only in the imagination.
By the way:
A mobius strip as a mathematical formula CAN generate x,y,z (three dimensional) data.
It's a safe bet that other formulas can be used to generate data in N-dimensions. The basis of determining dimensionality is generally accepted to be the minimum number of coordinates necessary to specify a point on an object. Even an imaginary one. In reality.
Carried over from Thread Central, which seems to have moved on:
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.
This a cute thread to come back to after summer break.
Trffl, gual - it bothers me that people aren't distinguishing between physical space and abstract space. Are we really jumping from 'dimension' in affine/vector spaces to 'dimension' in general relativity, or am I still on vacation?
Is someone trying to construct a Mobius strip building? Can that please not happen?
calamist -- "Are we really jumping from 'dimension' in affine/vector spaces to 'dimension' in general relativity, or am I still on vacation?" yeah, it almost seems like they are. But I don't really understand what they're trying to talk about because nothing is defined properly.
mobius strip
i would have clicked 'most important thread topic ever' but that isn't an option, so i suppose this goes in 'random.'
it's a 2d object in a 3d environment. if you choose to describe a point on a mobius strip with 3 dimensions instead of the requisite 2, you can do that. you could probably explain it's existence in 4-dimensional space too if you're clever enough. use as many dimensions you want, it's still 2d.
that is all.
I'm not quite sure what you're getting at, OP. I'd call it a 2d manifold, you can embed it in 3d.
I've seen numerous examples of mobius strip topology being used in envelope and overall building form, but does anyone have other examples of how it's used in building components, program, etc..?
There's the mobius house from UNStudio, but from what I understand it's mostly symbolic.
i agree trffl. quondam, as you can see, thinks that a mobius strip can only be thought of as a 3-dimensional entity. he says "one way to represent the mobius strip..." and concludes "mobius strip requires 3 dimensions." not an exact quote, but you can refer to above.
toaster, i don't think this has anything to do with architecture. i could be wrong though.
Well, if you embed in 3d then you can describe it with three coordinates. Quondam is just pasting the wikipedia explanation of how you do that. If we want to get specific here we should be using real math terminology.
Now that I see the discussion you guys are having in thread central... The problem is you are all not using the agreed-upon definitions for things and abusing terminology, that is why you are disagreeing about this stuff (or making very general statements that don't really mean anything).
so is it 2d, or 3d, or both?
i would say it could be either. i also think, if it can be either, my viewpoint of referring to a mobius strip as a 2d object is an accurate representation and not "wrong." i'm pretty sure quondam and tammuz are saying i'm "wrong."
All three factors in Quondam's first formula are constructed out of two data points: u and v. The parameter u runs around the strip while v moves from one edge to the other.
Polar coordinates are also a two coordinate system: radial (r) and angular (θ).
Thus a surface that can be described by either system can also be described as topologically two-dimensional. Which does not prevent you from mathematically describing such a construct in more dimensions, which does not make the object described more dimensional.
Einstein was very careful to construct his thought experiments from a clearly defined point of view, which is why it is called relativity. Everything is relative to the time/space location of the observer. Thus we return to Flatland.
This thread should have been titled, "Dunning-Kruger Effect Runs Wild."
^ I'm picturing research psychologists in wet T-shirts.
More seriously, you guys are arguing about this in a really bizarre way. While it is true that, if you disregard everything except the object itself and treat it in isolation as a non-Euclidian manifold, a Mobius strip can be thought of as a two-dimensional object, this is akin to the old anecdote about economists and spherical cows. It's a trivial result that illuminates nothing of importance. Similarly, re-framing the description of the shape using UV coordinates simply formalizes the same error for the sake of expedience: ignoring everything outside the object itself. It's a handy way to manipulate objects abstractly, and essential for manipulating them at the computational abstraction level, but it doesn't tell you anything about the objects as they relate to everything else.
Back here in reality, a Mobius strip, while it has some fun properties due to its peculiar topology and construction, is still a three-dimensional object (esp. if you actually cut a strip of paper and make one). Arguing about whether or not this is true is an example of the common error of mistaking abstractions for reality. When you draw a building elevation, it's a highly abstracted representation in two dimensions of a three-dimensional object, not the object itself. Abstractions are great for focusing attention on important characteristics, describing complex things in simple terms, and cognitive efficiency, but they aren't real. Continuing with the building elevation example, it is a distressingly common conceptual failing among architects that we get so absorbed in thinking about building design via two-dimensional symbolic abstractions (plan, elevation, section, detail, etc.) that many of us become completely separated from the reality of three-dimensional building and space. I am convinced this is a big part of the reason most modern architecture, as produced by rank and file architects, is shit. It's all abstraction and very little reality.
You can mathematically describe a Mobius strip as a continuous, one-faced, two-dimensional warped planar manifold, but that doesn't mean it's really a two-dimensional shape. The math is still just an abstraction: a map, not the territory. Same goes for "fractal" geometry too. You can fill space with one- and two-dimensional objects. By doing so, you make the conjunction of those shapes three-dimensional even though the resulting configuration is not solid. Abstracting the geometric relationships and describing them in terms of fractional dimensionality helps us consider some of their properties differentially (structural open-ness, recursive structure, etc.), but it doesn't make the space-filling result anything less than three-dimensional.
tl;dr ... this thread is now about psychologists in wet t-shirts. Does anyone have those spring break pics of Sigmund Freud handy?
i don't think your abstract to concrete fallacy applies here. it's not a logic jump.
first of all, abstract thought is a good thing for our profession. being able to see beyond things such as a 2d elevation to the actual building it represents is a good thing, and practicing this sort of abstract thinking is a good thing. ignoring the obvious because your narrow point of observation is limited to certain objects is not good.
to your logic fail, you seem to be suggesting that since thinking of a mobius strip as 2-dimensional is abstract, that makes it not real. an example of the abstract to concrete fallacy is that a map is not the topography it represents. the map is an abstract representation of the topography. if i talk about a map, that doesn't mean the topography doesn't exist.
to simplify what you're saying g, the mobius strip is a non-euclidean manifold, thus it is 2-dimensional. however, in your current frame of reference, the world around you operates in 3-dimensions, and with euclidean coordinates. therefore, since your point of observation is in 3-dimensions, everything else must be in 3-dimensions, including mobius strips. those crazy physicists who talk about extra-dimensional space are just morons who can't understand physics as well as an architect, who's physics education likely stopped at statically determinate beams.
from wikipedia (where you got your previous math quondam) - this is under 'topology' around half way down:
"The Möbius strip is a two-dimensional compact manifold (i.e. a surface) with boundary. It is a standard example of a surface which is not orientable. In fact, the Möbius strip is the epitome of the topological phenomenon of nonorientability. This is because 1) two-dimensional shapes (surfaces) are the lowest-dimensional shapes for which nonorientability is possible, and 2) the Möbius strip is the only surface that is topologically a subspace of every non-orientable surface. As a result, any surface is non-orientable if and only if it contains a Möbius band as a subspace."
but that's not the 'statically determinate beam' sort of math we learn in architecture schools. also, there are what they call 3-manifolds, which is a manifold that operates in 3-dimensions.
so anyway, if you're looking for 'math' that creates a 2-dimensional mobius strip in a 3-dimensional euclidean coordinate system, i think you're generally asking the wrong question.
curtkram, you say you're not engaging in the reification fallacy, but then you go right ahead and try to reify your abstraction of a mobius strip as really being two-dimensional when in fact it is not. You can describe it that way by abstracting away its third dimension, but that doesn't mean it doesn't actually have a third dimension. Non-euclidian geometry is simply a convention for thinking about topological problems using a set of mathematical conventions. Just like Euclidian geometry is also. But anything that fills space possesses three dimensionality. Now, that space (or coordinate system) may itself have a shape (be non-isometric, which is the fundamental assumption of all non-Euclidian analysis), but that doesn't really change anything. You're still just manipulating abstractions.
And none of that has anything to do with whether or not there are other dimensional axes within which the object exists. If we take change through time as fourth dimension, then a mobius strip is also a four-dimensional object. It did not exist before I tape the ends of the twisted strip of paper together, and the form of its existence may change or be bounded in the future (I could burn it because the sophistry being displayed in this thread enrages me so, move (translate) it around by carrying it somewhere, or maybe change its form and topology by crumpling it up and using it as the design model for a Gehry-esque monstrosity). It may have other n-dimensional properties as well. So what? None of that changes because I tell you that, no matter how much you try to reify your abstraction of a mobius strip as a two-dimensional object it still possesses three-dimensionality.
Anyway, that's not what this thread is about anymore.
still not reification. are you suggesting all abstract thought is a logical fallacy? you're using words to describe the thoughts you're trying to convey. those words are essentially an abstract representation of your thoughts. in that sense, all communication is a reification fallacy. that wouldn't make sense.
i'm not saying a mobius strip is 2-d, then abstracting away a third dimension. i'm simply saying it's 2-d.
time is not the 4th dimension. that's kind of ridiculous. if i have a dot that is constrained to a single axis (1-dimensional) and move that dot, it moves in time. time and space are both contained in all dimensions. the 4th dimension would include both time and space, as all other dimensions do. unless the 4th dimension behaves different for some reason. i can't see the 4th dimension, what with being constrained to a 3-d environment and all (or, if i can see the 4th dimension, i'm only seeing a shadow of sorts).
you can't build a 2-dimensional shape with studs and drwall. not even ghery could do that. not even if he had a metal skin instead of drywall. it doesn't work that way. my guess is that your assumption of what 2-d space is is incorrect, and that's why your trying to place a logical fallacy on an argument that was never a logic construct to begin with.
You guys keep jumping between topology and how you'd represent the thing in "real life" (sometimes within the same post). You should pick one or the other to talk about as they are rather different.
If you are indeed trying to talk about this from a "mathy" standpoint, it would also help if people stopped using vague analogies. Get formal.
In reality a mobius strip is not a three dimensional object: it is an imaginary object with properties that define it two-dimensionally. It is represented in three dimensional space by a twisted paper model. At any given point the model has two sides - a mobius strip has only one. Some seem to be taking the model literally instead of as an abstraction. This is real the reification fallacy: treating an abstraction as if it were a real thing.
The point about architects falling in live with their abstractions and losing sight of what they represent is entirely correct, and It's so prevalent that I'm assuming that this is what they are trained to do in school.
that's a narrow view on "imaginary."
Quondam, you're trying too hard to win a battle that only you are fighting, and now you're reduced the discussion to using semantics to create contradictions where none exist.
But if that's where you want to go:
Conscious thought is imagination, which has no reality. Therefore you do not exist.
Imagination does not mean lacking physical reality, it means lacking factual reality. In reality (the world or the state of things as they actually exist) there are things that exist only in the imagination.
By the way:
A mobius strip as a mathematical formula CAN generate x,y,z (three dimensional) data.
It's a safe bet that other formulas can be used to generate data in N-dimensions. The basis of determining dimensionality is generally accepted to be the minimum number of coordinates necessary to specify a point on an object. Even an imaginary one. In reality.
Carried over from Thread Central, which seems to have moved on:
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.
Are breasts two-dimensional? A brassiere has two topological dimensions: circumference and cup. Is 2D the same as DD?
Also a brassiere can be twisted into a single-surface mobius strip.
And that's not two-dimensional thinking - it's one dimensional.
This a cute thread to come back to after summer break.
Trffl, gual - it bothers me that people aren't distinguishing between physical space and abstract space. Are we really jumping from 'dimension' in affine/vector spaces to 'dimension' in general relativity, or am I still on vacation?
Is someone trying to construct a Mobius strip building? Can that please not happen?
un studios already constructed the mobius house.
Janjaap Ruijssenaars thinks he's going to 'print' a mobius strip house with a very big 3-d printer.
of course, these architects took some liberties with the definition.
Topologically....
Getting back to architecture:
calamist -- "Are we really jumping from 'dimension' in affine/vector spaces to 'dimension' in general relativity, or am I still on vacation?" yeah, it almost seems like they are. But I don't really understand what they're trying to talk about because nothing is defined properly.
Relating this to architecture: Mobius Buildings and Bridges
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