Beauty in Mathematics: Modular Multiplication Tables

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

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

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

Re: Beauty in Mathematics: Modular Multiplication Tables

#6

Post by sp332 » Tue, Oct 03, 2017, 6:28 PM UTC

dahart wrote: post #5

These 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…

It reminded me of a moiré pattern. https://en.wikipedia.org/wiki/Moir%C3%A9_pattern#/media/File...

Re: Beauty in Mathematics: Modular Multiplication Tables

#7

Post by dahart » Tue, Oct 03, 2017, 6:30 PM UTC

tabtab wrote: post #4

Parametric 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.

Your burger, sir: 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...

Re: Beauty in Mathematics: Modular Multiplication Tables

#8

Post by dahart » Tue, Oct 03, 2017, 6:33 PM UTC

sp332 wrote: post #6

dahart wrote: post #5

These 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…

It reminded me of a moiré pattern. https://en.wikipedia.org/wiki/Moir%C3%A9_pattern#/media/File...

That'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.

Re: Beauty in Mathematics: Modular Multiplication Tables

#9

Post by chillingeffect » Tue, Oct 03, 2017, 6:35 PM UTC

A 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/

Re: Beauty in Mathematics: Modular Multiplication Tables

#10

Post by madez » Tue, Oct 03, 2017, 6:38 PM UTC

dahart wrote: post #5

These 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…

Could you elaborate on the signal processing and the filter function?