

Show HN: Explained Visually - vicapow
http://setosa.io/ev/

======
JackFr
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.

~~~
tedsanders
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!

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

~~~
sopooneo
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.

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loco5niner
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.

~~~
minikites
[http://betterexplained.com/articles/intuitive-
trigonometry/](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...](http://betterexplained.com/articles/an-intuitive-guide-to-exponential-
functions-e/)

~~~
loco5niner
cool! I will check it out. thank you

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BHSPitMonkey
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](http://math.stackexchange.com/q/733754)

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bernardom
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...

~~~
lewis500
watch for one on eigenvalues and vectors soon

~~~
twistedTightly
Yes!! I cannot wait for the eigenvalues one.

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barbudorojo
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.

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saosebastiao
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](http://yudkowsky.net/rational/bayes)

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varunjuice
Bret Victor is doing some amazing work in this area.
[http://worrydream.com/#!/InteractiveExplorationOfADynamicalS...](http://worrydream.com/#!/InteractiveExplorationOfADynamicalSystem)

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computerjunkie
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.

~~~
twistedTightly
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.

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

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bgschiller
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](http://bost.ocks.org/mike/scroll/#4)

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barbudorojo
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.

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krat0sprakhar
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/](http://montepie.herokuapp.com/)

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softdev12
It would be interesting to see if you could explain something like galois set
theory in purely visual terms.

~~~
smorrow
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.

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DonGateley
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.

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

~~~
krat0sprakhar
+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!

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chatmasta
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.

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j2kun
Older version:
[https://news.ycombinator.com/item?id=8103240](https://news.ycombinator.com/item?id=8103240)

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

~~~
Yhippa
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.

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gordon_freeman
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?).

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

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RichardZite
Man, this is gold. Kudos.

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amelius
They should offer this to Kahn Academy.

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zedadex
The future of education.

