What Is a Manifold?
bastian.rieck.me
bastian.rieck.me
Basically, a manifold means something that takes the flow of gases from a one-to-many or a many-to-one.
An intake manifold takes one single entry point for air feeding the engine and splits up into a separate input for each cylinder.
An exhaust manifold takes the hot exhaust from each cylinder separately and combines them into one big pipe.
Though, there are exceptions to the "one" part, like intake manifolds with two inlets (dual plane is common), exhaust manifolds with two outlets (common on inline 6 engines), and also some really odd setups: https://speedmaster79.com/media/catalog/product/cache/1/thum...
I was thinking I was going to find a series of pipes or tubes fitted together.
Instead I found just two pieces of pretty thick sheet metal, close to plate thickness. They just went through a metal bender to form the pipes, and bolted together with a gasket in between.
Maybe that's how it got started. A plane, wrapped around something.
Example: http://fireflowtechnology.com/wp-content/uploads/2015/04/100...
A perfectly sensible meaning given the construction of the word. There is something about mathematics and linguistics (and to some extent CS) that encourages the creation of confusing, meaningless names like "accusative (case)", "(algebraic) ideal" and "(geometric) manifold".
I think there are less ambiguities in common language because they share so much context and compete; where as there is enough separation between certain disciplines for the semantics of esoteric words to evolve and coexist independently without issue until viewed externally where it appears ambiguous - this is even true for mathematical notations.
Confusing is hard to argue, but meaningless is, I think, hard to defend. These all have meanings; I could speak to the latter two, and I'm sure a linguist could speak to the former. They may not be obvious meanings, but that's not the same as saying they're meaningless. (I regard much business jargon, for example, as literally meaningless, defined only in terms of other words that also seem meaningless to me; but I'm sure an MBA would take issue with that characterisation.) I don't know who coined 'manifold', but 'ideal', for example, was literally Kummer's word coined to describe things that behaved like, but weren't quite, numbers in the ordinary sense (https://en.wikipedia.org/wiki/Ideal_number )—much like the "ideal points" of hyperbolic geometry (https://en.wikipedia.org/wiki/Ideal_point).
(EDIT: Fortunately no_identd knows more about the history of 'manifold' than I do (https://news.ycombinator.com/item?id=19659571).)
Does that have anything to do with “many-fold?”
https://www.youtube.com/watch?v=CEXSSz0gZI4
It is made by a physicist, not mathematician, and it's a great combination of informal explanations and carefully covering the actual math definitions.
I feel that his approach works way better than making analogies.
There is another course from Wildberger. A bit more hardcore, but also has some interesting perspectives:
https://www.youtube.com/watch?v=Ap2c1dPyIVo&list=PL6763F57A6...
It's interesting to expand on this idea and realize that maybe we are making the same mistake again, and that from our local perspective the universe is 3D, when in reality, it's a 3D manifold of a higher dimensional space.
One thing I am still unclear of though, is that isn't this proven to be the case? Is it not true that we are provably living in at least a 4D space, where time is the fourth dimension? We can observe its existence but cannot move freely through it and are confined to free transformations only in 3D space? In this way, aren't we living in a 3D manifold in a 4D space? So maybe then the question is are we living in a 3D manifold of a +4D space?
The basic background is that Einstein wrote a paper taking Lorentz more seriously than Lorentz took his own work: and that paper suggested that just maybe, when you accelerate in any given direction by an acceleration A, you see all of the clocks ahead of you some distance z tick faster by a factor A z / c², where c is the speed of light in vacuum, and behind you they tick slower with the corresponding negative z until a wall of death at z = -c² / A where clocks do not tick at all and time appears to stand still. This is in fact the only new fact that special relativity adds. Lots of people got very confused about the philosophical implications, but the mathematical implication is that time and space can be mixed together by these accelerations and must be treated as one unified geometrical entity.
Einstein then went one further on the whole 4D thing, because arguably we are always accelerating in this whole gravitational field of the Earth. You have a lot of options to choose from. So this part took Einstein many many years to work out. Maybe the easiest is to say that we standing on Earth are a non-accelerating reference frame, and then anybody who falls must see a wall of death somewhere out in space. That turns out to be a very boring approach, and also wrong. What Einstein suggested instead was that you are in a non-accelerating reference frame with no wall of death if you are in free-fall, and we standing on the Earth would see a wall of death beneath our feet, except we can't see beneath our feet. But if a body were more massive, maybe we could see the wall of death from orbit. And now we have a photo of a black hole to prove it! But even before that, the essential point is that if I put a clock up somewhere high (on a tower, in a plane, or at the top of a mountain) and I am standing on the ground, then I am accelerating towards that clock relative to free-fall: so that clock must be ticking faster than my clocks are. And we have had direct observation of that “gravitational time dilation” for a long time.
Mathematically, this means that we are in a manifold that looks locally 4 dimensional, in this weird way of coupling the four dimensions that couples accelerations with the ticking of clocks. We say, going back to this guy who worked out all of the mathematics before Einstein, that the manifold is locally Lorentzian, as opposed to Euclidean. But the manifold is four dimensional, not three dimensional. It has to have this coupling between time and space locally. But then globally it can have these interesting features like black holes.
Now, whether we can embed this curvy universe that we inhabit into a larger dimensional flat space, is not necessarily a given. I don't know many physicists who are deeply interested in that sort of question. Certainly to have the structure that we need, it needs to have Lorentzian timelike dimensions in it, one of which we use as our time dimension. Certainly also, the people working on string theory use these extra-dimensional possibilities to solve certain mathematical inadequacies that their string theories otherwise have: but usually those dimensions are locally available, so we would see them; so there is some sort of hand-waving about how they must be curled up into such a small length scale that we cannot actually observe them. But there is certainly a branch of string theory called M-theory that I do not personally know too much about which has something to do with viewing our universe as a geometric entity in a larger space.
Thankfully, the mathematics does not require this. The essential point of a manifold that the article somehow leaves out, is that I am no longer going to rely on global coordinates. I only care about local coordinates. So on the sphere, it is a two dimensional object, even though it lives in a three dimensional space, and that's because depending on where I am, I can uniquely identify points near me on the sphere by either their x & y coordinates, or their y & z coordinates, or their x & z coordinates. This fails for points that are not nearby me, because projecting a globe on to a flat surface this way will project two hemispheres onto the same point. But I can always choose one of those three and find myself in the middle of a hemisphere, and describe everything else on that hemisphere with those coordinates. So the whole point of a manifold is that I don't need global coordinates, and therefore it doesn't matter much whether or not I am embedded in a larger flat space or not.
I've not noticed before that special relativity implies a 'wall of death' as you put it but I see now you can get that from the Lorentz transform t'=(t-vx/c^2)/sqrt(1-v^2/c^2) combined with v=at.
Does the z=-c^2/a relation still hold in general relativity or is it modified by other terms?
Also
> we can't see beneath our feet
I presume you mean that we can, but for the values of A and z we experience here on earth, the radius of the Earth is much smaller than z so a point at a distance z "beneath our feet" doesn't exist as it's up and out the other side of the gravitational well
> And now we have a photo of a black hole to prove it!
I'm guessing you mean the event horizon is exactly such a "wall of death"?
The z = -c² / A relation does hold even when you transfer from the first-order-in-β transform to the full Lorentz transform; for an outworking from the inimitable John Baez, see [2].
I did mean that we can’t see beneath our feet, but I see your point: at g = 10 N/kg this term c² / g is something like 10^16 meters away, way way outside of the Solar System, which is only in the billions of km large. I was purely thinking about our Schwarzschild radius, which is millimeters away from the center of the Earth: therefore we cannot see the Schwarzschild event horizon because it is nonexistent; it is “underground” but so far that then most of the mass is outside of that, so you have to recalculate and get an even smaller amount, but that then needs another recalculation... and so on. It vanishes to zero because the mass is not located in a condensed enough space.
[1] http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.758...
[2] http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/r...
(also, it appears that in string theory, they are considering even higher-dimensional models for our universe...so it feels that 3/4 is rather a lower estimate)
Are there any experiments to rigorously prove or disprove the existence of further spatial dimensions?
Anyway, highly recommended.
https://mathoverflow.net/a/95908
[0] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathema...
Besides the gimbal lock problem mentioned in another problem, Euler angles are not easy to compose. It is also not obvious how to calculate the misorientation between two rotations with Euler angles, but it is trivial with quaternions.
Other approaches have their own problems. For example, matrices are more expensive to use in a computer (normalizing a quaternion to avoid rounding errors is much faster than normalizing the equivalent matrix, and quaternions need less memory), and Rodrigues vectors, while usually very useful, present the problem of infinite values for rotations of 180 degrees (which, in my opinion, should not be a problem, but it is in many programming languages).
At the end of the day, quaternions present the best compromise.
S^2 is curved, but locally you can't tell it's not (say) RP^2.
Of course, this is assuming S^2 is getting its geometry from a certain embedding into R^3 that comes to mind. You could define different Riemann curvature tensors over S^2 that may have zero curvature in certain places, like squishing a balloon against a flat table for example.
I guess my point was that topology and curvature are different things (like you pointed out with your comment about RP^2), and saying that "a sphere looks flat locally" is missing the point!
Also, it didn't occur to me that this notion of curvature doesn't make sense in 1d, i.e. for S^1 like you said.
In 2D there is only scalar curvature, i.e. you don't need the whole Riemann tensor, just one number (at a given point).
In 3D you need the Ricci tensor, but still not the full curvature tensor. This is still much simpler, for instance IIRC this is why you cannot have gravitational waves in 3 (meaning 2+1) dimensional spacetime. In 4D you get the full complication.
If one would translate Riemann's original German word for them, "Mannigfaltigkeit", it would translate to "manifoldyhead", or, more understandable to the speaker of Modern English:
Manyfoldyhood.
(Think of -head as in "Godhead", not as in "Brotherhood", in the same sense that the "ring" in "algebraic ring" refers to "ring" in the sense of "smuggler ring", not in the sense of "gold ring". The two suffixes merged in English, and have very different and rather complex etymological origins.)
While this word sounds somewhat ridiculous and perhaps a bit infantile in English, it, in my opinion, conveys what we mean by them significantly better.
Clifford tried to accommodate Riemann's highly specific word choice in his translation, as Jost notes:
"The English of Clifford may appear somewhat old-fashioned for a modern reader. For instance, he writes “manifoldness” instead of the simpler modern translation “manifold” of Riemann’s term “Mannigfaltigkeit”. But Riemann’s German sounds likewise somewhat old-fashioned, and for that matter, “manifoldness” is the more accurate translation of Riemann’s term. In any case, for historical reasons, I have selected that translation here."
Unfortunately, neither Jost (a native German speaker!) nor Clifford realized that English can and does accommodate Riemann's exact meaning directly.
To make a comparison which might require quite a bit of German knowledge beyond high school education to describe in exact linguistic terms, but which native speakers should hopefully find intuitive to distinguish (I at least, do):
Translating "Mannigfaltigkeit" as "Manifoldness" seems equal to mistranslating "Geheimnis" as "Secrethood" and "Geheimheit" as "Secretness", whereas the opposite pairing would yield an accurate (albeit not necessarily immediately apparent—in terms of the differences between the two—to the English native speaker, unless explicitly pointed out) translation. So much for the last suffix, but that still leaves insertion of the one before it (' * foldy * ' instead of ' * fold[( * )]' unclear.)
One may observe the difference involved there by a converse example, also involving "Geheim", as well as the root word of it, "Heim", by dragging forth a rather rare and archaic—but none the less highly likely intuitive to the native speaker—word:"Geheimig". Whose meaning starkly differs from both "Geheimnis" and "Geheimheit".
For further language related hijinks related to /Manyfoldyhoods/, see:
A) The Dutch word for them, which would back translate to "Variety"; and
B) This quote by Poincaré:
"I prefer the translation of Mannigfaltigkeit by multiplicity, because the two words have the same etymological meaning. The word set is more adapted to the Mannigfaltigkeiten considered by Mr. Cantor and which are discrete. It would be less adapted to those which I consider and which are discontinuous."
(As I don't speak French, I can't make much of any statements about the accuracy about the etymological claim by Poincaré, so I'll close with another Poincaré quote instead:
" Mathematics is the art of giving the same name to different things."
—Henri Poincaré)you say that
>Think of -head as in "Godhead", not as in "Brotherhood"
and
>Translating "Mannigfaltigkeit" as "Manifoldness" seems equal to mistranslating "Geheimnis" as "Secrethood" and "Geheimheit" as "Secretness",
but
https://www.etymonline.com/word/-head#etymonline_v_50781
indicates to me that -ness is exactly the correct translation of Mannigfaltigkeit, since in the instance of godhead a accurate synonym would be godliness (and in the instance maidenhead maybe maidenly).
regardless in english manifold already exists as an adjective and probably a good translation of the original german (if i'm to understand you correctly) would be to simply describe a space as manifold rather than a manifold.
Strictly speaking, if one entertains the distinction involved here, "Godhead", as a noun, serves as a hypernym to "God" and "Goddess"; and "Godheads" as a hypernym to "Gods" and "Goddesses", which neither "godness", "godessness", "godliness" nor "godessliness" do. This makes sense, as "Heit" used to function (and in some very rare German dialects supposedly still does) as a separate noun, unlike "-nis" and "-ness". Does that help make the distinction between the two suffixes clearer?
(Note: You left open the matter of -ig and -y.)
>would be to simply describe a space as manifold rather than a manifold
Correct, you got that right, however, I think that talking about spaces in this way doesn't so much serve as a substitutive translation but as a consequence of the distinction involved - coming hand in hand, basically.
Is there any way to distinguish between orientable or nonorientable surfaces when you're simply walking on it? If the bug was also 2d, I might have had some ideas hut I'm totally blanking for 3d.
Draw a circle on the surface and orient it. If the surface is non-orientable then there's a way to take a long walk, return to the circle, and find that the direction of the orientation has changed.
If the surface is non-orientable then I can draw a very small circle, put an orientation on it, then there's a path I can take that brings me back to find the orientation has changed.
Let's be more explicit.
Take a non-self-intersecting embedding of RP^2 in R^4, a point P in our RP^2, and a sufficiently small epsilon e. Then take three points on the circle of size e centred on P, and think of them as going in order, thus defining an orientation. Now take an appropriate walk around RP^2 and return to the circle. For an appropriate walk I will now find that the order of the points on my circle has reversed.
That's a more precise way of saying what I originally intended, and in that context your comment doesn't make sense to me. Can you expand on it?
ColinWright's point is that drawing such an arrow on a tiny circle is equivalent to drawing the letter R on the 2D manifold. It gives a local orientation to the 2D surface. (If you wish you may think of this as an arrow into some 3D embedding space, but you don't have to.)
If the 2D space is orientable, then when you take a copy of this little circle (or letter R) and go for a long walk, when you get home your copy will always match the original. That's all that orientable means. In the standard usage, it's a property of the 2D manifold, not of the particular walks you take. I think this is the point of confusion here.
(If you do the experiment with paper, the bug would be on the other side of the paper from where it started, but I don't think this concept of "side" exists when the strip is described as a manifold. The bug lives in the strip, it's not walking on top of it. It helps to imagine the strip to be transparent.)
For a Klein bottle the same would happen, but there the bug could also find the world mirrored along a different direction, depending on how it travelled.
If, instead, you twist the edges 180 degrees before gluing, we then get the Klein Bottle. I.e. when you warp from, say, the left edge to the right edge, you also get flipped vertically.
I'm guessing you're already familiar with that, but by a similar construction, we can start with a (hyper-)cube and glue faces together to get higher dimensional analogues.
To my knoweldge the easiest visualization of a non-orientable 3D manifold is just by thinking of a cube that warps around to the opposite face. However, in addition, it also reflects you into your mirror image. This is essentially a Klein bottle in one higher dimension.
It's then not too hard to start grappling with the 4D case and beyond!
It depends on which mathematicians! Plenty of differential geometers allow manifolds to have boundary, and say "closed manifold" (https://en.wikipedia.org/wiki/Closed_manifold) to emphasise when they are dealing with a (compact) manifold without boundary (or, as you point out, really a manifold whose boundary is empty).
Some are. A Koch curve for instance.
E.g., asking for a "differentiable manifold" now asks for manifolds with enough additional structure to do calculus on top of.
Poincaré Conjecture is yeah interesting to explain though to be honest I vaguely recall an article in similar style to your post that explained it.
Possibly something like describing gradient descent on a manifold is interesting to this audience? Or maybe a post on Flatland? Many possibilities really on good follow ups.
Can a point be a manifold?
(not that there are different 'classes' of manifolds out there; I am not sure if the point would qualify as a Riemannian manifold, for example)
f(0) = 1
https://en.m.wikipedia.org/wiki/Whitney_embedding_theorem
Basically says that many manifolds can be described as a lower dimensional submanifold of some euclidean space.
The article also gets some details a bit wrong/fuzzy: it says you can tell if you are on a sphere by walking in a straight line infinitely far and seeing if you ever cross yourself. But this property also follows on the surface of a (rounded at the top) cone. Even if you require this property in all directions you get problems on eg a torus.
I will try to be more precise about this in a subsequent article!
You don't want to be precise. You want to increase the resolution of the key ideas being seen. This is also why parables are so popular for teaching: they're literally not true, but they resolve to an image of something that is true. Please don't introduce homotopy/homology. Lines and angles are perfect.
This is related to people measuring the curvature of the universe right?
Sub-spaces of R^3 are a useful way of generating and picturing examples of 2D manifolds. But these things exist by themselves. As the article mentions with angles of a triangle, you can tell that a 2D manifold is curved from living inside. Likewise you can tell whether a 3D manifold is curved of not, without any mention of a 4th dimension.
Questions of curvature of the universe are a step harder, as we are talking about 3+1-dimensional space-time. But the slice of constant time is a 3D manifold, and as far as we can tell right now, it appears to be flat.
[0]: https://www.supplyhouse.com/Bluefin-CM4-75PXZ-PXV-3-4-PEX-Cr...