8 ms·
Beauty in Mathematics: Modular Multiplication Tables
- gfody 9y agothe primes look similar to sqrt(x*y)
- dnautics 9y agohow about general galois fields?
- enriquto 9y agoon a field you have two tables, one for sum and the other one for product, which one are you talking about ? each of these tables is the table of a group, thus they will look like the tables of other finite groups
- dnautics 9y agoAs with prime fields, The additive group will look pretty boring. Are the multiplicative groups necessarily isomorphic to the multiplicative groups on (z/nz)*?
- enriquto 9y agoThe multiplicative group is a finite abelian group, so it is certainly as "boring" as the additive group. The interesting pictures appear when you take into account a particular ordering of the elements of the group.
- dnautics 9y agoYes of course lexicographic ordering is assumed, and the pattern will depend on the polynomial used to mod out multiplication. The boringness of the additive group presumes lexicographic ordering.
- tabtab 9y agoParametric equation for a hamburger: x(t) = Sin(Tan(t)) y(t) = Cos(t) Add a multiplier to x to get a submarine sandwich. Courtesy of Uncyclopedia.
- dahart 9y agoYour burger, sir: https://www.dropbox.com/s/3iyakwegodne9gl/burger.jpg?dl=0 https://www.dropbox.com/s/3iyakwegodne9gl/burger.jpg?dl=0 A fun side-note, zoom out to see this is an infinite burger stack, where every other burger is upside-down: https://www.dropbox.com/s/lz53ybz3pmchh9s/burgerStack.jpg?dl=0 https://www.dropbox.com/s/lz53ybz3pmchh9s/burgerStack.jpg?dl...
- tabtab 9y agoCool! That's more of a 3D look. When I plot it I get a line drawing of a burger.
- dahart 9y agoThese are definitely interesting. They make me wonder what the underlying continuous function is, and if it's the same function for all the pictures. These mostly look like aliasing to me. Which makes sense, perhaps is nearly obvious, because that's what you get when you plot the mod of a multiply on a grid. It's pretty easy to reproduce something close to the large image (prime 9973), by just plotting the continuous function "(x * y) % 2", or even sin(x * y): https://www.dropbox.com/s/3eayuknh1urayay/001.jpg?dl=0 https://www.dropbox.com/s/3eayuknh1urayay/001.jpg?dl=0 As a graphics person, what I normally do is try to remove that aliasing by using more samples. https://www.dropbox.com/s/imx6njdek3wello/002.jpg?dl=0 https://www.dropbox.com/s/imx6njdek3wello/002.jpg?dl=0 And for nasty functions like this one, you find out that more samples doesn't fix the aliasing. See the hints of it still there far away from the center? That's when you get into the fun signal processing math and have to use a better filter function to remove all aliasing: https://www.dropbox.com/s/ysoibo3v8j9b2gg/003_gauss.png?dl=0 https://www.dropbox.com/s/ysoibo3v8j9b2gg/003_gauss.png?dl=0
- sp332 9y agoIt reminded me of a moiré pattern. https://en.wikipedia.org/wiki/Moir%C3%A9_pattern#/media/File:Moire_Lines.svg https://en.wikipedia.org/wiki/Moir%C3%A9_pattern#/media/File...
- dahart 9y agoThat's exactly what it is! You need two patterns for moire, so in my case one is the continuous function, the other is the pixels (or sampling). If I sample every other pixel, I get a completely different moiré.
- madez 9y agoCould you elaborate on the signal processing and the filter function?
- tapatio 9y agoThis video will give you a quick overview of aliasing and an anti-aliasing filter: https://www.youtube.com/watch?v=v7qjeUFxVwQ https://www.youtube.com/watch?v=v7qjeUFxVwQ
- chillingeffect 9y agoA while back, Noah Vawter was using this to make ultra-fast math at the cost of accuracy. [1] He computed each bit of an output function such as addition and multiplication as a function of any two of the N inputs of both operands. Then generated approximations using single-operator functions (and, or, nand, etc.) that minimized error. The result was single-cycle approximations. For example, he claims 44% accuracy (not sure how this is measured) for 7-bit multiplies in a single cycle. [1] http://gweep.net/~shifty/ http://gweep.net/~shifty/
- gp2000 9y agoDo you have a more direct link? That one just goes to Noah's home page and for the life of me I couldn't navigate to the ultra-fast math page nor find it via Google. Edit: I think I've found it: http://www.gweep.net/~shifty/portfolio/oddsvf/index.html http://www.gweep.net/~shifty/portfolio/oddsvf/index.html
- JadeNB 9y ago> [1] http://gweep.net/~shifty/ http://gweep.net/~shifty/ Eep, that's an eye-burning page. Where should I go for discussion of the fast-but-inaccurate techniques you mention?
- Karrot_Kream 9y agoTao's undergrad analysis book has a great derivation of integers and rationals from the basics of Peano axioms for natural numbers. Highly reccommend it for somebody who wants to learn some basic but elegant mathematics.
- jaf7 9y agoThank you.
- 3JPLW 9y agoSite is down; archive/cache/mirror/ctrl-f: http://archive.is/NHBkq http://archive.is/NHBkq
- nekopa 9y agoSome of these remind me of Chladni patterns.