
Visually stunning math concepts which are easy to explain - aaronbrethorst
http://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain
======
pyduan
For those who are hungry for more, Lucas Vieira Barbosa (LucasVB) has made a
lot of great such illustrations for Wikipedia over the years:

[http://en.wikipedia.org/wiki/User:LucasVB/Gallery](http://en.wikipedia.org/wiki/User:LucasVB/Gallery)

~~~
jamessb
Lucas also has an interesting blog:
[http://1ucasvb.tumblr.com/](http://1ucasvb.tumblr.com/)

And Matt Henderson has some good animations too:
[http://blog.matthen.com/](http://blog.matthen.com/)

------
Spittie
I wish my teacher showed me stuff like that in high school instead of just
scary numbers. While I do understand that eventually you need to get down to
the numbers, my brain seems to pick up the overall concept much faster seeing
nice visual representations like those.

~~~
jordigh
It's not just your brain. It's everyone's brain. Human reasoning is
fundamentally geometric. What happens when you see people who apparently are
able to understand "just scary numbers" is that they are building in their own
head these same images. They've had previous exposure to such images and know
how to build their own. But in the end, everyone ends up reasoning with
drawings.

~~~
dllthomas
That doesn't conform very well to my understanding or experience. Citation?

~~~
Double_Cast
I too believe that math is most easily understood geometrically. I've
suspected this since highschool, when I personally derived many of these gifs
during trig class.

I don't have anything more than anecdotes to support this hypothesis, but it
shouldn't be surprising given that an entire brain lobe (occipital) is devoted
to visual processing [1]. And remember how our first introduction to numbers
was "the number line"? The reals are isomorphic to a line, but _defining_ the
reals as a line isn't feasible since a line doesn't differentiate the
rationals from the reals (or anything in between). But on the other hand,
showing kids a line is easier to grok than defining numbers as "an ordered
field", isn't it? Also consider that Newton invented calculus using
infinitesimals, which made sense to him spatially but didn't find rigorous
footing until Weierstrass [2]. Additionally, the Greeks used to refer to
finding a figure's area as "quadrature" [3], i.e. finding the area of an
equivalent square. If not universal, I'd say geometric interpretations were at
least pretty widespread.

[1]
[http://en.wikipedia.org/wiki/Occipital_lobe](http://en.wikipedia.org/wiki/Occipital_lobe)

[2]
[http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_...](http://en.wikipedia.org/wiki/\(%CE%B5,_%CE%B4\)-definition_of_limit)

[3]
[http://en.wikipedia.org/wiki/Quadrature_(mathematics)](http://en.wikipedia.org/wiki/Quadrature_\(mathematics\))

~~~
dllthomas
_" If not universal, I'd say geometric interpretations were at least pretty
widespread."_

I'd have had no objection to that. Clearly, _many_ people _do_ learn visually
and clearly many people are good at manipulating mental imagery. Support them!
Just be careful you're not ignoring those that don't (or at least know that
that's what you're doing). If you're going to make a case that there _are no
such people_ (or a negligible number) then _that_ needs support.

The story here about imagination seems highly relevant (along with the broader
point):
[http://lesswrong.com/lw/dr/generalizing_from_one_example](http://lesswrong.com/lw/dr/generalizing_from_one_example)

~~~
Double_Cast
I'm familiar with the typical-mind-fallacy posts and glad to see a LW
reference on HN. Just the other day, I read Eliezer's article on Algernon's
Law [1]. It's tangentially related. Incidentally, it precedes LW by a decade!
His thesis sounds reasonable to me. But I thought MBTI test sounded reasonable
too, which I learned doesn't have any empirical backing. So my intuitions may
be less than reliable. meh.

> _According to Galton, people incapable of forming images were
> overrepresented in math and science. I 've since heard that this idea has
> been challenged, but I can't access the study._

So yes, I guess some lack a visual imagination. But I've seen several people
enlightened by diagrams, and never seen anyone enlightened via plug-&-chug
formulas. So I'd be surprised if a visual imagination did'n confer some kind
of advantage in math. I highly doubt that a visual imagination confers a
disadvantage, given that Euler had photographic memory. But I concede that the
null hypothesis is certainly likely. Yvain says he can't access Galton's
study. After a few minutes Googling, I gave up too after I hit a paywal.

[1]
[http://web.archive.org/web/200102021712/http://sysopmind.com...](http://web.archive.org/web/200102021712/http://sysopmind.com/algernon.html)

~~~
dllthomas
_" But I've seen several people enlightened by diagrams, and never seen anyone
enlightened via plug-&-chug formulas."_

I've personally experienced being enlightened by both, with respect to
different things. I probably lean more toward the visual, but don't expect
everyone does (without more, carefully gathered, evidence).

 _" I highly doubt that a visual imagination confers a disadvantage"_

That would surprise me as well, assuming nothing was sacrificed for that
visual imagination (and even then I expect a visual imagination to be more
useful than many things).

~~~
Double_Cast
I'm sorry to dredge up this thread again, but this boggles my mind.

/u/Someone contests that visualizing things like the Monster Group, M-Theory,
or 7 Touching Cylinders is practically impossible [1]. I agree. To reason
about the Monster Group, I imagine even professional mathematicians manipulate
symbols with functions, operations, et al. But, to quote Eliezer, "Does this
person [Ph.D economist] really understand expected utility, on a gut level? Or
have they just been trained to perform certain algebra tricks?" [2]

Clearly, the economist does not understand his craft. Were he able to
visualize the _substance_ rather than merely Plug & Chug the _symbols_ [3],
then surely he wouldn't have bought the lotto ticket. And since symbols are
mere abstractions (lossy compressions) of the substance, Plug & Chugging the
symbols will rarely trump visualizing the substance. Therefore, I suspect my
inability to visualize the Monster Group reflects the limitations of my brain
rather than an intrinsic disadvantage of visualization. I.e. I would prefer
visualization of the Monster Group to Plug & Chug if only I were smart enough.

(Disclaimer: I have no idea what the Monster Group is. But I do remember
seeing an old youtube clip about visualizing 11 dimensions. I still don't get
it. If you're interested, [4])

> _I 've personally experienced being enlightened by both_

The way I see things, visualization is necessary. However! Here you are saying
enlightenment _is_ possible without visualization... So please share with me,
exactly how were you enlightened without visualization? Can you give examples?
Does it just happen, like that theory about how savants can crunch numbers
without knowing what they're doing on a conscious level? Is it akin to
weighting parameters according to how strongly they affect a function's
output, like in neural networks? Am I missing something that's totally obvious
to you, like how some philosophers argued that imagination didn't exist?

[1]
[https://news.ycombinator.com/item?id=7549869](https://news.ycombinator.com/item?id=7549869)

[2]
[http://lesswrong.com/lw/gv/outside_the_laboratory/](http://lesswrong.com/lw/gv/outside_the_laboratory/)

[3]
[http://lesswrong.com/lw/nv/replace_the_symbol_with_the_subst...](http://lesswrong.com/lw/nv/replace_the_symbol_with_the_substance/)

[4]
[http://www.youtube.com/watch?v=JkxieS-6WuA#aid=P-eR3HseAzw](http://www.youtube.com/watch?v=JkxieS-6WuA#aid=P-eR3HseAzw)

------
greenyoda
I'm surprised that nobody posted the visual proof for the countability of the
rational numbers:

[https://en.wikipedia.org/wiki/File:Diagonal_argument.svg](https://en.wikipedia.org/wiki/File:Diagonal_argument.svg)

~~~
daturkel
In my point-set topology class we used the diagonal argument as the root of a
proof for the following question:

Suppose a submarine is moving in a straight line at a constant speed in the
plane such that at each hour the submarine is at a lattice point. Suppose at
each hour you can explode one depth charge at a lattice point that will kill
the submarine if it is there. You do not know where the submarine is nor do
you know where or when it started. Prove that you can explode depth charges in
such a way that you will be guaranteed to eventually kill the submarine.

If there's interest, I'll post my solution. My proof actually gives an
overview of what order to bomb points in, but I have no idea what it would
look like if you plotted out, say, the first 100 or 1000 points. I'd be
curious to see someone implement it.

~~~
rkaplan
I'm interested. What's the solution?

~~~
cousin_it
The set of all possible submarine routes is countable (velocity vector +
position at time 0). Let's assume we have an enumeration of that set. Now it's
easy to bomb them all. At time 1 we bomb the point where submarine 1 is at
time 1, at time 2 we bomb the point where submarine 2 is at time 2, and so on.

~~~
daturkel
Yep, this is how the proof works!

~~~
Someone
How does that use the diagonal argument? If the diagonal argument could be
used it would be to prove that the number of submarine routes isn't countable.

Edit: just looked at the SVG called "Diagonal Argument.svg". That's not what I
know as the diagonal argument
([https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument),
[http://mathworld.wolfram.com/CantorDiagonalMethod.html](http://mathworld.wolfram.com/CantorDiagonalMethod.html))

------
PhasmaFelis
I've often seen hypercubes pictured or described like this:
[http://en.wikipedia.org/wiki/File:Hypercube.svg](http://en.wikipedia.org/wiki/File:Hypercube.svg)

...which doesn't really explain much. Then I saw this animation of a rotating
hypercube and suddenly it made so much more sense:
[http://en.wikipedia.org/wiki/File:Tesseract.gif](http://en.wikipedia.org/wiki/File:Tesseract.gif)

The takeaway for me was that, in that static depiction, the inner cube is a
cube, and the outer cube is a cube, and each of the six (apparent) truncated
pyramids is _also_ a cube, just a visually distorted one. There are eight
cubes in the hypercube and each shares a face with six others, just as the six
squares in a cube each share an edge with four others. You could have told me
all of that and I wouldn't have understood it, but after seeing the animation
I was able to work it out for myself.

------
cubancigar11
While the most up-voted Pythagoras theorem proof [1] might be fun to see, it
is rather the proof of the spectacular failure of our education - that nobody
remembers anything taught in our school. The similarity of right triangles in
a circle is proven using Pythagoras theorem in the first place!

Thankfully a better proof is present [2] which depends on distributive
property and algebra.

1\.
[http://math.stackexchange.com/a/733765/141120](http://math.stackexchange.com/a/733765/141120)
2\.
[http://math.stackexchange.com/a/734887/141120](http://math.stackexchange.com/a/734887/141120)

------
jloughry
This animated gif of the Fourier transformation from the time domain to the
frequency domain (from the original post on stackexchange.com) is just
stunning:

[http://math.stackexchange.com/a/738048](http://math.stackexchange.com/a/738048)

 _Where_ were these when I was in school?

~~~
tptacek
Help me understand why? I feel like I understood everything this GIF was
trying to communicate, but also no closer to understanding how the transform
itself works.

~~~
zokier
I agree. I understand time and frequency domains reasonably well, but I still
do not have a clue _how_ Fourier transformation gets me from one to the
another. How do the elements in the animation correspond to the terms in eg
this equation
[http://www.texpaste.com/n/v6sjue00](http://www.texpaste.com/n/v6sjue00)
(which is the Wikipedia definition for discrete Fourier transformation)? I
don't even understand what the equation itself is supposed to represent, last
time I checked a sum produced a single number, how does that represent a
function in frequency domain? I guess the subscript k has some significance,
_but the animation does not help a single bit here_

~~~
0x09
k is the output index (the "bin" in the frequency domain), or the notch in the
rotated space in that graphic. Every output frequency coefficient contains a
full sum of the input function, hence a direct translation of that formula is
O(n^2) -- which is why the FFT, with O(nlogn) complexity is so important.

The graphic doesn't really help show how the analysis itself happens, it just
presents the result, which is a series of waves that add up to f. The actual
process of obtaining a frequency coefficient from a time-domain function is
easy to describe: multiply the function by a (co)sine wave with a particular
frequency and sum together the result. But it's not very intuitive why that
works until you consider that one period of a sine wave sums to zero. By
multiplying the sine with the function, you perturb the shape of the sine with
just the amount of energy that the function contains at that given frequency.
So that instead of summing to zero, the sum measures "how much" of that
particular wave is present. That's Fourier analysis.

Fourier synthesis is more easily visualized (I think anyway). Simply multiply
a sine wave at each frequency by the corresponding coefficient derived above,
and sum those weighted waveforms together elementwise to recover your
function.

~~~
zokier
> But it's not very intuitive why that works until you consider that one
> period of a sine wave sums to zero. By multiplying the sine with the
> function, you perturb the shape of the sine with just the amount of energy
> that the function contains at that given frequency. So that instead of
> summing to zero, the sum measures "how much" of that particular wave is
> present. That's Fourier analysis

Thank you. This really helped.

Based on that explanation, wouldn't the formula be more like:
[http://www.texpaste.com/n/nphm1fgp](http://www.texpaste.com/n/nphm1fgp) ? I
suppose there is some further magic which allows for phase differences or
something.

~~~
0x09
Right, you need both sine and cosine parts, hence e^ix, in order to fully
describe the function. Unless the function is completely odd (or even), in
which case the transform really is equivalent to `x[n] * sin(2 _pi_ k*x/N)`
(or cos). To see why imagine analyzing a function W that is just a plain
cosine wave (any frequency and amplitude). If we only use the sine part of the
Fourier transform, F(W) is indistinguishable from F(-W). In fact both are zero
everywhere.

Transforms that use only sines or cosines (like the DCT) provide a complete
basis by increasing in frequency by only a half cycle (pi) rather than by
integer cycles (2pi). Essentially trading half a transform of sines and half a
transform of cosines for one transform of half-cosines (or sines).

------
cruise02
I love these. I posted several on my blog a few years ago.

Six Visual Proofs: [http://www.billthelizard.com/2009/07/six-visual-
proofs_25.ht...](http://www.billthelizard.com/2009/07/six-visual-
proofs_25.html)

Visualization of (X + 1)^2: [http://www.billthelizard.com/2009/12/math-
visualization-x-1-...](http://www.billthelizard.com/2009/12/math-
visualization-x-1-2.html)

------
rndmize
Trinomial cube:
[https://www.google.com/search?q=binomial+cube&source=lnms&tb...](https://www.google.com/search?q=binomial+cube&source=lnms&tbm=isch&sa=X&ei=Ih9CU8rKCcrkyAGggIHgDg&ved=0CAYQ_AUoAQ&biw=1197&bih=937#q=trinomial%20cube&revid=1326263989&tbm=isch&imgdii=_)

------
sphericalgames
Animated / interactive Bezier curves: [http://www.jasondavies.com/animated-
bezier/](http://www.jasondavies.com/animated-bezier/)

------
NAFV_P
I remember the example that explains how to derive the area of a circle from
several years ago, but there is also an analogue for the surface area of a
sphere provided you know the volume...

The volume is

    
    
      4/3*pi*radius^3
    

the surface can be broken up into lots of spherical triangles. The vertices of
each of these triangles is joined to the sphere's centre to form a load of
tetrahedrons. As the number of these triangles increases and their size
decreases without limit, their total volume will asymptotically approach the
volume of the sphere. The volume of a tetrahedron is

    
    
      1/3*base*height.
    
      sum(tetrahedron_volume)=volume_of_sphere=4/3*pi*radius^3
    

The tetrahedrons are flattened to end up as triangular columns with equal
height (1/3 of the sphere's radius). The resulting shape should be a column
with height 1/3 the radius of the sphere whose cross-sectional area is equal
to the sphere's surface area. The height is

    
    
      1/3*r ...
    
      (4/3*pi*radius^3)/(1/3*r)=4*pi*radius^2
    
    

EDIT: bloomin' asterisks, I should have remembered.

------
yogrish
Another site that explains Mathematical concepts Intuitively. Trigonometry:
[http://betterexplained.com/articles/intuitive-
trigonometry/](http://betterexplained.com/articles/intuitive-trigonometry/)
Other Topics:
[http://betterexplained.com/archives/](http://betterexplained.com/archives/)

~~~
yogrish
One more link [http://world.mathigon.org/](http://world.mathigon.org/)

------
alok-g
Here is a great book along these lines. Not always visual, but always (and
often unexpectedly) simple.

[http://www.amazon.com/Q-E-D-Beauty-Mathematical-Proof-
Wooden...](http://www.amazon.com/Q-E-D-Beauty-Mathematical-Proof-
Wooden/dp/0802714315/)

------
sixothree
I've found these videos to be fairly interesting.

[https://www.youtube.com/watch?v=yJZP_-40KVw&list=PLN0wPs8UzD...](https://www.youtube.com/watch?v=yJZP_-40KVw&list=PLN0wPs8UzDMjy1-Ynp2ShLO2xTrhzJiGy)

------
ryannevius
How about a quick explanation of radians?

[http://24.media.tumblr.com/dd1b123f13f5578e11b04d7579df1fce/...](http://24.media.tumblr.com/dd1b123f13f5578e11b04d7579df1fce/tumblr_mzn47kFZqE1qk3kdro1_500.gif)

------
gabemart
This page just underlines my complete mathematical illiteracy. I don't really
understand any of it. I would like to get a basic grounding in math, but it
seems like so wide a field I have no idea where to start.

~~~
wdmeldon
[https://projecteuler.net/](https://projecteuler.net/) is really a great place
to start. Solving the first problem with any level of efficiency means
understanding Summation. I didn't even know what that was until I was staring
at the programmatic solution to the problem and trying to figure out why it
worked mathematically.

Teaching yourself math is far more difficult than teaching yourself
programming, but it is possible.

------
imodgames
Here's one for the integral of y=x^2:
[http://www.mathedpage.org/proof/integrating/](http://www.mathedpage.org/proof/integrating/)

------
caio1982
Really neat representations. I've learned a lot of new stuff tonight from the
original post, thanks for sharing! My favorite so far: Gibbs Phenomenon.

------
deckar01
I just posted a visualization of multiplication of the integers modulo n
[http://math.stackexchange.com/questions/733754/visually-
stun...](http://math.stackexchange.com/questions/733754/visually-stunning-
math-concepts-which-are-easy-to-explain/743542#743542) to this question.

This visualization makes it easy to notice the factors of n and the symmetry
of multiplication.

------
mattdeboard
Kind of a meta question but is all the latex markup on that page supposed to
be styled or do people just write it out like that out of habit

~~~
nateguchi2
I had this problem too, Turns out I have the HTTPS Everywhere plugin for
chrome so the page loaded over HTTPS, so mathjax wasn't allowed to load as it
was loading from a plain HTTP server

~~~
mattdeboard
Aha, I'm also using HTTPS Everywhere. Thanks.

------
devilshaircut
This may have already been posted in another thread, but it doesn't appear to
have been posted here. But this is a great visualization for the Pythagorean
Theorem.

[http://i.imgur.com/W8VJp.gif](http://i.imgur.com/W8VJp.gif)

Fun for kids, also, when they are learning about it.

------
PavlovsCat
Personally, I can always use some music to actually get _stunnned_ :)

[http://www.youtube.com/watch?v=kkGeOWYOFoA](http://www.youtube.com/watch?v=kkGeOWYOFoA)

------
Totient
The Fourier transform .gif was gorgeous. Anyone know of a similar one for
Laplace transforms? (I'm still having trouble building up intuition for the
Laplace transform.)

------
suyash
Would love to see a similar post for CS Core Concepts (think OOP, Data
Structure & Algorithms etc). Anyone know of such a url? Thanks in advance.

------
shitgoose
Very nice! My personal favorite is "The sum of the exterior angles of any
convex polygon will always add up to 360∘".

------
z3t4
The reason why you understand this better then numbers might be because you
are more visual orientated.

------
Gravityloss
Anybody up to visualizations about quaternions?

~~~
Zecc
[http://acko.net/blog/animate-your-way-to-glory-
pt2/](http://acko.net/blog/animate-your-way-to-glory-pt2/)

Check out under "Blowing up the Death Star". You'll need a decent graphic card
that works well with WebGL though.

~~~
Gravityloss
That's quite good, thank you very much! It's clear a huge amount of effort
went into creating it.

------
kimonos
Very interesting and easy-to-understand presentations. Thanks for sharing!

------
ilaksh
The last one on the page at the moment. Is this what I think it is? LOL.
[http://i.imgur.com/8m47tuJ.png](http://i.imgur.com/8m47tuJ.png)

~~~
dfc
If you thought it was tacky and sophomoric the answer is "yes."

------
dfc
The topology answer is a great example of the problem some mathematicians have
communicating ideas to lay people.

