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.)
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.
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.
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.
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 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.
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.
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).
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.
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.
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 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.
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?
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(I still have not made it all the way to the end of the video though, perhaps that is where they end up.)
I see what you did there.
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.
You can take the finite difference with eg np.diff(waveform) though.
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.
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.
> 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.
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.
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!
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.
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.
Everyone who watches your videos loves them and wants everyone else to watch them.
fascinating, and absurdly confusing, that there are multiple titles.
Or could you walk around in such a world? That would be a very cool concept for a game.
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).
His videos on Euler's formula inspired me to make a silly toy so I could play with it myself.
https://gitlab.com/aprentic/complex-viz/
[0] https://en.wikipedia.org/wiki/Print_Gallery_(M._C._Escher)
But, as with all jokes, the punchline isn't funny at all without the setup.
This kind of risks obscuring what's actually going on.
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.