
How mathematicians think in dimensions above 3 and 4 - mahipal
http://mathoverflow.net/questions/25983/intuitive-crutches-for-higher-dimensional-thinking
======
Dove
The comments at that link are beautiful. If you're the sort who likes to skip
comments, I encourage you to view them in this case.

I'd like to highlight a point of dissonance between the title ("How
mathematicians _think_...") and the actual request ("anything that makes it
easier to _see_ , for example, the linking of spheres") -- emphasis mine.

I find the identification between visualization and intuition revealing. As a
rule, mathematicians must be able to reason about things they cannot even
begin to visualize -- non-measurable sets, infinities so large they need
special names, infinite linear combinations of orthogonal functions.

That's not to devalue attempts at visualization. They're useful for developing
intuition. But the original joke works because the mathematician is perfectly
happy reasoning in hyperspace even though he cannot see it. The fourth
dimension is not particularly harder to describe than the nth.

~~~
JadeNB
> infinite linear combinations of orthogonal functions

I'm not sure it's quite accurate to say that these can't even begin to be
visualised. The theory of Fourier series means that someone picturing a
'reasonable' function on the circle has already begun the endeavour.

~~~
Dove
You're right. I thought, even while I was writing that, that it was a weak
example. Though I had Banach spaces in mind, not fourier series, and I have
always had trouble with them.

Still. I felt I needed three examples. "Pick something not from set theory," I
said to myself. I couldn't come up with a strong example. Maybe that's
telling.

Unmeasurable sets, though. When I try to visualize one, I see the letter E
because that's what we called it in the constructive proof.

~~~
gjm11
You're in good company. Supposedly, someone once asked J G Thompson (a very
eminent group theorist, winner of the Fields Medal, the Wolf Prize and the
Abel Prize) what mental picture he had in his head when thinking about a
difficult group theory problem. "A big black letter G", he said.

~~~
JadeNB
At James Arthur's 60th birthday conference, he said something like the
following (roughly paraphrased): "First …" and, unfortunately, I don't
remember what first was (but I agree with Dove
(<http://news.ycombinator.com/item?id=1385842>) that there should be 3 :-) ).

"Next, I studied the work of Langlands, and the group was called GL_2. Then, I
studied the work of Harish-Chandra, and the group was called G."

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Jun8
"For instance, the fact that most of the mass of a unit ball in high
dimensions lurks near the boundary of the ball can be interpreted as a
manifestation of the law of large numbers"

As usual Terry Tao's comment is wonderfully illuminating (at least for a non-
mathematician like me). It's common knowledge that the ratio of the volume of
a n-sphere to its circumscribed n-cube goes to zero as n->0
([http://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_are...](http://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_area)),
but I never thought of this as another result of the law of large numbers.

~~~
sesqu
That's actually what I considered to be the _cause_ for the law of large
numbers, when I tried to trace the normal distribution to axioms of three-
dimensional geometry. I failed to get much beyond that, for lack of ability to
reason about exponential functions, and abandoned the pursuit.

------
asdflkj
Correct but unhelpful answer: only through inordinate amount of practice with
solving problems.

A possibly helpful answer to the question "Why is it probably a good idea for
most people to give up on this?":

A dimension R^n is only a collection of points each of which is specified by
an n-tuple over R. This isn't hard to hold in your head, but it tells you very
little about how objects behave in any given space. By "think in", presumably
the inquirer wants to be able to predict the behaviors of objects. The trouble
is that there are many more different kinds of objects and behaviors than one
is naturally inclined to assume, because one doesn't normally think of his
ordinary 3D intuition as an insanely complex piece of specialized hardware
that it is. You can certainly learn to do in software small parts of it, one
by one. You would take those n-tuples, and do some particular bit of math on
them to get you where you want to go. This will be difficult, like all math.
Gather a big pile of these small pieces, train them until they're fast enough,
and eventually you've got something like a crude emulator. I would hazard that
unless you are interested in deep abstract (as in, non-visual) problems of the
relevant math branches, you won't have the discipline to carry all this out.

------
winthrowe
<http://www.dimensions-math.org/Dim_reg_AM.htm>

This video series was linked by one of the comments, parts 3 and 4 had some
interesting visualizations of various the 4d platonic solids.

~~~
henrikschroder
I liked part two about how 2D creatures would see and maybe visualize 3D
objects, that's the explanation that helped me most with how to visualize 4D
objects.

~~~
louislouis
I'm watching it now and it busting my brains.

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thefool
There were two great quotes from the math book i used last semester.

The first was the most mathematicians equate geometry with understanding. In
otherwords, if you can describe (draw) a picture that represents what you are
talking about, then you really understand it.

The second was that not all math is something you can visualize. In that some
of the beauty of math is that it can describe things beyond humans abilities
of perception, and that it is precisely in these cases that math is its most
useful.

------
ErrantX
The best part of the that joke (at least in my mind) is that generally
engineers are pretty good, if not the best, at visualising multiple dimensions
> 3.

------
rw
P.S. Terry Tao posted a comment on the OP. Worth a read.

~~~
rms
For fans of Tao, his Google Buzz is worth following.
<http://www.google.com/profiles/114134834346472219368>

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izendejas
A bit offtopic (from thinking about R^n to visualizing it), but I recommend
reading about parallel coordinate systems.
<http://en.wikipedia.org/wiki/Parallel_coordinates>

------
korch
Isn't it a little _too convenient_ that the Universe seems to prefer 3d for
physical processes, and that almost any kind of phenomena we need to
understand can be framed in a 3d model? Or is it us the whole time, unable to
see beyond our own limitations? When we see grandiose contrivances in a film
plot, we naturally suspend disbelief in order to enjoy the story. I think
those of us who have learned university math and who think we understand
higher dimension are doing likewise—telling ourselves a story so we don't have
to walk out of the theater.

I don't think we consciously "think" in any dimensions above 3. Like all life,
we are neurologically hardwired by default to think in 3d, since that's what
has worked for the evolution of all life on Earth. As for formal math, I am no
longer even sure there is a connection between the logical, axiomatic and
cultural edifice we've built up and call "mathematics", and our day-to-day
"neural navigation software" that allows each of us to get from point A to B
on the surface of a rotating oblate rock tethered to a nuclear fireball that
is hurtling through the infinite, continuous space we call the Universe. The
map is not the territory. The mathematical deductions of our neural experience
are not the same thing as the experience itself.

From all of the suggestions on Mathoverflow, almost all of them are variants
on projecting higher dimensional objects onto 3d and 2d objects, then
comparing all the different projections in a clever way to get a "feel" for
how the higher dimensional object changes. Even this is completely non-
intuitive, as our brain's visual apparatus is optimized to take in a total
picture and immediately spot the biggest changes. i.e. there's a predator
running at us from over there! That is afterall what eyeballs and visual
perception evolved for! If we can't even see the whole visual field at once,
but just slices of projections of it, then our finely tuned visual hardware is
thwarted and unable to detect and piece together the "shape" of objects. This
is why is say nobody can "think" higher than 3d—even if you are doing it, your
brain is still imperceptibly and behind the scenes translating your logical
construct into a 3d "sensory" construct.

With that said the best way I've come across for visualizing 4d objects is
from complex analysis, where you can use color gradients to represent a
dimension. It doesn't work so well going beyond 4d, but it's a great set of
training wheels. <http://www.mai.liu.se/~halun/complex/>
<http://www.nucalc.com/ComplexFunctions.html>

------
TallGuyShort
This video's really good for visualizing the higher dimensions:

<http://www.youtube.com/watch?v=aCQx9U6awFw>

~~~
RevRal
I've seen that video a few times over the years, and it has always struck me
as... how do I say it... not correct. Something in there just seems wrong. I
do like some of the ideas in there, though.

Hopefully someone here can elaborate because it's been on my mind since I
first saw it.

~~~
herdrick
Well, I'm no physicist, but I agree - it's bogus. The problem, I think, is
that it assumes that time is the fourth dimension. Instead it should be the
_last_ dimension. So if you're talking about 6-space, then time should be the
7th dimension. If you always call time the fourth dimension, then you get
these wacky results where in 5-space and up you can see and exist in all time
simultaneously.

This is the problem with calling time a dimension. It's useful, but it makes
it natural to talk moving forwards and backwards in time. Error.

~~~
henrikschroder
The root of the problem is that most people can't see the difference between
4D _space_ and 3D _spacetime_.

3D spacetime behaves mathematically exactly the same as 4D space, it is a
construct to make it easier to analyze and do physics calculations of 3D
objects over time, by setting time as the 4th dimension, thereby giving every
object a position in time as well.

But the visualization is completely different, and the things you can do are
completely different. If I have a 3D left-hand glove in a 4D space, I can
"turn" it in the 4th dimension and end up with a right-hand glove. But if I
have the same glove in spacetime, it doesn't matter how much I wait around
(i.e. move in the 4th dimension), my left-hand glove will never turn into a
right-hand glove.

