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
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.
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
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!
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!
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.
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;
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. Golden->lithium Silver->beryllium Bronze->natrium Copper->magnesium Nickel->aluminium
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.
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.
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.