One of the most incredible feelings is when you make that new connection of understanding on an idea. Sometimes it's when two things you knew get connected in a way you didn't think existed, and sometimes it's when a complicated idea all fits into place in your mind. It's probably the drug that keeps people programming despite all the configuration hell we have to deal with on any non-trivial project (and even most trivial ones).
Every single one of 3Blue1Brown's has given me a big hit of that new brain connection drug. If you enjoy this video, I recommend checking out 3Blue1Brown's video "What does genius look like in math? Where does it come from? (Dandelin spheres)", which deals with how and why ellipses and conic sections are related. https://www.youtube.com/watch?v=pQa_tWZmlGs
I love the feeling you're describing! The understanding of an abstraction (or in other words, a mental model) that links two parallel thought processes in an unexpected, and fun way.
I like the term Kensho [0], thought you might too. Interested in thoughts from other people as well.
I like the concept of kensho. I find the idea that certain ideas can only be accurately conveyed through experience to be interesting as well.
I'm working my way through a book called "The Book Of Secrets", which has 114 different meditative techniques for different kinds of minds. One of which, is practically certain to work for any given individual.
Moreover, I would recommend to watch most of their videos, at least to myself. No one could introduce me to so many ideas in so little hours without leaving any open question. This channel and its companions are fantastic if you somehow missed deep math sense at school. It is not “lets learn another seemingly useless theorem” study, it is kickstart introduction to selected interesting topics.
3Blue1Brown is one of the best math educators on YouTube. As a matter of fact, I would say he's the best I know of at the moment. I'm currently working my way through his series on diff-eq.
So... typically this is the part where there might be some kind of
sponsor message. But one thing I want to do with the channel moving
ahead is to stop doing sponsored content, and instead make things just
about the direct relationship with the audience.
I mean that not only in the sense of the funding model, with direct
support through Patreon, but also in the sense that I think these
videos can better accomplish their goal if each one feels like it's
just about you and me sharing in a love of math, with no other motive,
especially in the cases where viewers are students.
=====
He's really a good person. I'm glad that now he is able to do this full-time by just the support from his patreon.
Not only does he have a penchant for finding geometric analogues for mathematical ideas that are neat and beautiful, his animations clarify and increase my understanding of those concepts.
Was just going to ask this. Wish there were more of these tools all around, because I bet a lot of experts have developed intuitive pictures of their own that would be good to convey directly. A big barrier to mathematics is the abstraction cost of the language. In many cases mathematical expressions are fine, but not always in mathematical physics.
How are his videos on linear algebra? Could they serve as a complete resource to learn linear algebra or would you want something alongside/more material?
His videos on linear algebra are excellent, but they are not a complete resource. If you want a comprehensive resource, search for the LA course by "maththebeautiful" on YouTube. Also check out Gilbert Strang's MIT lectures.
Indeed, the MIT lectures by Prof Strang and the MathTheBeautiful videos by Prof Grinfeld provide perfect pairings to go with the enhanced visual intuition you acquire on 3Blue1Brown.
NB: MathTheBeautiful [1] is by MIT alum Pavel Grinfeld [2]. He approaches Linear Algebra from a geometric perspective as well, but with more emphasis on the mechanics of solving equations. He has a ton of videos organized into several courses, ranging from in-depth Intro to Linear Algebra courses to more advanced courses on PDEs and Tensor Calculus. Highly recommended.
search for the LA course by "maththebeautiful" on YouTube. Also check out Gilbert Strang's MIT lectures.
I just realized there's a deep connection among your recommendations...Gilbert Strang was Greenfield's PhD advisor: https://dspace.mit.edu/handle/1721.1/29345. Pavel has a clear and precise teaching style like Strang, and I've heard him reference Prof's Strang's courses before but didn't make the connection. Good trees produce good fruit. The small world graph and the truth of that never ceases to amaze. Good trees, good stuff.
They are absolutely amazing for the intuition but lack the mechanics, both of which are essential for linear algebra. The calculations are necessary, but without the intuition they will seem meaningless and the nuances will be difficult to remember.
I would pair with Gilbert Strang MIT open courseware and Khan Academy. The 3 together made linear algebra probably the most useful mathematics class Ive taken.
I loved his videos on linear algebra. Were far more intuitive than anything I studied in engineering. In particular, the way he describes what a determinant is, and what value it has, is excellent. I'm proud to be a Patreon supporter of 3Blue1Brown.
Grant (3Blue1Brown) sounds exactly like a friend of mine named Brad. Just absolutely uncannily sounds exactly like him, even in his intonations and quirks. I've been watching his videos for over a year and using his epiphany-driving videos as the triggers to dig into specific subtopics of math that I used to hate (\cough\calc2\cough\). Turns out I love math and not just discrete math!
Anyway, my funny story about my friend Brad. I have convinced our circle of friends that Brad is actually the guy doing the 3Blue1Brown videos. He's hilarious and just started playing along at dinner one night without even knowing what I was talking about. So most of our friends now think Brad's got a huge but secret Youtube channel where he teaches incredibly insightful math perspectives.
There are times when I watch his videos that I wonder whether he could be an alien from an advanced civilization sent here to help accelerate our mathematical understanding and reach the singularity sooner. His explanations are 5 standard deviations better than average.
A similar concept related to elliptical constants and proportional areas, is the curve formed by the cycloid [1] of a circle, which is the curve formed by tracing a point on the circumference of a circle as it rolls across a surface from point A to point B.
For example, the fastest distance between two points is not a straight line, it's the cycloid, specifically the Brachistochrone curve [2]. This is the path light follows.
One common misconception of the cycloid is related to its arc-length. At first glance many assume the arc-length of the cycloid is equal to circumference of the circle, but this is not the case. The line-of-sight distance between the cycloid starting point A and ending point B is equal to 2πr, the circumference of the circle -- however the arc-length of the cycloid curve is 8r, which is an integer value given an integer radius. The cycloid curve is full of interesting properties, many yet to be discovered and all its implications are not yet fully understood.
Another interesting aspect of the cycloid is not only is it the fastest path, but no matter where two objects begin on the curve, they'll both traverse the curve at maximal/optimal speed and both will arrive at the bottom of the curve at the same time, regardless of the delta between their starting positions. This aspect of the cycloid is referred to as the Tautochrone curve [3].
So if you're looking for ways to distribute partitions or encode invariants in your models, data or otherwise, the geometrical aspects of elliptical and cycloidal curves are a good place to explore.
And Vsauce did one with Adam Savage on the Brachistochrone where they build a mechanical model of one that shows it's the fastest/optimal path among different curves, and their experiment also shows the cycloid Tautochrone invariant property where objects begin up the curve at different distances apart and yet all arrive together simultaneously in constant time. https://www.youtube.com/watch?v=skvnj67YGmw
NB: Consider this, two seperate impulses of light beginning at different distances away from the observer, both impulses of light traveling along the optimal path at the optimal speed, and both arriving at the observer simultaneously, without bending time. And as shown above, on a cycloidal curve, this phenomenon is not unique to light.
>For example, the fastest distance between two points is not a straight line, it's the cycloid, specifically the Brachistochrone curve [2]. This is the path light follows.
You may be conflating [Bernoulli's solution to the Brachistochrone curve](http://www.math.rug.nl/~broer/pdf/ws-ijbc.pdf) with the optimal path for light. [Fermat's Principle](https://en.wikipedia.org/wiki/Fermat%27s_principle) states that, when traveling between two points, light will always take the path that minimizes the time taken from the first point to the last. In a medium of constant refractive index (which includes free space), this results in a line.
>So if you're looking for ways to distribute partitions or encode invariants in your models, data or otherwise, the geometrical aspects of elliptical and cycloidal curves are a good place to explore.
How do you mean? What sort of data can be encoded this way and how?
> NB: Consider this, two seperate impulses of light beginning at different distances away from the observer, both impulses of light traveling along the optimal path at the optimal speed, and both arriving at the observer simultaneously, without bending time. And as shown above, on a cycloidal curve, this phenomenon is not unique to light.
Two separate impulses of light beginning at different distances away from a stationary observer will necessarily arrive at different times, otherwise you violate the basis of special relativity: the speed of light is constant and invariant of reference frame.
You can solve the brachistochrone problem using Fermat's principle, as shown in the 3blue1brown video GP is referring to. Since you're looking for the shortest time, if you can construct a lens where the speed of light is proportional to the speed of the bead on a wire, then the shortest-time wire is the path that light would take by Fermat's principle, and then you can use (an infinitesimal version of) Snell's law to find the direction of the wire at each height.
>For example, the fastest distance between two points is not a straight line, it's the cycloid, specifically the Brachistochrone curve [2]. This is the path light follows.
Which is not true for free space, or any space with a constant index of refraction.
Oh yes, they do seem to be confused! espeed, if you happen to read this, the brachistochrone is the fastest path for something accelerated by a constant force (e.g. gravity near the surface of the earth).
I think I need to quote guy who inspired 3blue1brown's video now too: "Physics is like sex: sure, it may give some practical results, but that's not why we do it." R.P. Feynamn, you are missed.
Every single one of 3Blue1Brown's has given me a big hit of that new brain connection drug. If you enjoy this video, I recommend checking out 3Blue1Brown's video "What does genius look like in math? Where does it come from? (Dandelin spheres)", which deals with how and why ellipses and conic sections are related. https://www.youtube.com/watch?v=pQa_tWZmlGs