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What Is a Manifold? (rieck.me)
283 points by Topolomancer 8 days ago | hide | past | web | favorite | 110 comments





It's funny, because in my much younger days I dropped out of college and worked as a mechanic for 3 years. In the automotive world and internal combustion World, manifold has a much different meaning, though related.

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.


A similar notion exists in the fire service, where a manifold is an appliance for combining or splitting water supply lines. A typical example of a fire service manifold would be an appliance with 3 2.5" threaded ports and a single 5" Storz port, which can be used to either merge multiple 2.5" or 3" supply lines into one 5" supply line, or to divide the flow from a 5" line up into multiple smaller lines.

Example: http://fireflowtechnology.com/wp-content/uploads/2015/04/100...


And on a smaller scale, ordinary water plumbing :-).

And I always think of the mathematical definition. It's nice to see the mechanical terminology explained. Thanks!

Sorry, I was not aware of that! It was not my intention to mislead you. Hope you enjoyed the article nonetheless :-)

Plenum is also used to describe automotive intakes, and has a physics meaning as well.

> An exhaust manifold takes the hot exhaust from each cylinder separately and combines them into one big pipe.

Does that have anything to do with “many-fold?”


Yes, the original Old English was Manigfeald which means exactly that.

That's what manifold means to me as well.

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...


That picture looks like an intake for a V8 and each trumpet is still for an individual cylinder; it's just that the intake ports on the heads are paired up for manufacturing and thermal efficiency.

My point was that manifolds aren't all many to one. This one is clearly many to many.

> Basically, a manifold means something that takes the flow of gases from a one-to-many or a many-to-one.

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".


> 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".

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).)


I don't think "encouragement" is quite right, ambiguities are somewhat the default - it's the opposite of encourage, an absence of something, there must be a competitive force present to reduce their likelihood.

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.



Isn't this like a multiplexer/demultiplexer in electronics/signal processing?

A multiplexer/demux takes one of many inputs and routes them to a single output, or vice versa. A manifold has no valving, so it's a mixer/adder instead.

PWM-switched mux/demux with signal input then :).

still not quite. more like a node where you join parallel circuits back together; the total volume of stuff moving, current, is the addition of everything coming in, rather than blocking and switching things.

You could at least say that the ground plane on a circuit board is an exhaust manifold

I actually clicked the link expecting exactly that explanation.

Sorry! It just goes to show that I am living in a filter bubble of mathematics. My intention was not to deceive you :-)

No need to apologize. Words have different meanings.

Not at all - I got more than I bargained for and learned something new!

This is also the general usage for chemists... As a chemistry and math major during my college days it was confusing to say the least.

I took apart a furnace once to replace an igniter.

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.


If you want more, there is an excellent course on "introductory" topology on YouTube:

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...


What a great read! I've never thought about how the way we experience the surface of the earth as 2D manifold of a 3D space, and how up until recently (relatively speaking) this was unknown.

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?


Sort of, not quite.

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.


Thanks for this, it's very clearly expressed.

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"?


Yes, the event horizon of a black hole is precisely this sort of wall of death. Indeed there is an active debate in the literature about whether outside observers ever see black holes because from the outside perspective infalling mass should become “frozen” on the surface, with Liu and Zhang very recently in 2009 arguing that if you consider a shell of nonzero thickness falling symmetrically into the black hole then as this mass approaches the event horizon the event horizon actually expands to gobble up some of the matter—this paper is then cited and refuted in a 2011 paper in the same journal by Penna, who argues that they got one little detail very wrong and that once you correct for this you do indeed see the matter “frozen” on the wall-of-death; see [1] for a PDF of this paper.

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...


Thanks! Honestly, about the implications in terms of physics I am really unsure. Maybe someone else from HN can chime in?

(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)


I'm always interested in this as well.

Are there any experiments to rigorously prove or disprove the existence of further spatial dimensions?


If anyone has questions (especially technical ones), the ##math channel on freenode is pretty phenomenal. It's quite active, with quite a few grad students, across a range of fields. The best part is how friendly and helpful they are to learners.

Anyway, highly recommended.


this in general accurate but I would warn people to have a thick skin in there because there are some seriously toxic people in there (TRW3W or whatever his/her name that was fairly knowledgeable but hung out in there I think purely to assert his/her superiority).

I still recognize that name/handle and I haven't logged into ##math for ~10 years, so I can certainly corroborate what you're saying.

This is good advice for existing at all in the world. They're not toxic except in high doses, like all sorts of people.

If the bug were truly mathematical, he would notice the curvature of S^1 and S^2 without walking around (since curvature is defined at each point), but that is obviously very pedantic of me and shows why I wouldn't write an article nearly as interesting as this one.

But S^1 isn't curved. At least if we mean intrinsic curvature, which is the kind that belongs to the manifold itself.

S^2 is curved, but locally you can't tell it's not (say) RP^2.


I was thinking of the Riemann curvature tensor which is just some measure of curvature at each point. If the bug were at some point of S^2 he would notice the curvature.

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.


Not only is there no intrinsic curvature in 1D, but the 2D and 3D versions are also simplified special cases.

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.


Another connection to make: quaternions can be represented as a Lie group, which is a (smooth differentiable) manifold. This means representing 3d orientation using quaternions works better for optimization than e.g Euler angles (roll, pitch, heading) in things like SLAM.

What do you mean “works better”? Symbolically simpler once you know the requisite math behind Lie groups? In terms if computation, why would one representation be any different than the other? Unless you mean there’s a subset of certain computations where that representation is better but also a subset where it is worse? Similar to how there are cases where a Hough transform allows a more efficient calculation, but many cases where you can’t efficiently extract other calculations out of the Hough transform with effectively inverting it entirely.

Representing rotations via Euler angles can lead to gimbal lock[0], whereas using quaternions doesn't. The best explanation I could find quickly is this one:

https://mathoverflow.net/a/95908

[0] https://en.wikipedia.org/wiki/Gimbal_lock#In_applied_mathema...


euler angles are sort of the odd one out, a particularly bad way to represent rotations. Orthonormal matrices, quaternions, skew-symmetric tensors with exponential map -- all represent SO(3) without gimbal lock, with different pros and cons for each.

They are more practical, specially when using a computer.

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.


"Quick" etymological fact about what mathematicians in English call Manifolds:

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)

i'm sorry i don't quite follow (even though i'm pretty keen on etymology in general and etymological origins of mathematical objects in particular).

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.


>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).

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.


i don't speak german at all (outside of what i've been exposed to in the sciences (ansatz, eigen, etc.) so the nuanced distinction is not clear to me (albeit i can take it on faith visavis your translation to english) but i'll say that i appreciate your bit about rings and varieties (names of objects whose relationship to the objects i've always been curious about - in the case of algebraic ring i always assumed it had something to do with closure). thank you.

All is well until the poor bug finds out it's on a Klein Bottle, or worse yet RP2.

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.


Are you familiar with the construction of a torus by taking a square and "gluing the edges together"? It's effectively the game Astroids, where going off one edge of the screen wraps around to the opposite edge.

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!


I took that idea to its extreme and released "Geodesic Asteroids" on iOS (free) a couple of years ago. You can play on the 2D map or an embedded torus. There is also an old blog post I wrote at the time: http://nbodyphysics.com/blog/2015/03/06/asteroids-on-a-torus...

Okay. That's pretty darn cool. I ended up trawling your blog a bit too. Really interesting stuff! Wish there were an RSS feed I could subscribe to!

Maybe it would be interesting to have somewhat of a switch between the 2D map or the torus map, in that while playing 2D you can enter a challenge state where the map switches.

For a Mobiüs strip, the bug would find the world had been mirrored along the width of the strip if it made a round trip and came back to the same point. Other bugs that stayed behind would claim that the traveller itself had become mirrored.

(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.


I'm pretty sure the property of sidedness comes up when you embed the manifold in euclidian space. The property seems to be related to orientability.

Möbius. Not Mobiüs.

> Is there any way to distinguish between orientable or nonorientable surfaces when you're simply walking on it?

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.


This depends upon the circle you chose. Nonorientable manifolds still have many orientable circles (any sufficiently small loop is orientable).

You confuse me.

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?


I think when they say "circle" they're talking about your path, not your orientation reference. That's probably wrong terminology, but it makes some sense if you're thinking about how the path has to be a closed loop.

Yes, I took "circle" to mean any closed path since the difference is immaterial in generic manifold. You wouldn't normally be talking about "orienting a circle" in the manner the original poster did, but I see now what they mean. I took it to mean "choose an orientation along a closed path" which would be impossible to do for some (but not all) closed paths in a non-orientable surface.

A closed path is S^1, which is certainly orientable, no matter what it's embedded into. You just draw an arrow on the line.

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.


Yes, you can always choose an orientation of the path but not necessarily of the manifold along the path. I understand what he was saying now and misunderstood since it is not the usual way one talks about specifying an orientation at a point and in fact had imagined he was specifically specifying a "big" circle, like a nontrivial loop on the torus. You'd more typically talk about choosing a basis of the tangent space at a point - presumably ColinWright wanted to avoid involving extra definitions like tangent space and so chose a visual definition that wasn't quite as precise. I don't think there's any further confusion.

Cool, I think we're on the same page. I know what you mean by "orientation of the manifold along the path" but this isn't precisely the standard usage. (My vague memory is that you can define an orientation without needing a tangent space, but I can't think of an example, so could be wrong here.)

My background is in riemannian geometry so I never had to worry about lack of a tangent space. Certainly you can define orientability of some topological spaces that aren't even manifolds. I'd forgotten but you can even define a local orientation at a point for a general topological manifold in terms of it's top-dimensional homology. I think that's the most general situation it makes sense in, you need a well defined dimension to consider this.

what ain't a manifold at this point

Besides the other answers, things also start to get interesting when you add some qualifiers.

E.g., asking for a "differentiable manifold" now asks for manifolds with enough additional structure to do calculus on top of.


Good point; I originally wanted to talk about the Poincaré conjecture as well, but then I realized that this would make the post even longer. Do you have some ideas about other interesting topics?

To be honest it's not even my field so I know a lot less than I wish I did but yeah agreed it has no shortage of interesting things to talk about :)

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.


> Good point

Can a point be a manifold?


Yes, but it's a simple one of dimension zero. There's not much to do here---every 'neighbourhood' of the point is exactly that: just the point.

(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)


Simplest example of what not a manifold is the shape of the letter “Y”. Technically for a topological space to be a manifold, every small neighborhood around each point should look the same. But “Y” has a special point at the center.

A Y-shaped "line".

A fractal isnt a manifold. Discrete sets are an edge case. They are considered to be 0 dimensional manifolds to make them fit, but there is a not much that manifold theory has to say about them.

> A fractal isnt a manifold

Some are. A Koch curve for instance.


I believe you are correct. Koch curve is a continuous curve so its topological dimension would be 1 (not its fractal dimension).

Things with edges, like disks and finite lines.

Both of those are typically referred to as manifolds with boundary. The setting of Stokes Theorem is often manifolds with boundary.

Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).

> Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).

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).


I thought a manifold with a boundary would still be a manifold, but its boundary has to satisfy a dimensionality condition. For example, the 2D disk is a 2-manifold with a 1-dimensional boundary. Strictly speaking, this is a _topological manifold with a boundary_, though.

Nothing wrong with them, they can be manifolds.

f(x) = 0 (x < 1, x > 1)

f(0) = 1


At the end of the article he mentions Jeffrey Weeks... I searched youtube and found this very interesting vid: https://www.youtube.com/watch?v=j3BlLo1QfmU

Is the conclusion that a manifold is a space that looks like a lower dimensional space that it really is?

Yes and no. The definition doesn't require that what you have is embedded in a higher dimensional space. Though that's a good source of examples.

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.


no. there's no requirement for lower dimension. like another person that responded to you R^3 is 3 dimensional manifold. a (smooth) manifold is a space you can do calculus on. that means you have a consistent notion of distance (so that you can figure out when points are close together, ie you approach limits). the set of charts and continuous transformations between them is what encodes this constraint.

You can have a thing in some higher dimensional space that still "looks like" a thing in lower dimensional space (like the surface of the sphere in R^3 being two dimensional), but R^3 itself is a 3 dimensional manifold that "looks" 3 dimensional as well.

To some extent, yes! It needs to have a 'nice' structure, though---this is why we have the condition that is should resemble, say, an Euclidean space.

Nice article, it would have been nice to add a word or two about manifold embedding though.

I plan on adding this in a sequel :) Thanks for the suggestion!

I think the natural next question is how does a bug in R3 check if its part of a 4-dimensional manifold?

This is related to people measuring the curvature of the universe right?


The article squeezes together two notions, of whether a manifold is curved, and whether it's a submanifold of some higher-dimensional one.

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.


I think everything said here can applies to just topologies.

That isn’t really true. Dimension isn’t so well defined in topology but is reasonably straightforward with a manifold. The article also touched on geometry (sum of angles of a triangle). To be topological a bunch of things must change. The bug walking example doesn’t work and is demoted to an analogy. All notion of distance and direction is lost. If the article were about topology then it would probably be talking about distinguishing shapes by homotopy/homology rather than lines and walks and angles.

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.


If I'm not mistaken, dimension can be defined easily for a topological manifold, which is actually a more fundamental structure than a differential manifold. (The former requires that chart overlaps be homeomorphisms, i.e. continuous bijections), while the latter requires that they be diffeomorphisms, i.e. smooth bijections). Smooth manifolds don't form a nice category for technical reasons, but one can think of a forgetful functor from differential manifolds to topological manifolds, and dimension being defined in the latter.

I don’t disagree that one can define dimension for a manifold (topological or differentiable). I was replying to the parent comment and so I was pointing out that dimension isn’t really well defined for a general topological space.

Thanks for pointing out the fuzziness! I did not want to extend this further so as not to confuse readers who have no familiarity with the subject. Of course, the fact that you cross your own path at some point is _not_ indicative of living on a sphere; there are numerous other manifolds that satisfy this (not to speak of the torus, as you yourself mentioned).

I will try to be more precise about this in a subsequent article!


Don't. Precision obfuscates more than it increases resolution.

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.


Ah, and here I was imagining this[0] is a manifold. I must be spending too much time with PEX lately.

[0]: https://www.supplyhouse.com/Bluefin-CM4-75PXZ-PXV-3-4-PEX-Cr...


f(x) = 0. Move along it by moving perpendicular to the gradient of f.

So a manifold is a surface?

it can be, I think of a manifold (probably naively) as the mathematical abstraction that you end up with if you start from Euclidean space and fiddle with different features - e.g. if you're in a Euclidean space how do you get from one point to another - what are the minimum requirements? What if we want calculus to work what are the minimum requirements? If we're on a surface and we walk around is there any way to tell what shape that surface is?

A generalized surface in arbitrary dimensions. A common English "surface" maps naturally to a 2D manifold embedded in 3D space.

Ok, so a manifold is simply a hypersurface. At least I‘m pretty sure in machine learning it means exactly that.

A hypersurface is usually a codimension one object, while a manifold can have bigger codimension or even not be embedded in an higher dimensional space at all

Ok, a line in 3D space can also be called a manifold? Maybe a better question is then to ask, what is not a manifold?

Love me some explanation innovation. More of this and the coding world will reach a plateau.



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