A most unexpected answer to a counting puzzle [video](youtube.com)
youtube.com
A most unexpected answer to a counting puzzle [video]
https://www.youtube.com/watch?v=HEfHFsfGXjs
27 comments
I understand maybe 30% of 3blue1brown’s videos, mostly due to his excellent visualizations, but he really gets me excited for math in a way I’ve not experienced before.
How does he make such awesome visualizations? Does anyone know what tools are being used?
He built a python library for his visualizations called manim available on his github [1].
[1] https://github.com/3b1b/manim
[1] https://github.com/3b1b/manim
A few tools, detailed in his FAQ page.
https://www.3blue1brown.com/faq/
He has written a python library called "manim" for his visualizations https://github.com/3b1b/manim
The explanation video: https://www.youtube.com/watch?v=jsYwFizhncE
An interesting response to the explanation video from u/functor7 on reddit:
https://www.reddit.com/r/math/comments/ahz8k3/so_why_do_coll...
https://www.reddit.com/r/math/comments/ahz8k3/so_why_do_coll...
This is kind of a plea for help - how can I learn maths 'via' geometry?
I just can't get my head around 'numbers'. I can't seem to really visualise them or understand what they are. I know this sounds weird. With language or abstract things like 'problems' I can almost see them in my head. I can analyse a problem and all it's flows of cause and effect in a visual way. That's literally what happens in my head.
But with numbers I can't understand them at a basic level. Like - what are they? How do I visualise them?
When it comes to geometry it's almost like it's the 'language' of my brain, how I think, in terms of shape and form and translation and dimension.
I'll give you an example. I learnt Pythagoras theorem aged about 13 or something. But didn't truly understand it until I was about 20 something when I saw a diagram showing graphically the squares of the sides like this[0].
I really need to start at the absolute beginning, like build of a whole foundation of what maths is based on geometry. Does that even exist?
This might not be possible, but if I can even get started it would be great.
[0] https://mysteriesexplored.files.wordpress.com/2011/08/pythag...
I just can't get my head around 'numbers'. I can't seem to really visualise them or understand what they are. I know this sounds weird. With language or abstract things like 'problems' I can almost see them in my head. I can analyse a problem and all it's flows of cause and effect in a visual way. That's literally what happens in my head.
But with numbers I can't understand them at a basic level. Like - what are they? How do I visualise them?
When it comes to geometry it's almost like it's the 'language' of my brain, how I think, in terms of shape and form and translation and dimension.
I'll give you an example. I learnt Pythagoras theorem aged about 13 or something. But didn't truly understand it until I was about 20 something when I saw a diagram showing graphically the squares of the sides like this[0].
I really need to start at the absolute beginning, like build of a whole foundation of what maths is based on geometry. Does that even exist?
This might not be possible, but if I can even get started it would be great.
[0] https://mysteriesexplored.files.wordpress.com/2011/08/pythag...
If you managed to survive high school algebra, you might be able to 'redo' your basic math education with a geometric bent from Stillwell's book 'Numbers and Geometry', which teaches the classical link between the two subjects that has been eliminated from elementary education to some extent. However, the kind of intuition he presents is still semi-abstract - it's not a super duper picture heavy book - but I think it's about as close to what you're asking for as you are likely to find.
https://www.amazon.com/Numbers-Geometry-Undergraduate-Texts-...
https://www.amazon.com/Numbers-Geometry-Undergraduate-Texts-...
I think I know what you are getting at. You may have something like dyscalculia[1], but I'm not a doctor.
There are several mental models used for picturing numbers. I think the most popular is the number line [2]. Common Core uses this in a clever way. I've never found it useful personally.
I suspect some people might see numbers as purely symbolic and perform purely symbolic operations. When the symbol 2 and the symbol 3 have the operation 'add' applied, it yields the symbol 5. Or something.
I think I mostly visualize the numbers as a quantity of solid objects. So I think of 3 as 3 things. This has limitations. I can break objects apart but how do you visualize 3.767? So some of it is intuitive.
There is evidence that the human mind can only encode exact quantities up to four [3], after that they use an estimate. [4] So maybe your situation is the norm. What I mean is, sometimes it can feel like everyone else has 'got it' and you're the one lacking, when in reality everyone is just as bad.
[1] https://en.wikipedia.org/wiki/Dyscalculia [2] https://en.wikipedia.org/wiki/Number_line [3] https://en.wikipedia.org/wiki/Parallel_individuation_system [4] https://en.wikipedia.org/wiki/Approximate_number_system
There are several mental models used for picturing numbers. I think the most popular is the number line [2]. Common Core uses this in a clever way. I've never found it useful personally.
I suspect some people might see numbers as purely symbolic and perform purely symbolic operations. When the symbol 2 and the symbol 3 have the operation 'add' applied, it yields the symbol 5. Or something.
I think I mostly visualize the numbers as a quantity of solid objects. So I think of 3 as 3 things. This has limitations. I can break objects apart but how do you visualize 3.767? So some of it is intuitive.
There is evidence that the human mind can only encode exact quantities up to four [3], after that they use an estimate. [4] So maybe your situation is the norm. What I mean is, sometimes it can feel like everyone else has 'got it' and you're the one lacking, when in reality everyone is just as bad.
[1] https://en.wikipedia.org/wiki/Dyscalculia [2] https://en.wikipedia.org/wiki/Number_line [3] https://en.wikipedia.org/wiki/Parallel_individuation_system [4] https://en.wikipedia.org/wiki/Approximate_number_system
Just one suggestion, but https://www.euclidea.xyz/ is a great place to start. Visual Group Theory sounds appropriate too: https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/0...
I think this is a better image for the Pythagorean theorem:
https://i.imgur.com/UxZ3fQR.png
https://i.imgur.com/UxZ3fQR.png
Have you watched the khan academy videos? He leans on geometry and visual intuition quite a lot.
Geometric proofs of problems that do not seem to be geometric are the most satisfying in my experience.
I wrote about this nearly four years ago:
https://aperiodical.com/2015/03/%cf%80-phase-space-and-bounc...
It doesn't have nifty visualisations, but the (mythical) interested reader might find this a useful complement to the video and explanation provided there.
Edit: Thought so - I did submit my write-up at the time:
https://news.ycombinator.com/item?id=9202913
https://aperiodical.com/2015/03/%cf%80-phase-space-and-bounc...
It doesn't have nifty visualisations, but the (mythical) interested reader might find this a useful complement to the video and explanation provided there.
Edit: Thought so - I did submit my write-up at the time:
https://news.ycombinator.com/item?id=9202913
Your links to the original paper are broken in that writeup.
Bother.
Thanks for the catch, I'll get onto the hosts and get them to fix them.
Cheers.
Thanks for the catch, I'll get onto the hosts and get them to fix them.
Cheers.
And for completeness - now fixed. Thanks again.
I wrote a small Java program to verify the collision count, just because I was bored and laying on the sofa:
> https://github.com/Qualsiasi85/collision-pi
warning: code is a mess - and may have all sorts of nasty bugs, but after all this has been done for fun and doesn't need to be "enterprisey" :)
I do like this kind of videos, they tickle my fantasy and keep alive my passion in programming (that is slowly dying because of work! :) )
> https://github.com/Qualsiasi85/collision-pi
warning: code is a mess - and may have all sorts of nasty bugs, but after all this has been done for fun and doesn't need to be "enterprisey" :)
I do like this kind of videos, they tickle my fantasy and keep alive my passion in programming (that is slowly dying because of work! :) )
There was an ACM problem a few years ago where it was asked to count the collisions. We implemented the simulation and noticed the digits of PI but we didn't actually proven the fact. Got AC.
Nice, but I miss the more fundamental series of 3Blue1Brown like "the essence of linear algebra" series, the "essence of calculus" series, and the unfinished "deep learning" series from more than a year ago
Shame that this awesome visualization method he uses isn't used more for more fundamental concepts to learn from the ground up rather than specific puzzles like lately, I learned a lot from the "essence" series!
Shame that this awesome visualization method he uses isn't used more for more fundamental concepts to learn from the ground up rather than specific puzzles like lately, I learned a lot from the "essence" series!
The essence of linear algebra is vector spaces and linear transformations. It's not entirely impossible (although probably way more difficult) to learn it without reference to Euclidean spaces and geometric reasoning.
Should be called "The intuition for linear algebra". But heck, the whole reason we have and teach the mathematical method is that intuition either breaks down in interesting cases (e.g. stochastic integrais versus "an area under a curve" integrals) or limits you (linear algebra in function spaces, etc).
It's a nitpick. I love 3b1b.
Should be called "The intuition for linear algebra". But heck, the whole reason we have and teach the mathematical method is that intuition either breaks down in interesting cases (e.g. stochastic integrais versus "an area under a curve" integrals) or limits you (linear algebra in function spaces, etc).
It's a nitpick. I love 3b1b.
The solution he posted is about using phase spaces, not really about this problem in particular.
> Shame that this awesome visualization method he uses isn't used more ...
The animation engine is on GitHub so you can make your own: https://github.com/3b1b/manim
The animation engine is on GitHub so you can make your own: https://github.com/3b1b/manim
I think the are in the works, but it's just a guess. There's a subreddit for fans, you might post your wish list :)
https://www.reddit.com/r/3Blue1Brown/
https://www.reddit.com/r/3Blue1Brown/
By the way, I don't want to sound unhappy about the videos, the answer for sure is interesting indeed, I totally don't want to say it's bad content at all, it's awesome in every way, graphics, audio, explanation, ... :D