How and why to take a logarithm of an image [video]
youtube.comThis video is an absolute tour de force of communicating a complex concept.
All of 3Blue1Brown is - hoghly highly recommend
I've seen most! Highlighting this one out of them all. Exemplary! : D
Seems like you could apply the clever transforms to generate a displacement map (that then allows you to move it across any source image and quickly get the Droste effect).
(I still have not made it all the way to the end of the video though, perhaps that is where they end up.)
"Complex concept"
I see what you did there.
Similarly, it's possible to take the derivative of a song. You can use a Fourier transform to express the song's waveform as a series of sin and cosine functions, then take the derivative.
Imagine, for the sake of simplicity, you could express the song's waveform with the function 13 * sin(41x).
The derivative of this function is 533 * cos(41x).
Cosine, of course, is just a phase shifted sine, and the constant coefficient inside the function stays the same. So you're not changing anything about the shape of the wave, just stretching it vertically.
This has the effect of mimicking a "high pass filter," amplifying the volume of the highs.
Well, you get the frequency domain derivative. This is the same as scaling the time domain by a linear ramp. Not exactly hugely useful, unless you happen to be in radar.
You can take the finite difference with eg np.diff(waveform) though.
The title I get when I click on this is, "How (and why) to take a logarithm of an image"
YouTube has A/B testing features that allow videos to have multiple titles and/or thumbnails.
Right. So I thought it would be helpful to share the more-descriptive title that I got.
This is what I use DeArrow for, crowdsourced titles and thumbnails (from the maker of SponsorBlock): https://github.com/ajayyy/DeArrow
I'm sorry, what? Can people now see different titles? Insanity, if true.
It has been that way for a while now. I see Veritasium video titles and thumbnails change quite often, it can be quite annoying as it sometimes gives the appearance of it being a whole new video.
A/B testing a title feels wrong to me, its almost as bad as A/B testing a UUID. Just pick a title and stick to it unless you need to fix a factual error.
Titles and thumbnails have a huge impact on video performance, and when it's your main income it seems reasonable to try to marginalise the impact.
Right, but then there's this thing called "shared reality" and once you break it, all kinds of bad consequences happen.
This is even worse, as it also breaks temporal continuity for individual reality. E.g. I expect that if I saw a video titled X today, I'll be able to find it under title X tomorrow, and if I can't, it's one of the rare/marginal cases when it got banned/deleted/retitled, or I just misremembered. Titles becoming unstable in the general case is a bad situation.
Gaslighting is now a government policy
And video performance = ad revenue.
Oh yes. Some channels cycle through many different ones as they test them. Veritasium is notorious for this.
> How (and why) to take a logarithm of an image
I watched it a few days ago and this descriptive title was part of the reason I clicked. I generally trust 3B1B anyway but normally a title like "This picture broke my brain" would put me off.
In case you're curious, when I ran that title/thumbnail AB test, the option "This picture broke my brain" did end up winning. I was a bit disappointed, because I didn't really _want_ it to win, but I did include it out of curiosity. Ultimately, I changed it to the other title, mostly because I like it better, and the margin was small.
I was genuinely torn about how to title this, because one of my aims is that it stands to be enjoyed by people outside the usual online-math-viewing circles, especially the first 12 minutes, and leaning into the idea of a complex log risks alienating some of those.
That makes me wonder: do you see a difference in when viewers drop off between using a more math-y title versus a more accessible one?
The "broke my brain" title originally put me off from watching. I caved after a few days; I think the video is one of your best!
That level of granularity would be interesting. For what it's worth, the metric they go by is not click-through rate; it's expected total watch time. For example, if you have two thumbnails, A and B, and for every 100 impressions of A, there are 51 total minutes of watch time, and for every 100 impressions of B, there are 49 total, then what you'd see in the dashboard is "51% A, 49% B". More total clicks with less engagement will not necessarily win out.
I generally agree that it's a pretty wild choice to just let creators put up multiple titles. That said, it's hard not to play with the shiny toy when it's sitting right there, especially if you know it may mean the lesson reaches more people. In this case, I genuinely don't know what the "right" title is, even setting engagement aside. Is it fundamentally about analyzing an Escher piece? Is it fundamentally a lesson on complex analysis, and complex logs in particular? It's both, but you don't always want to cram two stories into one title. This becomes all the more challenging when titles are, inescapably, marketing.
perhaps a bit inappropriate of me to say so here as it is off-topic, but i am going to take the opportunity anyways:
big thanks for all of your work making math both enjoyable and accessible. my kids (and i) love your videos. your positive impact extends far and wide.
You should be able to have different titles for different ages and education levels of users
As annoying as those titles are, the work that you (and few others, like Veritasium) do makes it well worth the tradeoff. Just keep reminding everyone that the annoying title gets the video into the brain of thousands of other people who aren't subscribed yet. It's a tiny price to pay for astounding value.
Everyone who watches your videos loves them and wants everyone else to watch them.
For me it is "Decoding Escher's most mind-bending piece"
Same for me and this is why I had not clicked on it before.
i see "Decoding Escher's most mind-bending piece".
fascinating, and absurdly confusing, that there are multiple titles.
It's a pretty common feature of youtube creator studio. https://www.theverge.com/news/840789/youtube-video-title-a-b...
i had no idea, thanks. at first glance it seems okay-ish for the creator, but only serves to be confusing for the users.
I've been wondering if you could do a similar thing for a Droste effect image containing two copies of itself. Packs of Laughing Cow cheese show a cow with two earrings, each of which is a pack of the cheese.
What "similar thing" are you asking for? The Laughing Cow image exists. The Print Gallery is an object itself existing at 2 zoom levels in the same place, but the cheese exists in different places. You can't have two copies of the same image in the same place - that's not a copy; it's just itself.
This kind of technique can be used in 3D space as well! The analysis here represents Escher's techniques as conformal maps in the complex plane. Conformal maps are also possible, though more limited, in R^3. This is something that I explored some years ago and wrote an article about it, though it focuses more on graphics than math: https://www.osar.fr/notes/logspherical/
So to do this same Droste effect in 3D you would need a self-similar volume? Though since we can't really see 3D, we could never have that "one circle zooms in" effect.
Or could you walk around in such a world? That would be a very cool concept for a game.
Though since we can't really see 3D, we could never have that "one circle zooms in" effect.
Well, the 3D structure just needs to be sufficiently "holey" for the effect to become apparent. For example a cage-like structure, or a house with no roof (when seen from above).
Those videos are awesome! 3B1Bs visualizations finally made e^(pi*i) make sense.
His videos on Euler's formula inspired me to make a silly toy so I could play with it myself.
I love 3B1B but generally don't have time to watch long videos. Can anyone sum up the punchline?
One of Dutch artist M.C. Escher's works is a man is admiring a piece of art that itself depicts the building the (very same) man is in [0]. Escher left out the middle bit of the painting, probably since it's fairly complicated, putting his signature there instead. The video itself is about the complex analysis used to fill in that missing middle, based on a paper ~20 years ago.
[0] https://en.wikipedia.org/wiki/Print_Gallery_(M._C._Escher)
I think the gap also has a compositional purpose: the viewer's eye is meant to travel around the image in a circle, and the gap helps anchor that in a way that the filled-in version might not.
Hm, since I can't edit my comment: link to paper [0]
[0] https://pub.math.leidenuniv.nl/~smitbde/papers/2003-de_smit-...
HN only allows editing comments for a short time
The whole point is the explanation... it's a bit like someone telling you to take a 2 week holidays somewhere and you'd just say: it's too long, can't someone just get me a plane ticket there and back the same day so I can compress the stay?
The punchline is that you can fill in the centre of Escher's piece by using complex analysis, and it produces a very satisfying, "obviously correct", solution.
But, as with all jokes, the punchline isn't funny at all without the setup.
The joke is that if you fill in the center, it shows the Droste effect of the image and kind of diminishes the magic of it.
The print gallery is just Aw^c in the complex plane
Answers that are only comprehensible to those who already know the answer:
Well he wanted video boiled down to the punchline. Ask a silly question, get a silly answer.
Well, maybe, but that seems like a deliberately uncharitable interpretation of the question, which I interpreted more as "Can you summarize the video in ~1 line?" - or at least closer to that than "Can you give me the answer the video comes to without specifying the question it asks?"
Even in those terms the answer given isn't really an answer because it just gives an expression with undefined variables.
The image is essentially a self-similar 'droste-effect' image in disguise. The warping of that image shifts that self- similarity into a visual loop, but the warped image still has a droste-style self-similarity in the center as well.
Awᶜ
This kind of risks obscuring what's actually going on.
Obscurity indeed. This morning shortly after I woke up I read your message and could only think that you were referring to Awk somehow. That didn’t make much sense to me. Now, hours later and after finally eating something, it does make sense! You really do have to have your faculties about you.
Clickbait title broke my brain.
So in other words you can take any Droste image and make an Escher zooming spiral effect. Neat.
Also curious what happens if you take Escher's painting and undo the effect. Probably not great since it wasn't in the video.
What a cool video.
It was in the video, actually.
I'm somewhat sure that was the recreated version.
I've been loking into how 3B1B builds their rendering pipeline, and it's honestly mind blowing. They use Python along with custom OpenGl shaders to handle most of geometric transformations, shich seems to be what creates those "brain breaking" visual effects.It's fascinating how our visual cortex tries to interpret overlapping geometric patterns and ends up producing such counterintuitive perceptions. Shat I still can't quite wrap my hand around is... to what extent are these effects caused by the rendering itself, and how much of it is just how our brain interprets the visual information?
there is at least one video, if not a series of videos, where he explains his process in detail, showing the coding process and everything. it is a collaboration with another person where, if i recall, he is teaching them.
i will see if i can find it.
edit: "How I animate 3Blue1Brown | A Manim demo with Ben Sparks" https://www.youtube.com/watch?v=rbu7Zu5X1zI
more on workflow: https://github.com/3b1b/videos?tab=readme-ov-file#workflow
code for videos: https://github.com/3b1b/videos
manim: https://github.com/3b1b/manim
manim community edition: https://github.com/ManimCommunity/manim/
This is great, thanks for digging these up. The workflow doc on Github is exactly what I wat looking for - didn't realize they had that level of documentation publicly available.
Makes me wonder how this would look/feel interactively if a game world was rendered like this
Manifold Garden isn't far off.
Clickbait title could use another pass. What is this about?
This was the title used when I came across the video. Apparently YouTube uses many different titles for A/B testing but this is the one I got. Can't edit it now, unfortunately.
EDIT: seems like dang or team took care of it, thanks!
It makes more sense when seen on YouTube where you get the thumbnail of one of M. C. Eschers famous drawings is shown.
It’s a drawing of a guy looking at a picture of a town with himself standing in the town, but it’s all twirled and twisted so it’s self repetition isn’t obvious.
I clicked on the link and the video title is "Decoding Escher's most mind-bending piece", which is a lot better. I also had no idea what "3B1B video" meant, apparently it's a channel called "3Blue1Brown".
It's about examining the mathematical methods MC Escher used in one of his recursive drawings.
> Examining the mathematical methods MC Escher used in one of his recursive drawings
This would be an excellent title :)
Depends how you define excellent. If the goal is to get more views then it's not all that great, and views are kind of the point of YouTube for many, especially if they are trying to make a living from it.
That's great for YouTube, but HN has some guidelines:
> please use the original title, unless it is misleading or linkbait
Probably he didn't use these techniques explicity: the video mentions but doesn't emphasise that he probably sketched out the map by feel instead of analytically, which is probably one reason why he didn't fill in the center.