1. What about the platinum ratio, from the zero-nacci sequence?
    Hence, P_n = 0*P_(n-1) + P_(n-2)
    As a result, the sequence is 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 …..
    The ratio doesn't approach anything but by the formula, it is (0+sqrt(0^2+4))/2, or 1.
    And sure enough, if you take 0 squares out of the platinum rectangle, you get a platinum rectangle ;)P

  2. can we use this to find larger primes ie if u find the largest prime set position to it and look at the number in the sequence. and repeat

  3. Couldn't we use the silver ratio to create infinitely big prime numbers? Like start at 3, the 3rd index is 5, the 5th index is 29, the 29th index is… Etc.?

  4. I love peregrine falcons too, Those and Snowy owls. Anyway, I think this doesn't get taught at schools is the lies to children. Its easier to just focus on one ratio so that we don't overload them. Really enjoyed this video. I guess there must be Fibonacci sequences which use the negative multiples I wonder if these ever occur in nature?I did a quick excel experiment and I think the spiral becomes a wavy path which tends to y=x as the arcs get shorter and shorter. This is the same for all metallic ratios just they basically get there faster as the ratio cuts down the bounding squares for the arcs quicker. Its just like a damped oscillation along the 45.Anyway very thought provoking, thanks.

  5. 6:39 It always confuses me when a variable is used twice with a different meaning in the same formula, could you please start using a/b instead of n/N? I think subtle weird usage patterns like this really throw a lot of people off when trying to follow maths problems, or copyingsomething over and not noticing the different uses of ‘n’ and ater really getting stuck trying to understand it all.
    I had a maths teacher who would write complex problems on the blackboard using ‘a’ for 2 different nested indexez and it was really really confusing :p

  6. I can't seem to find it on the internet, so i hope someone here can answer: is there a limit to the pitch angles as you increase N?

  7. the reason why we like geometric spirals in nature is the growth it represents, that it is alive. When we see it in art, it shows the artist's realization to make his art be alive.

  8. Something I noticed: the formula for the Nth metallic ratio [N+root(N^2+4)]/2 is actually the solution to the general quadratic equation x^2 – Nx – 1 = 0 if you solve it according to the quadratic formula. Not sure if that's relevant to any larger mathematical problem but I thought it was interesting to note!

  9. 3.3027 4.23606 5.19528 … is the sum of the fractional part (also the sum of the reciprocal) for this infinite series, infinite or is it bound

  10. Counting the fibonacci numbers backward into the negative: 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55

  11. I ended up doing this by accident (after step 1, when you have the 45 degree angle) when I was trying to figure out how much wood I would remove if I used successive cuts with the table saw to round off the corners of my kalimba. I didn't even know what I was doing. Neat!

  12. Interestingly…after playing with the Pell Sequence in a spreadsheet, i only see six (6) prime Pell numbers with indexes that are also prime. After that, the Pell sequence expands into large numbers with many zeros.

  13. Everyone on numberphile looks like a serial murderer, I think it's something to do with the passion they show for maths

  14. "We can easily work out how much you've cut off"

    You didn't have to explain anything for me to know the answer- too much.

  15. We can call the √2 ratio the platinum ratio, but in fact it's not a metallic ratio, so maybe the diamond ratio would be better.

  16. As a marine biologist, I love these. Forms like this pop up all over the undersea world, especially among invertebrates. Well done!

  17. "Copper, nickel… aluminium?" That one cracked me up.

    Awesome content as always. I'll have to use these metallic ratios in my photo cropping (I've used the golden rectangle but then defaulted to boring ratios like 1:2, 1:3, etc.)

  18. This is precisely what I used to do when I was younger and used scissors to cut my fingernails. They used to call me Wolverine.

  19. 101;2,002;30,003;400,004;5,000,005;60,000,006;700,000,007;8,000,000,008;90,000,000,009; and now for the magical Omni-presence of zero: considering that zero is nothing but an integer stuck between positive and negative it sure does make numbers larger… back to the sequence10,000,000,000,001;110,000,000,000,011;1,200,000,000,000,021;13,000,000,000,000,031;140,000,000,000,000,041;1,500,000,000,000,000,051;16,000,000,000,000,000,061;170,000,000,000,000,000,071;1,800,000,000,000,000,000,081;19,000,000,000,000,000,000,091;200,000,000,000,000,000,000,002;

  20. To name theese ratios, grab a periodic table, eliminate all non-metalls (and hydrogen), and go N steps forward to the higher atomic number to name the N ratio.

    metalloids are not used!

  21. Hold on is this the dude all grown up from the meme where the guy is straining his forehead and the vein is popping out!? Serious inquiry

  22. Wow! It's amazing…
    … that this absolute BS made it into numberphile.
    If a person cuts a nail, then leaves it alone until a later date, then cuts it again in the same manner, the resulting nail clipping is 2 identical curves separated by a distance.

    Nail clippings being that silly larger curve with a smaller curve within it is formed out of ignorance + imagination + stupidity.

  23. So the Pell Sequence is a pseudo prime number generator. I wonder why this isn't used to find those larger primes. Just look at the prime indices and you're done.

  24. Since the pell number approximately doubles at each step, can it be used to quickly generate extremely large primes?

  25. He constructs a spiral starring at 10:26. I assume it is an approximation, since it is made up of circular segments. Or did I miss something?

  26. 13:38 – Peregrine Falcons hunt by swooping on prey (mostly birds) at high speed. The swept-back wings are the most striking feature whilst flying. They hit birds from out of the sun, making it easier to see and consequently making it nearly impossible for the prey to see them coming. When striking at great speed, they hit the prey with their talons folded into a "fist", stunning it, then circle back to take the falling bird.

  27. So I was curious about something. I had no idea that the Golden Ratio was a result of a known equation: (N + sqrt(N*2 +4))/2, so I wanted to know what the ratio of ((N + sqrt(N*2 +4))/2)/N looked like. Turns out, it approaches y=x, it approaches N as N increases. And as N decreases or goes negative, it approaches 0. Don't know what that means, but it's fascinating.

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