[HN Gopher] Elliptic Curves as Art
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Elliptic Curves as Art
Author : nill0
Score : 196 points
Date : 2025-06-19 04:02 UTC (18 hours ago)
(HTM) web link (elliptic-curves.art)
(TXT) w3m dump (elliptic-curves.art)
| bedit wrote:
| 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.
| 6510 wrote:
| You should sell these cookies to mathematicians. I'm 100% sure
| they would love an elliptic curve.
| tempodox wrote:
| 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.
| larodi wrote:
| 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.
| broken_broken_ wrote:
| I low key want to buy t-shirts of these now.
| madcaptenor wrote:
| I would also buy those.
| gloosx wrote:
| Looking at these I can see how Nature is using a lot of elliptic
| curves to capture our attention. They are like flowers!
| MonkeyClub wrote:
| 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.
| madcaptenor wrote:
| I thought "oh, this is going to be expensive" (old book?
| about art?) but there's a $12 Dover paperback.
| ykonstant wrote:
| 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
| felineflock wrote:
| 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.
| yndoendo wrote:
| Could title the t-shirt "Topologist's coffee mug!"
| charlieyu1 wrote:
| Interesting but I don't understand how they draw elliptic curves
| over finite fields. Aren't finite fields supposed to be discrete?
| madcaptenor wrote:
| 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.
| ykonstant wrote:
| The prime field F[?] can be represented in the complex numbers
| as the set of roots of the polynomial x - 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 F[?].
|
| So all you have to do to map the finite field of size pn to the
| complex numbers is to find a "good" F[?]-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 F[?](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.
| cosmodev wrote:
| 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.
| HappMacDonald wrote:
| To me most of these look like the procedure for the topological
| inversion of a sphere stopped halfway.
| loxias wrote:
| Do you mean a sphere eversion? :)
|
| Shoutout to my fav math visualization BITD
| https://www.youtube.com/watch?v=wO61D9x6lNY
| knottedoak wrote:
| This is very beautiful!! Thank you. Sheds so much light on the
| Modularity Theorem and Fermat's Last Theorem too.
| Datagenerator wrote:
| Please let us reproduce these beautiful pictures, can you share
| the sources?
| dylan604 wrote:
| I'd love to see them iterating the values and show the animated
| versions!
| loxias wrote:
| These are SO, AMAZINGLY pretty! Are these blender renders? or...?
| matcha-video wrote:
| 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
| aanet wrote:
| Thanks for posting these.
|
| These are too pretty. <3 <3 <3
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