Elliptic Curves as Art | Hacker News Reader

Elliptic Curves as Art

elliptic-curves.art

228 pointsnill028 comments

Very nice. The rendering makes them look like physical objects. It might be possible to 3D-print some of these in a semi-transparent material. That would be an instabuy for me.

I’ve been working with zk proofs and elliptic curves for a while, and seeing them visualized like this is such a treat. Really enjoyed it! Visualized mathematical functions like these are true nerd art and I absolutely love it.

I was prepared for disappointment, and instead found the procedures and results both beautiful and useful. That is, the authors present a visualization that preserves most of each curve's characteristics---at least the geometric ones. The underlying paper is an absolute joy to read: https://arxiv.org/abs/2505.09627

interestingly reminds me of what you can do with Structure Synth (https://structuresynth.sourceforge.net/) and Context Free Art (https://www.contextfreeart.org/), perhaps is a mathematical connection between these grammar-based formalism and the elliptic curves.

Programmatic art is alive an well on the blockchain! One of my favorite artists in this space is Tyler Hobbs https://opensea.io/collection/fidenza-by-tyler-hobbs

The prices are listed as crypto currencies, so the payments are on a blockchain. Is the artwork itself sold as a kind of NFT on a blockchain as well? If so, what data is actually stored on the blockchain? The parameters that created the image?

That's a great question - these are sold as NFTs, and I think you only get the art, not the input parameters used to generate the art. I really like that idea though

It's also alive and well off the blockchain!

I thought of printing some of those in a t-shirt but someone would probably see it as branding for an extra-terrestrial donut shop.

The kind that would serve coffee in a Klein bottle.

Could title the t-shirt "Topologist's coffee mug!"

> The kind that would serve coffee in a Klein bottle.

For some definition of "in"

Interesting but I don’t understand how they draw elliptic curves over finite fields. Aren’t finite fields supposed to be discrete?

Their visualizations of elliptic curves over finite fields are the ones that consist of a bunch of discrete points. They then roll those up using some mapping from a complex torus to R^3. There was a time in my life when I might have understood what those words mean, but now I'm just cribbing from the paper.

The prime field Fₚ can be represented in the complex numbers as the set of roots of the polynomial xᵖ - x.

Now, to build a finite field of size pⁿ, you find an irreducible polynomial P(x) over that prime field and put a field structure on the roots, seen as an n-dimensional vector space over Fₚ.

So all you have to do to map the finite field of size pⁿ to the complex numbers is to find a "good" Fₚ-irreducible P(x) and plot its complex roots. Then you associate points on the curve with such pairs of complex numbers and map them on to the torus as you do with all the rest, marking them as "hey, those are the Fₚ(n)-points of the curve".

In principle, any polynomial P(x) will do; in practice, I suspect some polynomials will serve much better to illustrate the points on the curve than others. We must wait for the follow up paper to see what kind of choices they have made and why.

To me most of these look like the procedure for the topological inversion of a sphere stopped halfway.

Do you mean a sphere eversion? :)

Shoutout to my fav math visualization BITD https://www.youtube.com/watch?v=wO61D9x6lNY

This looks fascinating—love the idea of turning abstract math like elliptic curves into visual art. Looking forward to seeing how the site develops! The blend of aesthetics and deep mathematics is such a cool approach.

Thanks for posting these.

These are too pretty. <3 <3 <3

Looking at these I can see how Nature is using a lot of elliptic curves to capture our attention. They are like flowers!

You may also enjoy "The geometry of art and life", a 1946 book by Matila Ghyka.

Some texts in the field veer off into sacred geometry territory too swiftly, but I think Ghyka's offers pleasant discussions without.

I thought "oh, this is going to be expensive" (old book? about art?) but there's a $12 Dover paperback.

I low key want to buy t-shirts of these now.

I would also buy those.

You should sell these cookies to mathematicians. I'm 100% sure they would love an elliptic curve.

Please let us reproduce these beautiful pictures, can you share the sources?

I'd love to see them iterating the values and show the animated versions!

These are SO, AMAZINGLY pretty! Are these blender renders? or...?

This is very beautiful!! Thank you. Sheds so much light on the Modularity Theorem and Fermat's Last Theorem too.

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