objects that we'd like to hold, but can't in this universe

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gbagcn2
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objects that we'd like to hold, but can't in this universe

Postby gbagcn2 » Thu Jan 22, 2009 6:23 pm UTC

ax + b = y
ax^2 + bx + c = y

I am not looking for a link to a 2 dimensional graph. Since these equations use 4 and 5 variables the graphs should be 4 and 5 dimensions.

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby t0rajir0u » Thu Jan 22, 2009 7:19 pm UTC

What exactly do you want? A visualization or general theory? The subject that deals with objects of this sort is called algebraic geometry, although I don't think this is actually what you want.

You're also not quite correct on the dimension: the first "graph" (we call them varieties) sits in [imath]\mathbb{R}^4[/imath] but has dimension [imath]3[/imath] (it's something like a translation of a bunch of hyperbolas) and the second variety sits in [imath]\mathbb{R}^5[/imath] but has dimension [imath]4[/imath].

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby gbagcn2 » Thu Jan 22, 2009 7:23 pm UTC

I want a link to a picture of these graphs.

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby antonfire » Thu Jan 22, 2009 9:36 pm UTC

I bet you want to hold a hypercube, too.

This thread is now about objects that we'd like to see or hold, but can't without escaping from our mortal coil.

I'd like to have a model of the root system of E8. Also, a good-sized chunk of the hyperbolic plane to draw on. And, uh, higher dimensions too, while I'm at it.
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby Buttons » Thu Jan 22, 2009 10:07 pm UTC

Can someone give me a link to a picture of a nonvanishing continuous vector field on a sphere?

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby heyitsguay » Thu Jan 22, 2009 10:10 pm UTC

Easy. Hilbert Space

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby Ended » Thu Jan 22, 2009 10:25 pm UTC

Buttons wrote:Can someone give me a link to a picture of a nonvanishing continuous vector field on a sphere?

Yep, sure:
Spoiler:
nonvanish_cts_vectorfield_sphere.gif
nonvanish_cts_vectorfield_sphere.gif (2.94 KiB) Viewed 7278 times


@OP: At best, you will be able to plot some 3D cross-section of those objects. And I doubt that there is an easily-obtainable link to the exact plots that you want. Looks like a good opportunity to learn a programming language (I'd recommend MATLAB for this particular task, if you have access to a copy).
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby Bassoon » Thu Jan 22, 2009 11:23 pm UTC

I want to hold a black hole in one hand and a white dwarf in the other.

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby Talith » Fri Jan 23, 2009 3:34 pm UTC

I want a pint of beer in a klein bottle.

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby qinwamascot » Sun Jan 25, 2009 9:28 pm UTC

To the OP: It's impossible to plot in 4- or 5-space in a traditional manner, much like you can't draw a 3 dimensional graph on a 2 dimensional piece of paper. What you can do is project it onto a 2-dimensional plane, then have 2 or 3 dimensions modifiable via some interface. Unfortunately, my license for IDL, where I could do this for you reasonably easily, expired 6 months ago. It's doable in MATLAB, C, C++, Java, and any other language that's actually useful.

But before you attempt this yourself, it's important to realize that you'll lose a significant amount of data. It's unlikely that you can picture a tesseract or a 5-hypercube in your brain (it's possible that you know what a projection of it onto 3-space looks like, but the actual shape is not 3 dimensional). And the shapes you're looking at aren't as simple as a cube. Realistically, you'll glean more data by just holding 2 or 3 things constant and graphing in 2-space or 3-space, then modifying your constants. These traces of the larger graph are easy enough to understand and show how it behaves.

Finally, there are some online 4-dimensional graphing utilities. There are 2 basic strategies: employ 2 rotation angles, or use colors to indicate the 4th dimension. Google will find you some. At least one 5-dimensional plotting utility exists, but it isn't free and the plots look about average.
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby mudge » Mon Jan 26, 2009 2:16 am UTC

antonfire wrote:This thread is now about objects that we'd like to see or hold, but can't without escaping from our mortal coil.


Then I want to see an intelligent post from gbagcn2.





Oh, and a mature post from me.
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby MarshyMarsh » Mon Jan 26, 2009 9:53 pm UTC

I want a catagraphic map which minimum colour use is greater than 5! (no bridges).

I had a very interesting lecteur a few years ago. I forget the ins and outs but it was about some *not to intelligent* person who claimed to have drawn 4d objects in a 3d plain. The pictures where a garbled mess. I can try to find those if you want?

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby the tree » Mon Jan 26, 2009 11:04 pm UTC

antonfire wrote:Also, a good-sized chunk of the hyperbolic plane to draw on.
Why is that not possible? Wouldn't a bit of paper stuck to a literal saddle do?

I want a cantor set shaped knife, so beef will never pose a challenge to me again.

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby antonfire » Tue Jan 27, 2009 12:19 am UTC

Sure, approximately, but I wouldn't call that a good-sized chunk. I mean, how many regular septagons can you fit on there before you run out of room? Not many is how many.
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby fiyarburst » Tue Jan 27, 2009 3:11 am UTC

Can somebody link me to a picture of the internet? The whole thing. Nothing else.

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby black_drgn » Tue Jan 27, 2009 3:49 am UTC

fiyarburst wrote:Can somebody link me to a picture of the internet? The whole thing. Nothing else.


Apparently, suprisingly, this was attempted before and put on a wallpaper. The largest closeup I can find w/o the key is here.

Spoiler:
another reason why google is awesome. By the way, this is from before or around 2005, so it's not up to date.

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby chapel » Tue Jan 27, 2009 10:21 am UTC

Talith wrote:I want a pint of beer in a klein bottle.


Well, since the inside is also the outside then so long as there exists a Klein bottle and a pint of beer then it would technically be inside.

But while we are at it. http://www.kleinbottle.com/images/Klein ... turama.png

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby antonfire » Tue Jan 27, 2009 11:12 am UTC

I never understood why people think that the fact that the inside of a Klein bottle is the same as the outside is so special. The same is true for a regular bottle!
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby SlyReaper » Tue Jan 27, 2009 1:58 pm UTC

Besides, it's perfectly possible to hold liquid in a Klein bottle anyway. You just have to turn it over to take a swig from it. Non-orientable surfaces: they're actually not that interesting.
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby t0rajir0u » Tue Jan 27, 2009 6:22 pm UTC

antonfire wrote:I never understood why people think that the fact that the inside of a Klein bottle is the same as the outside is so special. The same is true for a regular bottle!

This is not at all true if sameness is defined properly. An actual bottle has thickness, so when you say the inside of a regular bottle is the same as the outside I think you mean that the inner surface and the outer surface are connected by, say, the lip of the bottle. This is true, but topologically it's true because both the outer surface and the "inner" surface here are part of the exterior, in a topological sense. The interior is between those surfaces, where the glass is, and it is this "inside" that is distinct from the outside in a normal bottle.

I can't say much more to defend the Klein bottle, but I will say that it's quite curious that it's the only counterexample to the Heawood conjecture.

SlyReaper wrote:Besides, it's perfectly possible to hold liquid in a Klein bottle anyway.

I can't even wrap my mind around this. A Klein bottle can't be realized as an actual surface (by which I mean a subset of [imath]\mathbb{R}^n[/imath]) in dimension less than [imath]4[/imath], so we're talking about a 2D object embedded in 4D space "holding" a 3D volume... in what sense does a circle in space hold the disc it contains?

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby gnuoym » Tue Jan 27, 2009 6:53 pm UTC

How has noone pointed to these? For all your non-orientable drinking requirements.

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby SlyReaper » Tue Jan 27, 2009 8:41 pm UTC

t0rajir0u wrote:
antonfire wrote:I never understood why people think that the fact that the inside of a Klein bottle is the same as the outside is so special. The same is true for a regular bottle!

This is not at all true if sameness is defined properly. An actual bottle has thickness, so when you say the inside of a regular bottle is the same as the outside I think you mean that the inner surface and the outer surface are connected by, say, the lip of the bottle. This is true, but topologically it's true because both the outer surface and the "inner" surface here are part of the exterior, in a topological sense. The interior is between those surfaces, where the glass is, and it is this "inside" that is distinct from the outside in a normal bottle.

I can't say much more to defend the Klein bottle, but I will say that it's quite curious that it's the only counterexample to the Heawood conjecture.

SlyReaper wrote:Besides, it's perfectly possible to hold liquid in a Klein bottle anyway.

I can't even wrap my mind around this. A Klein bottle can't be realized as an actual surface (by which I mean a subset of [imath]\mathbb{R}^n[/imath]) in dimension less than [imath]4[/imath], so we're talking about a 2D object embedded in 4D space "holding" a 3D volume... in what sense does a circle in space hold the disc it contains?


Well it can be realised in 3D space if you allow self-intersection. I've seen little plastic models of Klein bottles that self intersect. They hold liquid perfectly well.
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby Buttons » Wed Jan 28, 2009 2:17 am UTC

gnuoym wrote:How has noone pointed to these? For all your non-orientable drinking requirements.

Has anyone else here met that guy? He hawks his goods at math meetings all the time, and let me just say: totally insane. And hilarious.

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby t0rajir0u » Wed Jan 28, 2009 2:57 am UTC

SlyReaper wrote:Klein bottles that self intersect

aren't Klein bottles. Klein bottles are smooth manifolds and any surface that self-intersects can't be smooth. Those are immersions of Klein bottles into a lower dimension. (That's like drawing a Mobius strip on paper - sure, it looks like a Mobius strip...)

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby SlyReaper » Wed Jan 28, 2009 8:54 am UTC

t0rajir0u wrote:
SlyReaper wrote:Klein bottles that self intersect

aren't Klein bottles. Klein bottles are smooth manifolds and any surface that self-intersects can't be smooth. Those are immersions of Klein bottles into a lower dimension. (That's like drawing a Mobius strip on paper - sure, it looks like a Mobius strip...)


Well, technically true. But you can make the size of the intersection as small as you like, so you can all but ignore it. Topologically speaking, you can think of the self intersecting pseudo-Klein-bottle as a Klein bottle if you conveniently overlook the intersection. It serves well enough to visualise the object.
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Re: objects that we'd like to hold, but can't in this universe

Postby skeptical scientist » Wed Jan 28, 2009 5:53 pm UTC

I'd like to hold the real projective plane. While it's true that the immersions of the Klein bottle in R3 aren't actually Klein bottles, they make it very easy to imagine the real thing. On the other hand, I've yet to see a good way of visualizing RP2.
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Re: objects that we'd like to hold, but can't in this universe

Postby Buttons » Wed Jan 28, 2009 6:09 pm UTC

Here's one. It's pretty fun to play with, too.

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Re: objects that we'd like to hold, but can't in this universe

Postby Token » Wed Jan 28, 2009 6:15 pm UTC

I always thought the old "glue a disc around the edge of a Mobius strip" worked pretty well for visualising RP2. Maybe I'm just visualising it wrong.

[No, I've never successfully glued a disc to a Mobius strip in the correct manner. Only in my head.]
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby doogly » Wed Jan 28, 2009 6:25 pm UTC

SlyReaper wrote:Well, technically true. But you can make the size of the intersection as small as you like, so you can all but ignore it. Topologically speaking, you can think of the self intersecting pseudo-Klein-bottle as a Klein bottle if you conveniently overlook the intersection. It serves well enough to visualise the object.


Unless you let that visualisation exercise convince you that an actual Klein bottle would hold liquid.
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby Token » Wed Jan 28, 2009 6:31 pm UTC

doogly wrote:
SlyReaper wrote:Well, technically true. But you can make the size of the intersection as small as you like, so you can all but ignore it. Topologically speaking, you can think of the self intersecting pseudo-Klein-bottle as a Klein bottle if you conveniently overlook the intersection. It serves well enough to visualise the object.


Unless you let that visualisation exercise convince you that an actual Klein bottle would hold liquid.

Would that be something like expecting a 1-manifold embedded in 3-space to hold liquid?
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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby doogly » Wed Jan 28, 2009 6:42 pm UTC

Token wrote:
doogly wrote:
SlyReaper wrote:Well, technically true. But you can make the size of the intersection as small as you like, so you can all but ignore it. Topologically speaking, you can think of the self intersecting pseudo-Klein-bottle as a Klein bottle if you conveniently overlook the intersection. It serves well enough to visualise the object.


Unless you let that visualisation exercise convince you that an actual Klein bottle would hold liquid.

Would that be something like expecting a 1-manifold embedded in 3-space to hold liquid?


Yeah, I dunno. I think maybe the problem is that liquid is a slippery concept.
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Re: objects that we'd like to hold, but can't in this universe

Postby Chet » Wed Jan 28, 2009 9:51 pm UTC

The closest thing to a 5 dimensional graph that you're going to get is a 3d plot with varying opacity and color to represent the other two dimensions IMO.

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Re: Looking for a link for these 4 and 5 dimensional graphs

Postby t0rajir0u » Thu Jan 29, 2009 3:00 am UTC

SlyReaper wrote:you can make the size of the intersection as small as you like, so you can all but ignore it. Topologically speaking,

the size of the intersection isn't relevant; it's the fact that it exists. I'm making a distinction between what is a good model for the Klein bottle and what is a Klein bottle. If you wouldn't say that a drawing of a Mobius strip in R^2 is a Mobius strip, you shouldn't say that a "Klein bottle with intersection in R^3" is a Klein bottle.

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Re: objects that we'd like to hold, but can't in this universe

Postby Sir_Elderberry » Thu Jan 29, 2009 3:02 am UTC

objects that we'd like to hold, but can't in this universe


Her. :(


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Re: objects that we'd like to hold, but can't in this universe

Postby PM 2Ring » Thu Jan 29, 2009 5:39 pm UTC

It's quite easy to make a flat model of a Klein bottle from paper & adhesive tape. Sure, it still self intersects, but you can cut it into a pair of Moebius strips with a pair of scissors.

And regarding the projective plane, there are programs around that let you play with hyperbolic tessellations, I even wrote one myself (back in the heyday of the Amiga). I tried to write a general hyperbolic sketching program, but decided to give it up before my brain exploded. :)

Another approach to hyperbolic tessellation is to crochet them: Crocheting the Hyperbolic Plane.

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Re: objects that we'd like to hold, but can't in this universe

Postby Govalant » Thu Jan 29, 2009 7:27 pm UTC

I'd like to hold an object with vastly different gravitational and inertial mass.
It might be the thing I want most in my life.
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Re: objects that we'd like to hold, but can't in this universe

Postby gorcee » Thu Jan 29, 2009 8:07 pm UTC

I'd like to hold Natalie Portman.

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Re: objects that we'd like to hold, but can't in this universe

Postby Xanthir » Fri Jan 30, 2009 3:46 am UTC

Govalant wrote:I'd like to hold an object with vastly different gravitational and inertial mass.
It might be the thing I want most in my life.

To be precise, you probably want to hold something with normal inertial mass, but vastly smaller gravitional mass. All the awesome bits of playing with something in micro gravity, none of the muscle atrophy.

Though come to think of it, the opposite *would* be cool, if it has enough gravitational mass to actually attract things. It'd get pretty dirty, though.
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Re: objects that we'd like to hold, but can't in this universe

Postby quintopia » Fri Jan 30, 2009 7:03 pm UTC

gorcee wrote:I'd like to hold Natalie Portman.


She wants to f*** you too.

Xanthir wrote:Though come to think of it, the opposite *would* be cool, if it has enough gravitational mass to actually attract things. It'd get pretty dirty, though.


Yeah, imagine playing indoor soccer with that! Of course, it would take away all the cool trick kicks and stuff since it couldn't leave the ground. . .but the fact that it would decelerate at an unbelievable speed would be awesome. It would go from like 100mph when you kicked it full force to a full stop in just a few seconds (since friction is proportional to normal force).

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Re: objects that we'd like to hold, but can't in this universe

Postby jazznaz » Sat Jan 31, 2009 12:32 am UTC

I'd love to take a run around a flat designed by Escher...

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