Visually stunning math concepts which are easy to explain | Hacker News Reader

Visually stunning math concepts which are easy to explain

math.stackexchange.com

560 pointsaaronbrethorst78 comments

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

Claudio Rocchini [1] has also made impressive contributions to the field of mathematical visualization.

[1] https://en.wikipedia.org/wiki/User:Rocchini

Lucas also has an interesting blog: http://1ucasvb.tumblr.com/

And Matt Henderson has some good animations too: http://blog.matthen.com/

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.

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.

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

You're a special little snowflake then. Here's your citation: [SOURCE]

What?

As a person who is generally considered to be good with scary numbers, I can confirm that this is exactly how it works for me.

For the record, I have no trouble at all believing that it works that way for many people. The claim was that this is how it works for everyone, and that I find questionable - in my experience, some people work better verbally than visually. I'm pretty good visually, but it definitely feels like I'm engaging different systems when I am reasoning using words than when I am reasoning using images - and I don't expect this generalizes.

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

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

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

"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

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...

"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).

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

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

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

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

"A formula is worth a thousand pictures." [1]

Also, draw a picture of the monster group (http://en.wikipedia.org/wiki/Monster_group), 11 dimensions of space time (http://en.wikipedia.org/wiki/M-theory), or explain how a picture of seven infinitely long cylinders can illustrate that each pair of those cylinders touches each other.

So, I disagree that "everyone ends up reasoning with drawings".

[1] https://www.utexas.edu/faculty/council/2002-2003/memorials/D...

There's a formalism that we manipulate to reason, but we're always aided by geometric and spatial analogies. The Monster is a permutation group, which means it moves things around, and we visualise 11 dimensions by considering 3 dimensional analogues or by projecting the shadow of 11-dimensional objects onto 2 or 3 dimensions.

> But in the end, everyone ends up reasoning with drawings.

That is a big generalisation, I have solved problems without using visualisation.

How would you interpret your comment if someone was congenitally blind?

The only blind mathematician I ever knew worked with fractals and was able to draw fractals on the computer and then accurately describe what these fractals looked like.

Newton actually released most of his work as graphical proofs, not calculus, partly for this reason (and partly because he was an asshole who wanted to keep his methods a trade secret).

Before Newton, math was graphical. The problem is, it's hard to come up with graphical proofs. It's a lot easier to reason symbolically. If you want to solve mathematical problems, equations may be better. If you want to explain your reasoning, then diagrams can be better.

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

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.

There's interest. I've taken point-set topology, so hopefully I'll be able to follow the proof without too much trouble. If so, building a simulation shouldn't be too difficult.

See my reply to rkaplan's comment here: https://news.ycombinator.com/item?id=7544735

I'm interested. What's the solution?

Here's a link to the proof I typed up for my problem set:

https://www.sharelatex.com/project/534230c0234f079f3ce526fa?...

Theorem 1.10 states "If S andT are countable sets, the set S ×T ={(s,t) : s∈S,t∈T} is countable." and is proven using a diagonal argument.

Makes perfect sense. I'll code something up next weekend.

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.

Yep, this is how the proof works!

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, http://mathworld.wolfram.com/CantorDiagonalMethod.html)

I've often seen hypercubes pictured or described like this: 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

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.

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 2. http://math.stackexchange.com/a/734887/141120

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

Where were these when I was in school?

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.

Give this a shot: http://betterexplained.com/articles/an-interactive-guide-to-...

In the beginning, the red graph f is a square wave. It's a little lumpy, because it was made by adding together a bunch of different sine waves of different frequencies and amplitudes (by the way, that slight overshoot/undershoot at the edges of the square wave? That's the Gibbs phenomenon). When we look at a square wave in this form, with time running along the x-axis (like the display of an oscilloscope), we call it the "time domain".

It's hard to visualize the individual sine waves when they all overlap like that, so let's spread them out. The logical "direction" in which to spread them is by frequency, so imagine spreading them out in the frequency dimension "behind" the square wave. When the graph rotates into three dimensions, we can see all the individual sine waves: in fact, there are six of them. (If we used more than six, the square wave would be less lumpy; fewer sine waves, more lumpy.)

Notice that the sine waves are different amplitudes, and discrete frequencies. We represent them by spikes on a graph where frequency runs along the x-axis (like the display of a frequency analyzer) and the height of each spike is the amplitude of the sine wave at that frequency.

We've transformed a square wave f in the time domain, into a spectrum f-hat with spikes of different positions and heights in the frequency domain. You can read the blue graph as "six equally spaced frequencies with decreasing amplitudes". That's the fourier transform of the original (red) signal. Because it's a transform, it works in the other direction, too: begin with half a dozen signal generators, set their frequencies and amplitudes according to the spikes on the blue graph, add them together, and the result will be a square wave (or a reasonable facsimile thereof).

It can be done on signals of more than one dimension, too: take a cat photo, transform it into something that looks a bit like a starburst, then transform the starburst back into a photo of a cat.

Note: interesting and oftentimes useful things happen when you transform a time-domain signal into the frequency domain, erase some of the spikes, and then transform back into the time domain.

While the visualization is attractive, my fear is that it cannot convey the deeper reason why those sine and cosine waves magically sum to the desired function.

Leaving rigour aside for the moment: think of functions f : R -> R as infinite-dimensional vectors. The integer harmonics of sine and cosine comprise a set of orthonormal "vectors" that form a basis for all functions on R (some fine print goes here).

Now compute the inner product of your desired function with every element of that basis. Each such inner product is a real number which we will call a coefficient. The list of nonzero coefficients, once you have computed them, is a complete description of your function.

Now it is clear why those sine and cosine functions "magically" add up to your desired function, since we are simply multiplying each of them by their corresponding coefficient that we computed above.

That visualization is no more (or less!) amazing than the fact that (1, 2, 3) = 1(1, 0, 0) + 2(0, 1, 0) + 3(0, 0, 1).

My attempt to try to help you:

1st picture: you see a red signal and “f” which means that the signal is in the “time world”. The x axis represents the time.

2nd picture: The “f” disappears and a lot of signals in blue color appear. If you add up all these blue signals the result is the red signal. As you can see there are a lot of different blue signals with different frequencies.

3rd picture: they do that 3D breakdown where you can see each signal and on the right appears another graph/plot which represents the Fourier Transform. This graph/plot is in the “frequency world” and the x axis represents frequency. Each frequency of each blue signal is represented as a straight line. This line is a only blue signal located in its frequency and the more larger is in the y axis the more representative is in the final result (which is the red signal). The largest straight line is the one on the left because is the one with more energy in the “time world” and for that is the more representative in the frequency plot. The f with that arch is the Fourier Transform.

4th picture: they show you the signal on the “time world” and the signal in the “frequency world”.

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 (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

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.

> 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 ? I suppose there is some further magic which allows for phase differences or something.

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(2pik*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).

Does anyone know if the author is the first person to use this visualization? The series of rotating circles is incredibly illuminating.

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...

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

Trinomial cube: https://www.google.com/search?q=binomial+cube&source=lnms&tb...

Animated / interactive Bezier curves: http://www.jasondavies.com/animated-bezier/

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.

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

One more link http://world.mathigon.org/

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...

I've found these videos to be fairly interesting.

https://www.youtube.com/watch?v=yJZP_-40KVw&list=PLN0wPs8UzD...

How about a quick explanation of radians?

http://24.media.tumblr.com/dd1b123f13f5578e11b04d7579df1fce/...

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.

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.

Here's one for the integral of y=x^2: http://www.mathedpage.org/proof/integrating/

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.

I just posted a visualization of multiplication of the integers modulo n http://math.stackexchange.com/questions/733754/visually-stun... to this question.

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

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

Are you blocking js? It should be formatted with mathjax to look like latex output.

It isn't for me either and I don't think I'm blocking js.

That is strange. How about these demos from mathjax:

tex: http://www.mathjax.org/demos/tex-samples/

mathml: http://www.mathjax.org/demos/mathml-samples/

I discovered it was https everywhere doing it.

I hate to say it but I have ended up turning off a lot of the mixed-content/partial site rules for this reason.

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

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

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

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

Personally, I can always use some music to actually get stunnned :)

http://www.youtube.com/watch?v=kkGeOWYOFoA

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.)

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.

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

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

Anybody up to visualizations about quaternions?

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.

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

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

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

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

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

Reply on news.ycombinator.com