This is almost surreal to say,
but the channel recently passed 1 million subscribers!
And I know what you're thinking:
"just 48,576 more to go before the next big milestone!"
And indeed, I'll hold off proper celebrations until then.
But, you know, seeing that seventh digit really does make you reflect on what led to this point.
And if I'm being honest,
a lot of the growth for this channel just had to do
with some very kind people sharing and promoting the content,
both in terms of certain specific shoutouts from creators with big audiences,
and in terms of individuals just sharing with their friends,
so what I want to do now is take a moment to let you all know about a few other math creators online
who I think you would like a lot.
There's just so much good creation that happens under a lot of people's radars.
First up, we have the channel "Think Twice".
I honestly have no idea why this channel isn't better known among online math communities.
It contains these really beautifully done math animations,
and if you like this channel,
you're definitely gonna like it.
You absolutely get the sense looking at any one of these
that it comes from someone who thinks very visually about math.
Naturally, this often takes the form in some very pretty geometry
or some algebraic results that are explained geometrically,
and the focus tends to be a little bit more short form and bite-sized pieces,
so if you ever want a quick little reminder of why math is beautiful,
This is just a great channel to pop over to.
Here, let me just let a couple more of these animations just play out.
Next up, I'd recommend you check out LeiosOS, run by James Schloss.
what I like about James is that he seems to think first and foremost about community,
and creating content seems to be more of a vehicle for bringing people together online.
Let me turn things over to him to explain what it's all about, both on and off YouTube.
James Schloss: LeiosOS is fundamentally a channel about algorithms and works with the Arcane Algorithm Archive,
an open source and collaborative effort to document every algorithm in existence in every language possible.
We're just getting started and it's obviously an impossible goal,
but that's half the fun.
This means that we cover a range of different topics and here are some clips of the videos.
Any light that enters a lens leaves on the other side of the lens with the same angle it came in with.
it's almost as if the light never entered the lens at all,
and hence the name, the invisible lens.
If we take any arbitrary matrix, say, the identity matrix with zeros along the 𝑤-dimension,
the rotation doesn't look quite right.
This is because most tesseract depictions use stereographic projections,
which resemble the act of holding a light source behind an object
and checking a shadow against the screen.
Here, we divide our points into smaller packages and use the Graham Scan to find their convex holes.
We then take the vertices from the holes and plug them into the Jarvis March
to create a wrapped gift of wrapped gifts.
This is, of course, just a small subset of the types of videos on our channel.
Quickly, I would like to thank Grant for making such amazing videos on 3Blue1Brown.
It's inspiring to see such a large and dedicated community
attempting to understand the true beauty of mathematics.
You guys are honestly amazing,
so even if I don't see you guys again,
please keep being such an awesome community.
Grant Sanderson: For my part, I think it would be awesome to see this algorithm's archive grow more.
both of the last two, at least on YouTube, have shorter form content,
So, next, let's switch to a channel with more of a long-form focus:
Welch Labs, run by Stephen Welch.
Now, I suspect many of you already know about Stephen's work, for example,
the excellent series "Imaginary Numbers are Real".
But there are many other great playlists from the channel that you should absolutely take a look at.
In fact, one of the things I like most about this channel is that
Stephen thinks in terms of series rather than in terms of individual videos.
And each one of them includes clear points where the viewer can engage
with the materials more than just through a passive viewing experience,
often including workbooks and associated PDFs,
so the focus really is on learning more so than just entertainment.
To get a feel here,
let me just play some snippets from the start and then from the end of his "Imaginary Numbers are Real" series,
which should give you a pretty good feel for the style in the scope,
and I'll also throw in a snippet from his series on machine learning.
Stephen Welch: Algebraically, this new dimension has everything to do
with a problem that was considered impossible for over 2,000 years:
The square root of negative one.
When we include this missing dimension in our analysis,
our parabola gets WAY more interesting.
Now that our input numbers are in their full two dimensional form,
We can see how our function x²+1 really behaves.
Our function does cross the x-axis,
We were just looking in the wrong dimension.
So why is this extra dimension that numbers possessed not common knowledge?
Part of the reason is that it has been given a terrible, terrible name.
A name that suggests that these numbers aren't even real!
All right, ready?
We'll draw the same exact paths on 𝑤
and see how they show up on our Riemann surface as they are mapped to our 𝑧-plane.
[upbeat techno music]
So, why does our green path start out at one location on our 𝑧-plane only to end up in another?
Simply because our green path leads us to the other layer of our surface.
From the perspective of our 𝑤-plane, it appears that we've returned exactly to our starting point.
But actually, we haven't.
The 𝑤-plane is just a projection; a shadow.
In reality, our path has led us to a completely different branch of our function.
[typing]
This is a decision tree.
Right now, it's learning.
When it's done, it'll have learned to do something very human,
something that everyone knows how to do, but no one can quite explain how they do it,
something that is fooled generations of brilliant scientists
with its apparent simplicity, but practical complexity.
something that has only become possible in the last few decades
thanks to creative solutions in the face of huge complexity:
it's learning to see.
Let's test it out.
With the help of a camera and a computer,
our decision tree sees exactly how many fingers were holding up.
Now, depending on who you are,
the fact that a machine can perform this task may be completely mind-blowing,
not all that out of the ordinary,
or may not even be all that impressive.
The good news here is whichever camp you're in,
you're in good company.
Our finger counting machine is a simple example that belongs to a deceptively complex class of problems,
problems that arise from thinking about how we think.
Grand Sanderson: And finally, turning away from the YouTube world,
I want to highlight the blog "Infinity Plus One".
The tone of the writing and the associated visuals are both delightfully playful,
but the author James Dilts manages to get in some pretty substantive math all while keeping it accessible.
A particularly good sequence of posts is the one he did on relativity, both special and general,
and it's a little hard to directly feature text posts and video forms,
but I'll leave you some links in the description to some of my favorite posts,
and, of course, the description also has links for the other three creators mentioned here.