
Beauty in Mathematics: Modular Multiplication Tables - happy-go-lucky
https://friendlyfieldsandopenmaps.com/2017/09/07/beauty-in-mathematics-modular-multiplication-tables/
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dahart
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 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)

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

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tapatio
This 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)

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tapatio
If you have more time, this is more thorough:
[https://www.youtube.com/watch?v=yWqrx08UeUs](https://www.youtube.com/watch?v=yWqrx08UeUs)

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chillingeffect
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/](http://gweep.net/~shifty/)

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gp2000
Do 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)

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Karrot_Kream
Tao'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.

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jaf7
Thank you.

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3JPLW
Site is down; archive/cache/mirror/ctrl-f:
[http://archive.is/NHBkq](http://archive.is/NHBkq)

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tabtab
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.

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dahart
Your 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...](https://www.dropbox.com/s/lz53ybz3pmchh9s/burgerStack.jpg?dl=0)

~~~
tabtab
Cool! That's more of a 3D look. When I plot it I get a line drawing of a
burger.

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nekopa
Some of these remind me of Chladni patterns.

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gfody
the primes look similar to sqrt(x*y)

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dnautics
how about general galois fields?

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enriquto
on 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

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dnautics
As with prime fields, The additive group will look pretty boring. Are the
multiplicative groups necessarily isomorphic to the multiplicative groups on
(z/nz)*?

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enriquto
The 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.

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dnautics
Yes 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.

