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- Uber HQ headed to San Francisco's Mission Bay
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- Cooper Union graduates stage tuition protest at Commencement ceremony
- Frank Lloyd Wright's La Miniatura finally sells in Pasadena
- The real-life architecture of "Ex Machina"

- Stop the presses: Paul Goldberger's take on critical relevance in the social media age
- One-Night Stand LA titillates, but leaves you wanting more
- There are tons of architecture job openings these days. Why aren't you hired yet?
- Art + Architecture: Andreas Angelidakis between the monumental and the particular
- How Architects Can Help Nepal (And Learn From Past Disastrous Mistakes/Successes)

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## Thread Central

Miles, as much as you want to contort the issue, I stand by all that I have said. You're mixing several different logics (planetary scale, curved space--yes between galaxies, relativity, perception) trying very hard to prove me wrong. All I'm doing is standing by basic geometry.

Anyhow, what you are saying is only further proof that what I've been saying all along is correct, ie, the mobius strip is a three-dimensional entity. Whether we percieve it that way or not, its surface is curved three-dimensionally, just like the surface of a globe is curved three-dimensionally (we're just too small relative to it to see it that way). If we were 1000 miles tall we'd see the three-dimensionality of the earth's sphere quite readily.

Plus, I'm not sure what the reference point to register movement is within the wiki example above of being able to move only in 2 dimensions on the surface of a sphere, because when the reference point is the center point of the sphere or the reference point is a plane (like a plane slicing through at the equator), then all movement along surface of the sphere would register a steady differential of three dimensions.

@observant - are you getting paid to post linkbait from msn?

@quondom - that survey drawing (and story) is lovely. thanks for sharing...

^^

No, I always read city, state, and country QOL rankings and hipsters annoy the hell out of me. So no.

Quondam:

I'm not mixing "logics", you're mixing reference points.

In basic geometry, any point on a planar system can be described by two coordinates: X and Y. The surface of a sphere has the exact same properties - any point on it can be described by two coordinates: latitude and longitude.

If you were constrained to a 2-dimensional surface - a plane, the surface of a sphere or a möbius strip - there would be no third dimension. Thus referencing the 2-dimensional surface of a sphere from the center of a 3-dimensional sphere, or anywhere not on the surface, is impossible because that dimension does not exist. It is an extrinsic point of view.

A möbius strip is not a 3-dimensional entity, it is an imaginary construct consisting of a single sided surface with a single boundary (edge). It is only the representation of it in the form of a model that you see as 3-dimensional.

@observant et al,

I happen to be family friends with the hipster from NYC who got attacked by a raccoon.

Also I will second the stippling is beautiful

Nam, are you really?!!

Sarah,

Yes really. Kind of weird...

Miles, the combination of latitude and longitude itself constitutes three dimensions.

I'm not at all sure what your mental problems are, but you seem to be confusing imaginary constructs with reality. You yourself have already said that a flat surface isn't really flat; it's actually curved (ie, 3-dimensional). Following that logic, a two-dimensional being on a flat surface (that is actually curved) is then also actually curved (ie, 3-dimensional). You then say that because this (so-called) 2D being on the actually curved surface percieves the curved surface as flat, then the curved surface actually is flat. That's where you make the mistake of accepting an imaginary construct (ie, percieving a curved surface as flat) as the reality. The perception of the surface of our planet as a flat plane does not in fact change the surface of our planet into a flat plane. The surface of our planet is three dimensional.

Your argument above begins with an imaginary construct: "If you were constrained to a 2-dimensional surface - a plane, the surface of a sphere or a möbius strip - there would be no third dimension." You are mistakingly accepting this imaginary construct as reality.

As an imaginary constructa möbius strip is a single sided surface with a single boundary (edge).In realitya möbius strip is a 2-dimensional strip that has been twisted into a three-dimensional curved surface with its two ends of now opposite orientation joined.That's it for me here on this subject.

a 2-dimensional strip that has been twisted into a 3-d surface is still a 2-d strip.

which is still imaginary, because we perceive pretty much everything in 3 dimensions (and ignore extra dimension, assuming there are extra dimensions). on a globe, you can describe a certain point on a 2-d plane with latitude and longitude (2 dimensions). but at a smaller scale, there is a 3rd dimension (elevation above sea level, or height).

everything is imaginary at a theoretical level.

Tout est dans la tête, vous savez.

@observant et al,I happen to be family friends with the hipster from NYC who got attacked by a raccoon.I see. Six degrees of separation at work. I don't want people getting attacked by raccoons or other animals, but hipsters and granolas aren't my cup of tea, so the headline grabbed me. They reflexively don't like me. They incorrectly assume I am the establishment. This chick did something to provoke the attack, per the article. She could have just walked around and past the altercation between the two raccoons. We periodically have them around and I might see one late at night when pulling in. I assumed they were the size of a cat. They are much larger.

Nam, it's crazy, and yet remarkable beautiful to think that our wide world is so connected and small.

And boys, 2d or 3d, doesn't really matter. Everyone knows you really never need more than a handful.

curtkram, a 2-dimensional strip that has been twisted into a 3-d surface is a 2-d strip that has been twisted into a 3-d surface. You can't just deny that the plane has been twisted into a three-dimensional surface. When a plane is distorted into a surface, it changes from requiring a 2 dimensional coordinate system to requiring a three dimensional coordinate system.

As to the two point coordinate system of latitude and longitude. The reason it is just two points is because its working with the hypothetical given that all ponts on the sphere surface are equidistant from the center of the sphere, thus making the third coordinate zero, generally sea-level. In reality, an exact GPS reading of any point on the Earth's surface would have three coordinates, latitude, longitude and distance from the Earth's center (or distance plus or minus sea-level).

Right now my brother and I are standing at the same exact point of latitude and longitude down the the finest calibration of reading, yet we are 20 feet apart. How can we be at the same exact point and yet still be 20 feet apart? Answer: I'm standing on the roof and he's on the ground floor. His coordinate is 40.000000, 76.000000, 0 and my coordinate is 40.000000, 76.000000, 20 and the bird stuck in midair overhead is at 40.000000, 76.000000, 100 and the secret hermit living in the cave below is at 40.000000, 76.000000, -30.

I only posted again because I wanted to provide a correct understanding of the latitude and longitude system. Remember too that latitude and longitude are magnitudes of degrees. Measuring by degrees is a radial system, thus every two point degree coordinate on a sphere surface by default has a third radius dimension.

but we all live in a multi-dimensional world even though we only perceive 3 dimensions. That is all our brains are capable of. It is perception, not coordinates, that makes 3 dimensions.

did you actually stand on a roof in the building your brother is in to make that point?

As to the two point coordinate system of latitude and longitude. The reason it is just two points is because its working with the hypothetical given that all ponts on the sphere surface are equidistant from the center of the sphere, thus making the third coordinate zero, generally sea-level.Incorrect.

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it (for example, the point at 5 on a number line). A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and its longitude). The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.sourceThus the surface of a sphere can be described as two-dimensional. If your brother moves inside or outside of the surface of the sphere (as measured by altitude) he is in 3-dimensional space, exactly the same as if he moved off the surface of an two-coordinate X / Y plane into the Z direction.

latitude and longitude are magnitudes of degrees. Measuring by degrees is a radial system, thus every two point degree coordinate on a sphere surface by default has a third radius dimension.Incorrect.

What you describe as a radial system is called polar coordinates. This two-dimensional system can be used to describe any point on a plane from any other point in the form of direction (degree) and distance. What you perceive as the 3-dimensional shape of the 2-dimensinoal plane is irrelevant.

Miles, according to what you are saying, then the distance between point (40N, 76W) and point (45N, 82W) on Earth is the same as the distance between point (40N, 76W) and point (45N, 82W) on the Moon.

Of course, the distance between identical latitudes and longitudes of the Earth and the Moon is not the same because the Earth and the Moon do not have the same radius. The distance between any two polar coordinates is relative to the radius.

The two surface entities in question are separate and distinct topologies independent of each other, just as they would be in comparison to a flat plane or a möbius strip.

Also, in the system you describe, how do you determine the difference in length between these these two sets of global positions: (40N, 76W to 40N, 77W) and (60N, 76W to 60N, 77W)?

So Miles, you're saying that the Moon is not a "sphere with a dimension of two because two coordinates are needed to specify a point on it." Please enlighten me then as to what the Moon is.

Green cheese, of course.

I love this conversation so much and, typical for me, I see both sides as equally valid. You could say I'm wishywashy, or you could say I'm magnanimous.*

*New argument: is liberty bell magnanimous, or just wishywashy?

Wishinanimous. Magnaniwashy.

Donna, you misposted. I think you want the political correctness thread.

what steven said.

morning(ish) all!

just picked up a bottle of Arran single malt whiskey with a sherry finish, from cask 226. It is going to raise some good money for a good cause. Auctioning it off for a non profit. Someone is going to have a delightful bottle at their booze cabinet.

I won't put up the link. Top 10 most dangerous cities in the world:

Only 1 American city was on the list (Detroit). Others included Rio de Janeiro, Baghdad, Capetown, and the "winner" was Caracas, Venezuela. What a shit hole. Also, I saw a movie filmed in Capetown that had Ryan Reynolds in it - how can this kid who looks like a frat boy in perpetuity be credible as a top spy? At any rate, that's another corrupt pit. I once met this European lady who was very content to have moved from South Africa to a desert community in the U.S.

Correction: add New Orleans (deemed worse than Detroit) and Ciudad Juarez, Mexico, where over 50% of the murders in Mexico occur.

I don't want people getting attacked by raccoons or other animals,but hipsters and granolas aren't my cup of tea, so the headline grabbed me. They reflexively don't like me. They incorrectly assume I am the establishment.I'm quoting myself. I spoke too soon. I helped a hipster last night, in a big way. I jump started his car. I went in to get a take-out pizza last night. As I exited the store, I saw this guy unsuccessfully trying to start his car with battery cables hooked up to an adjacent SUV. He then asked me "Is this your car?," pointing to my American car. I really wanted to get the pizza into the car and get on my way, preferring not to deal with a stranger. I had to do a quick analysis of the variables: car pulled up to the curb in front of the pizza place, nice neighborhood, daylight, hipster kid in his late 20s who was a Josh Groban-alike with some stubble and thus not likely a criminal, and, lastly, "what if that had happened to me." All right. I lifted the hood of my car, he hooked up the battery cables, and I revved my engine. His first 2 attempts at turning over his engine were unsuccessful, but his 3rd one was. I got him on his way. I felt like some priest or nun had spoken me to from above saying "help those who are not fortunate and who you may not like."

I then went home to google images of hipsters. Since he had a stocking cap, I wondered if he could have been a granola. I then saw that the hipster "uniform" can include a stocking cap.

"granola" as an identifier is a really dated term - makes it sound like you're an old person posting from the early 1990s. There is such a huge variety in diet lifestyles that you could have gone with: vegan, freegan, raw foodist, locavore, fruitarian, urban forager, etc...

also - maybe in the past year or two we've generally accepted that almost everyone between the ages of 20 and 45 is at least a little "hipster." The fact that you know who Josh Groban is makes you a hipster. Besides - the "original hipster" only really existed between 1999 and 2004 or so - everything that came afterward is what has been assimilated into general culture. people still playing the "identify the hipster!" game are just out of touch.

just to give you an example of how much "hipster" culture has permeated society - my 74 year old church-going aunt regularly rides a vintage schwinn to trader joes to buy shit like flax-seed granola - and she instragram's a "selfie" of herself eating said cereal to facebook. One of my best friends from middle school is now a pastor - he grows his own food, makes his own beer, and sports pretty impressive beard.

Well, that's about right. Granola was an 80s and 90s term and hipster came into being around the turn of the 21st century. There were relatively few granola enclaves in greater L.A. we poked fun at growing up, which have now gentrified because they were sitting on some prime land. For the most part, we pointed further northward, to the Bay Area and above, all the way to the Canadian border, for some dyed-in-the-wool granola living. People who were once hippies or granolas haven't morphed over to being hipsters. People still refer to the them as old hippies and old granolas. But, in the end, why align oneself with an identifiable moniker? If you walk into a place with jeans, a colored t-shirt, flip flops, a regular hair cut, and are clean shaven, they often won't know if you're a lawyer or a car mechanic on their day off. People can do what they want. But to sort of make a statement 24/7, even when running errands, seems kind of dumb to me. At any rate, I was a Good Samaritan last night. I guess I didn't sleep during that lesson in parochial school.

But, in the end,why align oneself with an identifiable moniker?If you walk into a place with jeans, a colored t-shirt, flip flops, a regular hair cut, and are clean shaven, they often won't know if you're a lawyer or a car mechanic on their day off.People can do what they want.But to sort of make a statement 24/7, even when running errands, seems kind of dumb to me.I think you answered your own question.

^

I see law school in your future. You are able to extract things that are said and contort them to make a point that suits you. I still think uniforms other than work attire are idiotic. I think that's pretty clear.

@observant - in most design-y and high-level professions you are networking/marketing yourself pretty much 24/7 - this means always being presentable and essentially "making a statement" about who you are and what you're about - even if you're leaving the house for an errand. You never know when and where your next project lead might come from. so - you think it's idiotic to market yourself? in a way that's what that hipster kid is doing.

Don't know what it is (full moon maybe, or surplus of fools?), but I'm feeling exceptionally snarky today. Even more than usual.

Just checked, new moon tonight. But I still think the parade of idiocy has something to do with it.

in a way that's what that hipster kid is doing.Who was the hipster kid marketing himself to? To a hipster chick? In the suburbs? And it was too hot to be wearing a stocking cap, so that wasn't too bright.

stocking cap in the summer? I thought that look died out years ago! where the hell do you live? geesh - even I sported that look back around 1999 or so (when I combined it with vintage golf shirts and my infamous sparkly blue blazer - I think I also wore a bowler sometimes - ah... art school).

yeah - he's doing some sartorial experimentation - he's learning his way. This kid will probably in a few years be wearing brogues and tweeds and riding a vintage dutch bike everywhere- at least he'll start getting closer....

can somebody argue about dimensions again?

I'm thinking about Saarinen's Miller House this morning. It's a masterpiece, no doubt. But it's now a house museum, and that actually distresses me. House museums are dead. The beauty of that house, and many important house, is that the inhabitants activate a backdrop setting of gorgeous design. With no inhabitants the place feels deprived.

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 trajectoryA 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

mobiusstrip.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 say3dorthogonal)orthogonal space.can't have one without the other"both times, i used the word

definebecause 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-componentsEvery 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/