
What Is a Manifold? - Topolomancer
https://bastian.rieck.me/blog/posts/2019/manifold/
======
joncrane
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.

~~~
adrianratnapala
> 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".

~~~
JadeNB
> 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](https://en.wikipedia.org/wiki/Ideal_number)
)—much like the "ideal points" of hyperbolic geometry
([https://en.wikipedia.org/wiki/Ideal_point](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](https://news.ycombinator.com/item?id=19659571)).)

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

[https://www.youtube.com/watch?v=CEXSSz0gZI4](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...](https://www.youtube.com/watch?v=Ap2c1dPyIVo&list=PL6763F57A61FE6FE8)

------
braxxox
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?

~~~
crdrost
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.

~~~
sideshowb
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"?

~~~
crdrost
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...](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.758.63&rep=rep1&type=pdf)

[2]
[http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/r...](http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html)

------
xelxebar
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.

~~~
mlevental
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).

~~~
hhmc
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.

------
krosaen
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.

~~~
mlthoughts2018
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.

~~~
ColinWright
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](https://mathoverflow.net/a/95908)

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

~~~
blt
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.

------
alcolade
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.

~~~
improbable22
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.

~~~
alcolade
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.

~~~
improbable22
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.

------
no_identd
"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)

~~~
mlevental
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](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.

~~~
no_identd
>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.

~~~
mlevental
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.

------
ulucs
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.

~~~
wcoenen
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.

~~~
falcrist
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.

------
termiefoo
what ain't a manifold at this point

~~~
username90
Things with edges, like disks and finite lines.

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

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

~~~
JadeNB
> 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](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).

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

~~~
patrickthebold
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](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.

------
725686
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](https://www.youtube.com/watch?v=j3BlLo1QfmU)

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

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

------
ErotemeObelus
I think everything said here can applies to just topologies.

~~~
dan-robertson
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.

~~~
senderista
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.

~~~
dan-robertson
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.

------
arduinomancer
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?

~~~
improbable22
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.

------
loeg
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...](https://www.supplyhouse.com/Bluefin-CM4-75PXZ-PXV-3-4-PEX-Crimp-x-
Closed-Copper-Manifold-w-1-2-PEX-Crimp-Ball-Valves-Lead-Free-4-Outlets)

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

------
amai
So a manifold is a surface?

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

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

~~~
ginnungagap
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

~~~
amai
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?

------
tw1010
Love me some explanation innovation. More of this and the coding world will
reach a plateau.

