Complex Projective 4-Space

Oh wow! I wasn’t expecting to see this on HN. I’m one of the authors of the paper, and I’m happy to answer any questions. (I’m sorry I didn’t see it earlier, it was posted in the middle of my night.) First of all, to clear up a possible misunderstanding: this new formula will not help you to compute determinants more quickly. The number of terms still grows a lot faster than polynomial, and if you just want to compute the determinant, there are polynomial-time algorithms for that. The best theoretical results on the complexity of computing determinants, that I know of, are in [0]. In practice you'd probably use Gaussian elimination, or maybe Bareiss for integers. Someone complained that there’s no code in the paper. Since it isn’t really useful for practical computation, I didn’t think code would be useful, but in fact it did start as a Mathematica program when I first discovered it. Adam Goucher (one of my co-authors) wrote a blog post on it [1] within a few days of its discovery, which links to my original Mathematica notebook. [0] https://users.cs.duke.edu/~elk27/bibliography/04/KaVi04_2697263.pdf [1] https://cp4space.hatsya.com/2022/07/02/a-combinatorial-proof-of-housto…

We also know the optimal circuits if you want to compute two boolean functions from four variables at the same time: https://cp4space.hatsya.com/2020/06/30/4-input-2-output-boolean-circui…

Prime numbers with practical applications do get discovered. Most notably, I suppose: 2^64 - 2^32 + 1: https://cp4space.hatsya.com/2021/09/01/an-efficient-prime-for-number-t… There are annual prizes on performing certain computations efficiently in prime fields for this prime: https://www.zprize.io/ But if there is a practical application to very large primes: I don’t know. 40 million digits sounds very inefficient unless you always operate in sub-groups.

"While all agreed "the hat" was the first einstein, its mirror image was required one in seven times to ensure that a pattern never repeated." The article has a small error here - it should say approximately 7. The ratio between the tiles and their mirrors is phi^4. https://cp4space.hatsya.com/2023/03/21/aperiodic-monotile/

If I recall (previously-submitted article - https://cp4space.hatsya.com/2023/03/21/aperiodic-monotile/ ), the shape has to be flipped upside down some fraction of the time. So you'd either need two tile shapes for a real bathroom, or your tiles would be a compromise between "both faces are easily cleaned" and "both faces stick firmly to the mortar".

yes https://cp4space.hatsya.com/2021/01/08/the-neural-network-of-the-stock…