The picture above shows an approximation to the Gosper curve, which was discovered by Bill Gosper in 1973. The Gosper curve is a space-filling curve, which means that it converges to a curve that completely fills in the interior of a shape. The shape in question (shown below) is called the
As impressive as space filling curves are in their aesthetics and simplicity of construction they also have more practical applications as [multidimensional data indices](https://arxiv.org/abs/1904.07700).
Since there is a one-to-one correspondence between a point in n-dimensional space and a point on the curve, a space filling curve provides a way to reference some point m,n,o,... to (effectively) a "distance" from the origin of the curve.
Therefore, instead of refering to point (5,4,2,1) we can refer to point 1465 along a space filling curve. And (even better) because of its shape, nearby points in the original space are also nearby on the index thus also offering a metric of similarity.
Those are great points; thanks! I was going to say something about the applications in the post, but it didn’t fit well and I deleted it. However, I didn’t know that there was a finite field version of all this, until I looked at the paper in your link.
That's very interesting. I need to read up more, but a lot of modern AI applications work with high dimensional vector data, called embeddings. The challenge is in retrieving these similar vectors, given a query vector. An memory efficient way to store them and yet being able to query them fast is needed. I wonder if this could be a potential way to do that.
Thanks for your comments, and for the link. I think I had read about the Ramsey theory breakthrough. Ramsey theory is a difficult topic to post about, because it quickly becomes terrifying. Here’s a story about Paul Erdős on the difficulty of computing Ramsey numbers.
“Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6, 6). In that case, he believes, we should attempt to destroy the aliens.”
As impressive as space filling curves are in their aesthetics and simplicity of construction they also have more practical applications as [multidimensional data indices](https://arxiv.org/abs/1904.07700).
Since there is a one-to-one correspondence between a point in n-dimensional space and a point on the curve, a space filling curve provides a way to reference some point m,n,o,... to (effectively) a "distance" from the origin of the curve.
Therefore, instead of refering to point (5,4,2,1) we can refer to point 1465 along a space filling curve. And (even better) because of its shape, nearby points in the original space are also nearby on the index thus also offering a metric of similarity.
Those are great points; thanks! I was going to say something about the applications in the post, but it didn’t fit well and I deleted it. However, I didn’t know that there was a finite field version of all this, until I looked at the paper in your link.
That's very interesting. I need to read up more, but a lot of modern AI applications work with high dimensional vector data, called embeddings. The challenge is in retrieving these similar vectors, given a query vector. An memory efficient way to store them and yet being able to query them fast is needed. I wonder if this could be a potential way to do that.
. Wonderful article. Love these posts. I came across this which may interest you.
https://newatlas.com/science/maths-problem-solved/
Thanks for your comments, and for the link. I think I had read about the Ramsey theory breakthrough. Ramsey theory is a difficult topic to post about, because it quickly becomes terrifying. Here’s a story about Paul Erdős on the difficulty of computing Ramsey numbers.
“Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6, 6). In that case, he believes, we should attempt to destroy the aliens.”