
The Fundamental Theorem of Algebra: A Visual Approach (2015) [pdf] - seycombi
https://link.springer.com/content/pdf/10.1007%2Fs00283-015-9572-7.pdf
======
vanderZwan
> _Points near the origin have dark colors, with the color assigned to a
> complex number z approaching black as z approaches 0. Points far from the
> origin are light, with the color of z approaching white as |z| approaches
> infinity. Every complex number has a different color in this picture, so a
> complex number can be uniquely specified by giving its color._

So I was wondering if this approach could also be useful in the context of
floats and integers. The implementation becomes simpler in many ways (we only
have a finite number of possible values to consider).

For example, start with a unique mapping of 8-bit integers to a continuous
colour ramp. I suppose the easiest option would be to start in the top left,
then go from left to right, top to bottom, resulting in a 16x16 image.

If we then take a formula that also returns an 8-bit value, map every possible
input to an output, and show the results as an image, it should make it easy
to see where we have an integer overflow, for example (if we see sudden jumps
from very bright to dark or the other way around).

Then I realised: I _have_ seen this in the wild! It's used all over the place
in the demo-scene. For example:

[https://www.youtube.com/watch?v=tCRPUv8V22o](https://www.youtube.com/watch?v=tCRPUv8V22o)

[http://wurstcaptures.untergrund.net/music/?oneliner=((t*(%22...](http://wurstcaptures.untergrund.net/music/?oneliner=\(\(t*\(%2236364689%22%5Bt%3E%3E13%267%5D%2615\)\)%2F12%26128\)%2B\(\(\(\(\(t%3E%3E12\)%5E\(t%3E%3E12\)-2\)%2511*t\)%2F4%7Ct%3E%3E13\)%26127\)&oneliner2=\(\(t*\(%2246893636%22%5Bt%3E%3E13%267%5D%2615\)\)%2F12%26128\)%2B\(\(\(\(\(t%3E%3E12\)%5E\(t%3E%3E12\)-2\)%2511*t\)%2F4%7Ct%3E%3E13\)%26127\)&t0=0&tmod=0&duration=30&separation=100&rate=44100)

Aside, this also reminds me of Gustafson's closure plots for comparing his
number encoding to standard floating points, although it's not really the same
thing:

[https://www.youtube.com/watch?v=aP0Y1uAA-2Y&t=33m47s](https://www.youtube.com/watch?v=aP0Y1uAA-2Y&t=33m47s)

------
stablemap
There's a big list of proofs on MathOverflow [1]. Not a lot of pictures in it
but there are many to be drawn.

My favorite is probably top answer: a polynomial f(z) gives you a map from the
Riemann sphere ℙ^1 to itself. Topology tells you it's closed and analysis
tells you it's open because locally a holomorphic map always looks like z^n.
It follows that the map is surjective.

[1] [http://mathoverflow.net/questions/10535/ways-to-prove-the-
fu...](http://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-
theorem-of-algebra)

