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What do numbers look like?

johnhw.github.io

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Re: What do numbers look like?

#51

Post by etangent »

superbatfish wrote: post #13

According to Dmitry Kobak, some details in these figures are merely convergence artifacts, and no longer produced when using more recent versions of UMAP. https://twitter.com/hippopedoid/status/1318917878364672001?l...

Of course they are artifacts -- everything in that picture is an artifact, by definition of how it is produced. But these artifacts are produced under specific conditions, which presents a certain window into the structure of what is being visualized. t-SNE is a different, older, method compared to UMAP, and it over-emphasizes local structure at the expense of global structure.

Example of what could cause a swirly chain in UMAP: if A related to B and B relates to C but A does not relate to C and so on. IMO that's a valid structure to visualize as a swirly chain. If re-run multiple times, of course you will get that chain in different locations and so on. But it is interesting that it is there.

Re: What do numbers look like?

#52

Post by teo_zero »

2-718-281-828 wrote: post #50
teo_zero wrote: post #47

Earlier quoted context omitted.

The structures that emerge from a series of transformations applied to an initial field (be it the natural numbers or the complex set), could be due to the transformations, or intrinsic of the underlying field. The parent comment stated that we're in the former case, i.e. we are seeing artifacts due to the transformation itself and we're not in front of some new properties of the numbers. It implied that this is a ba…

so, your point is that the mandelbrot set isn't merely an artifact, if i understand you correctly.

Wait... I'm not sure where this conversation is going.

I say that the beauty (or value or worthiness) of the pictures of the Mandelbrot set comes from the transformations we apply to uninteresting complex numbers.

Similarly, the beauty of the pictures in the article may come from some hidden properties of the underlying prime numbers, or from the transformations themselves, and I don't think that either case would be better than the other.

I said this in reply to a comment that seemingly stated "what a pity, these images are 'merely' due to the transformations". I was objecting to the tone of disappointment that I read in that message.

Re: What do numbers look like?

#53

Post by expazl »

teo_zero wrote: post #52
2-718-281-828 wrote: post #50

Earlier quoted context omitted.

so, your point is that the mandelbrot set isn't merely an artifact, if i understand you correctly.

Wait... I'm not sure where this conversation is going. I say that the beauty (or value or worthiness) of the pictures of the Mandelbrot set comes from the transformations we apply to uninteresting complex numbers. Similarly, the beauty of the pictures in the article may come from some hidden properties of the underlying prime numbers, or from the transformations themselves, and I don't think that either case would be…

Convergence artifacts are due to the algorithm converging poorly. This is why it's a "pity" the interesting structure you see in the picture has nothing to do with the distribution of numbers and is just a bug in the algorithm that happens to look nice.

So no, it's not like the Mandelbrot set. It's more like if you wrote a script to visualize the Mandelbrot set and created a bug that made part of the visual you created look like there was an interesting structure by accident, then shared your pictures with a wide audience going "look at this interesting structure I found in the mandelbrot set!" and then someone replies on Twitter with "you have bug in line 124 and when I correct it the structure disappears". Which is why it's a pity.

Re: What do numbers look like?

#54

Post by expazl »

glotchimo wrote: post #19
quakeguy wrote: post #16

Earlier quoted context omitted.

Sadly merely artifacts .

Can’t those artifacts be beautiful in and of themselves? Whether or not this represents some deep reality about dimensions, it’s cool to see something like this emerge from a program when working with a bunch of numbers.

Sure, but that's no more interesting than a ML generated picture of a cat with a speech bubble saying "I can has numbers?"
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