3 ms·

The Fundamental Theorem of Algebra: A Visual Approach (2015) [pdf]

88 points by seycombi 10y ago

vanderZwan 10y ago
> 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*(%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 http://wurstcaptures.untergrund.net/music/?oneliner=((t*(%22... 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 10y ago
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-fundamental-theorem-of-algebra http://mathoverflow.net/questions/10535/ways-to-prove-the-fu...

88 points by seycombi 10y ago

> 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*(%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 http://wurstcaptures.untergrund.net/music/?oneliner=((t*(%22... 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 10y ago

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-fundamental-theorem-of-algebra http://mathoverflow.net/questions/10535/ways-to-prove-the-fu...