To paraphrase that quote about hydrogen: Give gradient descent a few billion parameters, and it starts wondering where it came from and what does it all mean.
Constraints forces creativity. Some well chosen constraints are aesthetics rules that helps you land pleasing results. Poetry has a long history on that matter.
Another example of constrained creativity is early to mid nineties electronic music.
> Inspired by the Twitter logo, which is made from 13 perfect circles
Compared to that, the new logo doesn't have a circle (segment) anywhere to be seen (unless you consider straight lines as circle segments with the center located at infinity of course), and is simply the "mathematical double-struck capital X" from an unknown but probably pre-existing font (apparently Monotype's "Special Alphabets 4" comes close, but isn't identical, according to https://tweethunter.io/blog/how-to-write-twitter-x-iphone-ma...).
Im curious what the process looks like to implement this. It seems like it would be easiest to start with the animal using only perfectly(?) curved lines and then complete them into circles after the fact. Although that seems kind of pointless and I imagine they start with circles. And I guess it would hard to have a curve from a perfect circle without the circle?
I just have a hard time imagining you start with circles, lay them down (resize as needed) and continue. I mean I guess that doesnt sound so crazy after I say it... it just seems like it would add a lot of extra noise to the image that would make it much harder to draw.
I can't speak to this but I took a drawing class a long time ago. I'm not very good but it was a lot of drawing circles. When you see people freehand stuff it's kind of wild but that's not how people learn how to draw they're just very good at it from practice. Most of learning is drawing very basic shapes, usually circles, and erasing parts that don't make sense and continuing.
> drawing very basic shapes, usually circles, and erasing parts that don't make sense
There's a hilarious Spongebob bit [0] where Squidward is teaching an art class, and he starts off in that exact manner of trying to draw a perfect circle, only to have Spongebob subvert the entire idea. The whole episode is artistic gold IMO.
I have been practicing art a lot lately. You can draw just about anything using spheres, cubes, cylinders, and cones. You start off with the 2d versions.
"While sketching, I kept track of the number of circles I was using, counting one for every curve."
After sketching an animal, it should be easier to adjust the image by inserting/removing/moving circles.
I suppose the thing the circle is really informing is the "perfectness" of the curve. You cant just draw in curves and extend it to a circle (wont be perfect). I guess Im not sure how you get "perfect" curves.
I suspect its a stencil or something. So in some sense the circle does exist first, even if they only draw the curve from it initially (before marking it up with the full circle after the fact).
What's wild is how much clarity and personality you can get from that process. Instead of adding noise, it forces simplification, which actually helps with visual clarity
He was done a great deal of injustice when he was passed over for the Turing award that was given to Hinton, Bengio, LeCun.
Then there is this from his blog --
Dec 2024: Sadly, the Nobel Prize in Physics 2024 for Hopfield & Hinton is a Nobel Prize for plagiarism. They republished methodologies developed in Ukraine and Japan by Ivakhnenko and Amari in the 1960s & 1970s, as well as other techniques, without citing the original papers. Even in later surveys, they didn't credit the original inventors (thus turning what may have been unintentional plagiarism into a deliberate form). None of the important algorithms for modern Artificial Intelligence were created by Hopfield & Hinton. Details in the recent technical report, with lots of references, links, and facts.
I'm not sure whether or not he did this first, but it's very similar to an extremely impressive, but old and well-known illustration of the power of Fourier analysis in which you construct a "Fourier epicycle" (think: machine made of circular gears of different ratios) that can sketch any image. 3blue1brown has a great video on Fourier Epicycles but you can also get the idea here https://mathematica.stackexchange.com/questions/171755/how-c...
This is so true. I took a figure drawing class in college and we were instructed to draw with a cloth and charcoal dust. Easily some of the best work I made came out of that restriction.
Architecture too. The worst building come from architects given a blank page to start with. Constraints, and sympathy for the surrounding built environment produce great work.
I miss being creative, before I knew how to make front end UIs I had crazy ideas but then became grounded. This one isn't super crazy but I like those vertical buildings.
Tangent, with a dark/colorful theme in an editor the minimap looks like a city scape
This page feels like an AI traveled back in time and (faked) the date.
[edit] Nevermind. I'm being too harsh. The creator was obviously having fun and being creative. That's cool. I think if nothing else this just proves how jaded and skeptical about clever artwork I've become in the past few years.
My aunt grifted me “Animali Compassati” when I was a kid… A small book with instructions for animals that you could draw with a compass.
The site is unclear somehow… but the instructions were pretty great in the book.
If you pay closer attention, you can see that some of the designs rely on very deliberate placements of circles; for example, eyes of the monkey and owl, and the nose of the whale.
Those are just the obvious ones that I can immediately spot — there was probably a lot of careful consideration into the placement of circles in order to facilitate good looking arcs and circles that bring the animals to "life".
Just doing a Google search for "animals made from circles", you get the usual header full of "Images" and "Videos" crap, then in the actual results links, you have the usual Pinterest linkslop, Facebook linkslop, Reddit linkslop, a bunch of articles written by the designer (now we're getting somewhere). OP's link is finally on page 4 of the search results.
I was thinking that this would be low-hanging fruit for a model. The parameter space is so tiny compared to what a diffusion model already has to deal with...
More generally, any animal that cannot be drawn in 12 circles cannot be drawn in 13 circles when riding a bicycle. By recursion, no animal can be drawn when riding a stack of seven bicycles.
You also need to encode the painted areas somehow. They are not only intersections on K shapes, but sometimes exclusions as well (like (A^B)/C). Two ways come to mind:
(1) Listing closed curves by vertices. Each vertex of a painted area is an intersection of two or more circles, and delimits a section of a circle. So the section of circles that enclose a circle can be encoded each by the union of:
(1.1) A circle (index);
(1.2) A 2nd circle (index) that intersects the 1st on a first point;
(1.3) A bit identifying the (first) intersection (because there may be 2 possible);
(1.4) A 3rd circle (index) that intersects the 1st on a second point;
(1.5) A bit identifying the (second) intersection.
Note the base circle would be the first intersection of a subsequent section of this closed curve, and the 3rd circle would be the subsequent base circle. So 1/2/3 won't be necessary for subsequent curves. So only (K+2) indices + (K+1) bits are necessary for this encoding.
Total ~K log2(K)+K bits. I hypothesize (left to the reader :)) a closed curve should contain at most 2x13 points. There can be at most 2^13 distinct regions however, so each figure (Animal) can be encoded with less than that many curves per figure. So each figure (Animal) can be encoded with less than 2^13 x 26x(5+1) bits =~ 1.3Mbit.
But that's mostly pathological cases, if each Animal must be a fully connected area, then that might reduce (hypothesis above) to at most only
26x(5+1) bits = 156 bits, or 20 bytes!
I left out a problem which area shapes encoded within each other (like eyes). In that case you need at most another 156 bits per inner cutout shape.
(2) Alternatively, you could use boolean operations to encode each shape. Also left as a fun problem :)
I tend to wonder if stuff like this is an informative boundary on AI capabilities. I mean, you can't ask a LLM today to do that (AFAICT). "Here's a simply-specified but extremely broad search space, solve this problem in it" isn't something that fits the model. But it's a relatively common (if not "easy") task human beings like to show off.
What needs to change to enable this kind of exploration?
Nope, but this post is such a neat illustration of the richness of "life" that fits into 39 real parameters (each circle can be coordinatized as 3 real numbers: one for its radius and two for its center) that my first thought on seeing it was also "no surprise then that a matrix with a million entries can talk like an erudite person".
It should be obvious that this is entirely up to the reader. Take some responsibility for your own happiness. Nobody else is responsible for your enjoyment of anything.
Well, sort of? I mean, I've seen plenty of clever art in my life. I'm still figuring out AI. I posted that in the hope that someone in the community here would show up with something insightful to say.
Yes, it is impossible. People will think about things they find interesting regardless of your (dis-)approval. Who are you even calling "obsessive?" The collective of people who dare to mention algorithms you don't like? I mean, what the fuck?
Actually, the (in)famous "sparks of general intelligence paper" about GPT-4 included tasks such as "Draw a unicorn in TikZ" which really is not that far off from this task. There were also examples for drawing cars/trucks/cats etc with SVG.
But I do think that evolutionary algorithms or MCMC variants could do a better job of this, especially if paired with an auxiliary model for scoring their intermediate results.
Yes, this has been done in many forms with other algorithms. You score each generation with a model like CLIP, for example, and then you can evolve 'Mona Lisa made of triangles', say. A constraint like 'exactly 13 circles' will work fine. (And you might experiment with loosening it, like generating a lot of candidates with 5-30 circles each, as a 'library' or 'seeds', before shrinking them all towards 13, to see if you get novel animal designs which are find to find if you simply start the obvious way with 13 circles initialized to random points & sizes.)
I was thinking it could, actually, given a feedback loop. The tool use would a json that takes 13 circles, each with x, y position, radius, and whether it's filled in or empty, and output an image. It could look at the image and iterate.
Not exactly circles, but famously:
With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.
https://en.m.wikipedia.org/wiki/Von_Neumann%27s_elephant
Feels like the mathematical version of "just because you can doesn't mean you should."
"and with 20 billion I can make it hold a conversation."
To paraphrase that quote about hydrogen: Give gradient descent a few billion parameters, and it starts wondering where it came from and what does it all mean.
Bravo! HN Gold.
Related: 'A meeting with Enrico Fermi' https://www.nature.com/articles/427297a
Looks more like an amoeba to me...
It's really satisfying to create logomarks solely out of circles, idk why. A challenge, I guess.
I did a few back in my day as a designer:
1. https://dribbble.com/shots/1909369-Liberty-Eagle-Arms
2. https://dribbble.com/shots/1553151-Flint-mark-icons
That first one is some of my best work.
Constraints forces creativity. Some well chosen constraints are aesthetics rules that helps you land pleasing results. Poetry has a long history on that matter.
Another example of constrained creativity is early to mid nineties electronic music.
There's something oddly meditative about designing within strict constraints like circles
Yeah, those are all really nice. Good work.
> Inspired by the Twitter logo, which is made from 13 perfect circles
Compared to that, the new logo doesn't have a circle (segment) anywhere to be seen (unless you consider straight lines as circle segments with the center located at infinity of course), and is simply the "mathematical double-struck capital X" from an unknown but probably pre-existing font (apparently Monotype's "Special Alphabets 4" comes close, but isn't identical, according to https://tweethunter.io/blog/how-to-write-twitter-x-iphone-ma...).
It feels like I'm looking at the next so many Ubuntu backgrounds!
Im curious what the process looks like to implement this. It seems like it would be easiest to start with the animal using only perfectly(?) curved lines and then complete them into circles after the fact. Although that seems kind of pointless and I imagine they start with circles. And I guess it would hard to have a curve from a perfect circle without the circle?
I just have a hard time imagining you start with circles, lay them down (resize as needed) and continue. I mean I guess that doesnt sound so crazy after I say it... it just seems like it would add a lot of extra noise to the image that would make it much harder to draw.
I can't speak to this but I took a drawing class a long time ago. I'm not very good but it was a lot of drawing circles. When you see people freehand stuff it's kind of wild but that's not how people learn how to draw they're just very good at it from practice. Most of learning is drawing very basic shapes, usually circles, and erasing parts that don't make sense and continuing.
> drawing very basic shapes, usually circles, and erasing parts that don't make sense
There's a hilarious Spongebob bit [0] where Squidward is teaching an art class, and he starts off in that exact manner of trying to draw a perfect circle, only to have Spongebob subvert the entire idea. The whole episode is artistic gold IMO.
0: https://www.youtube.com/watch?v=vTlpFEvmxdM
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I have been practicing art a lot lately. You can draw just about anything using spheres, cubes, cylinders, and cones. You start off with the 2d versions.
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True, I did some amateur vector art (in Illustrator) and you basically have to compose objects out of basic shapes. It is truly highly meditative.
There's some information on: https://www.smashingmagazine.com/2017/01/illustrating-animal...
"While sketching, I kept track of the number of circles I was using, counting one for every curve." After sketching an animal, it should be easier to adjust the image by inserting/removing/moving circles.
Awesome, thank you!
There are some photos of sketches at the bottom of the page. Looks like they started with curves and turned them into circles later
I suppose the thing the circle is really informing is the "perfectness" of the curve. You cant just draw in curves and extend it to a circle (wont be perfect). I guess Im not sure how you get "perfect" curves.
I suspect its a stencil or something. So in some sense the circle does exist first, even if they only draw the curve from it initially (before marking it up with the full circle after the fact).
1 reply →
What's wild is how much clarity and personality you can get from that process. Instead of adding noise, it forces simplification, which actually helps with visual clarity
I remember some post that I can find now, that demonstrated the twitter bird logo is also made from circles. All I can find is this reddit post now.
https://www.reddit.com/r/interestingasfuck/comments/txdimd/t...
https://designshack.net/articles/graphics/twitters-new-logo-...
That was referenced in the post as the impetus for making these. Unfortunately it just links to a Google search.
Not sure how precise it really is, but it looks convincing enough to feel intentional
It’s mentioned in the article under the images as the inspiration for this work
Because circles there also need operations over them (union, intersection or subtraction), it is a good example of low complexity art [1].
[1] https://en.wikipedia.org/wiki/Low-complexity_art
My son is a big fan of bytebeat [2], which is also a low complexity art, but music.
[2] https://dollchan.net/bytebeat/#4AAAA+kUryC/X0CixswNhQyM1Q01N...
I did something similar 15+ years ago to use as an avatar in forums, twitters and some such - https://swapped.ch/#!/personal-mark
See also work from Schmidhuber in the mid/late 1990s https://people.idsia.ch/~juergen/locoart/node12.html
wow, that's beautiful - the whole site https://people.idsia.ch/~juergen/ is an amazing rabbit hole im gonna lose myself in.
The red button is an absolute delight, be sure not to press it.
He was done a great deal of injustice when he was passed over for the Turing award that was given to Hinton, Bengio, LeCun.
Then there is this from his blog --
Dec 2024: Sadly, the Nobel Prize in Physics 2024 for Hopfield & Hinton is a Nobel Prize for plagiarism. They republished methodologies developed in Ukraine and Japan by Ivakhnenko and Amari in the 1960s & 1970s, as well as other techniques, without citing the original papers. Even in later surveys, they didn't credit the original inventors (thus turning what may have been unintentional plagiarism into a deliberate form). None of the important algorithms for modern Artificial Intelligence were created by Hopfield & Hinton. Details in the recent technical report, with lots of references, links, and facts.
https://people.idsia.ch/~juergen/physics-nobel-2024-plagiari...
Agree.
Also, AlphaFold is great but hardly an innovation. David Baker deserved it 100%.
I thought this was a joke, but he actually did do this first. Impressive!
I'm not sure whether or not he did this first, but it's very similar to an extremely impressive, but old and well-known illustration of the power of Fourier analysis in which you construct a "Fourier epicycle" (think: machine made of circular gears of different ratios) that can sketch any image. 3blue1brown has a great video on Fourier Epicycles but you can also get the idea here https://mathematica.stackexchange.com/questions/171755/how-c...
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Japanese family crests (Mon: 紋) are almost entirely made of circles (and lines, but that's rarer)
Often depicting slices of vegetables, animals..
From few circles to hundreds
Art with restrictions can be more interesting than without.
This is so true. I took a figure drawing class in college and we were instructed to draw with a cloth and charcoal dust. Easily some of the best work I made came out of that restriction.
Architecture too. The worst building come from architects given a blank page to start with. Constraints, and sympathy for the surrounding built environment produce great work.
Reminds of 'Drawing with circles' in 3b1b classic on Fourier series: https://www.youtube.com/watch?v=r6sGWTCMz2k
So if we remove the condition of 13, everything is in fact made of circles only!
I really love the way these look. I'm imagining a short film with these characters, and it'd be a nice experiment to see how it turns out.
I miss being creative, before I knew how to make front end UIs I had crazy ideas but then became grounded. This one isn't super crazy but I like those vertical buildings.
Tangent, with a dark/colorful theme in an editor the minimap looks like a city scape
It bugs me than e.g. the owl's ears benefit from a dramatic change in color that isn't related to anything outlined by the circles.
This page feels like an AI traveled back in time and (faked) the date.
[edit] Nevermind. I'm being too harsh. The creator was obviously having fun and being creative. That's cool. I think if nothing else this just proves how jaded and skeptical about clever artwork I've become in the past few years.
My aunt grifted me “Animali Compassati” when I was a kid… A small book with instructions for animals that you could draw with a compass. The site is unclear somehow… but the instructions were pretty great in the book.
https://www.danielenannini.it/en/portfolio/animali-compassat...
https://www.compasses-zoo.net/compasses-zoo/index.php
Not circles, but arcs.
If you pay closer attention, you can see that some of the designs rely on very deliberate placements of circles; for example, eyes of the monkey and owl, and the nose of the whale.
Those are just the obvious ones that I can immediately spot — there was probably a lot of careful consideration into the placement of circles in order to facilitate good looking arcs and circles that bring the animals to "life".
Sure, you're not wrong. But a circle is just an arc of length 2pi * radius.
Reminds me of how some of the best ideas come out of working within restrictions, not in spite of them
Curious how well transforms on circles could be composed to animate these animals
I never really liked Twitter but I feel oddly nostalgic for the logo now.
Vaguely related and also fun: https://www.koalastothemax.com/ (2011)
2016…
This type of content is becoming rarer on the internet nowadays.
I don’t think less of this type of content exists. Its just harder to find when inundated with all other slop on the internet.
Just doing a Google search for "animals made from circles", you get the usual header full of "Images" and "Videos" crap, then in the actual results links, you have the usual Pinterest linkslop, Facebook linkslop, Reddit linkslop, a bunch of articles written by the designer (now we're getting somewhere). OP's link is finally on page 4 of the search results.
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Or we just don't look past twitter and such.
Could this be the next captcha challenge? "Draw an animal out of 13 circles to prove you are human".
I was thinking that this would be low-hanging fruit for a model. The parameter space is so tiny compared to what a diffusion model already has to deal with...
Reminds me of the time we made geometric art using a compass. https://homeschoolmath.blogspot.com/2013/02/geometric-art-pr...
Interesting.
What animals cannot be accurately depicted with 13 circles?
You can depict any animal swallowed by a pufferfish with 1 circle.
What is essential is invisible to the eye.
Pelican on a bicycle
More generally, any animal that cannot be drawn in 12 circles cannot be drawn in 13 circles when riding a bicycle. By recursion, no animal can be drawn when riding a stack of seven bicycles.
Corals?
minecraft sheep
centipede
I was going to suggest "octopus" but your answer is better.
I think so. With 13 circles, I can’t figure out how you could represent more than 26 legs (and other features would be lost).
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An owl?
Could be this https://chatgpt.com/canvas/shared/67ed7147fc708191be5b81ed4e...
But not that artsy as the OG.
1 reply →
class Animal {
Circle circles_[13];
}
You also need to encode the painted areas somehow. They are not only intersections on K shapes, but sometimes exclusions as well (like (A^B)/C). Two ways come to mind:
(1) Listing closed curves by vertices. Each vertex of a painted area is an intersection of two or more circles, and delimits a section of a circle. So the section of circles that enclose a circle can be encoded each by the union of:
(1.1) A circle (index); (1.2) A 2nd circle (index) that intersects the 1st on a first point; (1.3) A bit identifying the (first) intersection (because there may be 2 possible); (1.4) A 3rd circle (index) that intersects the 1st on a second point; (1.5) A bit identifying the (second) intersection.
Note the base circle would be the first intersection of a subsequent section of this closed curve, and the 3rd circle would be the subsequent base circle. So 1/2/3 won't be necessary for subsequent curves. So only (K+2) indices + (K+1) bits are necessary for this encoding.
Total ~K log2(K)+K bits. I hypothesize (left to the reader :)) a closed curve should contain at most 2x13 points. There can be at most 2^13 distinct regions however, so each figure (Animal) can be encoded with less than that many curves per figure. So each figure (Animal) can be encoded with less than 2^13 x 26x(5+1) bits =~ 1.3Mbit.
But that's mostly pathological cases, if each Animal must be a fully connected area, then that might reduce (hypothesis above) to at most only 26x(5+1) bits = 156 bits, or 20 bytes!
I left out a problem which area shapes encoded within each other (like eyes). In that case you need at most another 156 bits per inner cutout shape.
(2) Alternatively, you could use boolean operations to encode each shape. Also left as a fun problem :)
This guy is doing something similar for his game:
The Procedural Animation https://www.youtube.com/watch?v=qlfh_rv6khY
Gibbon: Beyond the Trees - Wolfire Games https://www.youtube.com/watch?v=KCKdGlpsdlo
Could an AI generate art like this and actually utilize perfect circles, to create whatever you ask?
[dead]
I tend to wonder if stuff like this is an informative boundary on AI capabilities. I mean, you can't ask a LLM today to do that (AFAICT). "Here's a simply-specified but extremely broad search space, solve this problem in it" isn't something that fits the model. But it's a relatively common (if not "easy") task human beings like to show off.
What needs to change to enable this kind of exploration?
Is it impossible, in this day and age, to enjoy a post without thinking about LLMs? It's like an obsession.
Nope, but this post is such a neat illustration of the richness of "life" that fits into 39 real parameters (each circle can be coordinatized as 3 real numbers: one for its radius and two for its center) that my first thought on seeing it was also "no surprise then that a matrix with a million entries can talk like an erudite person".
3 replies →
It should be obvious that this is entirely up to the reader. Take some responsibility for your own happiness. Nobody else is responsible for your enjoyment of anything.
Well, sort of? I mean, I've seen plenty of clever art in my life. I'm still figuring out AI. I posted that in the hope that someone in the community here would show up with something insightful to say.
Yes, it is impossible. People will think about things they find interesting regardless of your (dis-)approval. Who are you even calling "obsessive?" The collective of people who dare to mention algorithms you don't like? I mean, what the fuck?
1 reply →
Actually, the (in)famous "sparks of general intelligence paper" about GPT-4 included tasks such as "Draw a unicorn in TikZ" which really is not that far off from this task. There were also examples for drawing cars/trucks/cats etc with SVG.
But I do think that evolutionary algorithms or MCMC variants could do a better job of this, especially if paired with an auxiliary model for scoring their intermediate results.
Yes, this has been done in many forms with other algorithms. You score each generation with a model like CLIP, for example, and then you can evolve 'Mona Lisa made of triangles', say. A constraint like 'exactly 13 circles' will work fine. (And you might experiment with loosening it, like generating a lot of candidates with 5-30 circles each, as a 'library' or 'seeds', before shrinking them all towards 13, to see if you get novel animal designs which are find to find if you simply start the obvious way with 13 circles initialized to random points & sizes.)
I was thinking it could, actually, given a feedback loop. The tool use would a json that takes 13 circles, each with x, y position, radius, and whether it's filled in or empty, and output an image. It could look at the image and iterate.