Show HN: Explained Visually | Hacker News Reader

Show HN: Explained Visually

setosa.io

353 pointsvicapow45 comments

Don't love the network graph illustrating the spread of a virus.

It's deceptive and is counter to the purpose of explaining exponential growth. If the people infected are represented by relatively uniform points on a plane, and infection is shown as an ever expanding circle, the growth is geometric, not exponential. I understand that the exponential growth could be captured by the acceleration of the radius, but that is not intuitive.

Actually it shows polynomial growth, not geometric growth. Polynomial growth is of the form t^n whereas geometric/exponential growth is of the form n^t. (The only real difference between geometric and exponential growth is that geometric connotes discrete growth steps; otherwise they are the same.)

In any case, I totally agree with your point. The virus graph is quite misleading because it shows a growing circle. The plot beside the graph doesn't even look like exponential growth!

Yikes! I've been using "geometric growth" to mean "polynomial growth" my whole life.

Darn it, me too. I always thought geometric was a special case of polynomial where you only have a ax^2 term. I figured this was a reference to the fact that geometry is usually done in the plane, and linear increase corresponds to square increase in area.

I think I may have even pedantically "corrected" someone on this at some point. Awesome.

Trigonometry please (specifically Sin, Cos, Tan)! I can calculate the values when I need them, but I want to understand the 'why' and the 'how' on a fundamental level.

That's a great idea.

http://betterexplained.com/articles/intuitive-trigonometry/ and the whole Better Explained website is pretty great for things like this. Their article on the constant e is what finally helped me to understand what e was: http://betterexplained.com/articles/an-intuitive-guide-to-ex...

cool! I will check it out. thank you

For visual trig, there's a great interactive unit circle here:

http://www.touchmathematics.org/topics/trigonometry

Study the way the color-coded lengths vary as theta increases around the unit circle and you'll develop an intuitive sense of sin, cos, tan, etc.

Thanks for the link. I played around with it for a couple minutes and will be coming back to it later for sure.

This is from a fun thread that's been posted here before:

http://math.stackexchange.com/a/734790

With trigonometry, Some key ideas that I like are:

sin, cos and tan refer to right-angle triangles (RAT) - it seems amazing that so much math can be based on the simple idea that we start with a triangle where one of the angles = 90 degrees;

sin, cos and tan are the ratios of two of the sides of a RAT: for example, cos(60) = 0.5 says that the ratio of the two arms of the 60 degree angle of the RAT is always 1:2. Saying it again: if you have a RAT with one angle = 60 degrees (the other angles are 30 & 90 degrees), one arm of the 60 degree angle = 1 unit and the other arm will be twice as long, that is 2 units (this second arm is the hypotenuse, the longest side, of the triangle).

The key idea is that sin, cos and tan are RATIOS. (The word similarity of RAT, an abbreviation I devised for this comment, and RATios has only just struck me.)

The ancient Egyptians were faced with a dilemma: the annual floods of the Nile destroyed the boundary markers of the farmers, so they had to come up with a means of surveying the land. The solution: the right-angle triangle.

Oh hey, I just got back to HN and saw this. Thanks for getting the simple concept of "RATios of sides" into my head! Inching closer to understanding :-)

Here's a StackExchange thread full of similar visualizations (which has been shared here a few times in the past):

http://math.stackexchange.com/q/733754

I am jealous of the engineering students to come after me. Can you imagine when every advanced textbook is so clear and interactive? Eigenvalues, differential equations, relativity...

watch for one on eigenvalues and vectors soon

Yes!! I cannot wait for the eigenvalues one.

I will watch for it :-)

I'm even jealous of the engineering students that came before me! :P

Visualization is such a powerful tool. I didn't really understand waves until I saw "Similarities of Wave Behaviour" https://www.youtube.com/watch?v=DovunOxlY1k

This picture...

http://en.wikipedia.org/wiki/File:Eigenvectors.gif

...is pretty famous by now and helps a lot of people, but there's actually more information in it than the description below it lets on.

It's a two-dimensional plot, with two-dimensional vectors, and the number of eigenvectors is two. Well, the number of families of eigenvectors; the number of colours. By noticing that they're orthogonal to each other, you can imagine that if we were in three dimensions, there'd be a third one, orthogonal to the other two.

The number of eigenvalues is pretty obvious from the maths (the characteristic eqn will always have a lambda^n in it), but that there's the reason graphically. Most people only think of that picture as the behaviour of eigenvectors, but it also allows you to see the space they occupy and their relation to each other.

(oh, and dim(graph) == dim(matrix) because dim(graph)==dim(vectors)==dim(matrix))

While I'm on the subject, I think trying to visualise everything in two dimensions can set you up to make certain mistakes. Most people think that if lines aren't parallel, then they touch once and then diverge. This is true of planes, not lines, in general.

> orthogonal to each other

by which I meant: parallel and orthogonal to the linear transformation, not to each other. same thing for the third one.

An example of eigenvalues: Suppose you take a polynomial of x and then you change x for -x, you obtain a new polynomial. Some polynomials doesn't change they are associated to eigenvalue 1, other change sign they are associated to eigenvalue -1, those are the only ways in which a polynomial gets transformed into a multiple of itself when changing sign. And what happens with the others polynomials? Well, you can write any of them as a sum of those with even exponents (those that doesn't change when changing the sign of x) and those with odd exponents (those are the ones that change sign), so any polynomial can be expressed as the sum of an even polynomial and an odd polynomial in a unique way. In the same way, taken the conjugate of a complex number the part that doesn't change is the real part (associate to eigenvalue 1) and the part that change sign is the imaginary part (associate to eigenvalue -1) and every complex number is written as the sum of its real and imaginary part. Take a matrix and computes its transpose, the matrix can be expressed as a sum of a symmetric matrix (corresponding to eigenvalue 1, doesn't change with the transpose operator) and an antisymmetric matrix (change the sign, associated to eigenvalue -1). Finally take for example any function of x and consider the function obtained when you change x to -x, then any function can be expressed in a unique way as the sum of a even function (corresponding to eigenvalue 1, that is doesn't change with that transformation) and an odd function (associate to eigenvalue -1, that is change sign). For example our familiar function exponential of x is the sum of the hyperbolic cosiness and the hyperbolic sinus.

This way you are near the Euler Formula:

e^x = cosh(x) + sinh(x) (real case)

e^it = cos(t) + i sin(t) (complex case)

I must add that in this example the transformation satisfies that applied two times is the identity, that is called involution A^2=1 and the only eigenvalues K are those that K^2=-1 that is 1 and -1.

This is incredible. I still remember the first time I read through Eliezer Yudkowsky's Intuitive Explanation of Bayes Theorem [1], and how it was one of those moments where the lense with which I saw the world changed forever. I still refer people to that page whenever I have a tough time explaining it myself, as well as a handful of other pages (now including this one).

I know Show HN is typically used for getting constructive criticism, but I don't have much to say there other than to keep it coming.

http://yudkowsky.net/rational/bayes

Bret Victor is doing some amazing work in this area. http://worrydream.com/#!/InteractiveExplorationOfADynamicalS...

As a visual thinker, this is great! Especially for mathematics which I was great at until secondary school began, where I struggled a lot due to impatient math teachers poorly explaining concepts.

Congrats to the team for shipping this, I've subscribed and hope to see more stuff like this. It would be great if you could explain basic concepts (and gradually to complex formulas) and hopefully beginners who struggle due to bad teaching can bump into your website and keep their interest alive.

I couldn't agree more - as a highly visual person, I have struggled to grasp many mathematical concepts over the years, despite making it all the way through Calc 3.

There are many fundamentals I would love to brush up on / really understand, and this looks like it might be a great tool for that.

For Calc 3, Mathematica and Wolfram Alpha really helped me.

This is very cool!

As a piece of constructive feedback, consider waiting until a visualization is scrolled into view to start animation. I'm particularly thinking of the Exponentiation page. Mike Bostock wrote about this recently: http://bost.ocks.org/mike/scroll/#4

An example of a visual explanation I would like to see:

Suppose the length of your thumb doubles in each step then in 33 steps it would be equal to the distance from earth to the moon.

I would like to see such an animation, being able to touch the moon with my finger in 33 steps is to grow really fast.

This is awesome! Thanks for sharing. Can't wait for Eigenvalues to be covered! Shameless plug: I created a similar visualization for explaining Monte Carlo simulations by computing Pi. http://montepie.herokuapp.com/

It would be interesting to see if you could explain something like galois set theory in purely visual terms.

Galois theory is a thing, and set theory is a thing, but I'm pretty sure "Galois set theory" is not a thing.

Visual Group Theory by Nathan Carter? I don't know, I've only ever read it while procrastinating on something else. So I haven't seen the entire book.

On your main page I'd mention Bayes theorem on the entry for conditional probability since it might get anyone curious about Bayes theorem and satisfying it with a search engine to your page for the best explanation I've ever seen.

Great! Any chance of hosting this project on Github so more people can contribute?

+1. Have a couple of nice ideas on the new things I'm learning e.g. matrices, page rank etc. and I would love to contribute visualizations!

Awesome. I've been wanting to seed a project like this for a while. You should definitely be pushing this open source and soliciting contributions.

Older version: https://news.ycombinator.com/item?id=8103240

Wow. This is awesome. I never really, really understood Markov Chains. Really very well designed.

What's interesting to me is that even if the animations were gone and we were just shown static images I'd still be able to grok the explanation of Markov Chains based on the way he explained it. Very impressive.

you explained the concepts very well...I am especially impressed with the way you effectively used animation to bring the concepts to life. I would like to know what tools you used to create the analytics (R language?) and visualization(D3.js?).

You should add the raft visualization which has been out there for a while now.

Man, this is gold. Kudos.

They should offer this to Kahn Academy.

The future of education.

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