Interesting, because then if we take the square root
Hedjuck goldenratio plus#
And then you getĭifferent- phi squared is equal to phi plus 1. It into a quadratic equation that we can solve usingįairly traditional methods. Weird number, this weird ratio that we're beginning to explore? So let's see if we can turn But even this seems likeĪ pretty neat property. And this is a recursiveĭefinition of a variable, where it's defined
I could say that'sġ over 1 plus 1 over and just keep on going onĪnd on and on, forever. Well, that's just 1 plus 1 over and I could just write phi again Plus 1 over- instead of saying phi- I could say, We could say that phi isĮqual to 1 plus 1 over phi. We've defined phi in terms of 1 plus 1 over phi. Statement over here is kind of interesting because Neat property of any number that if I just You get phi minus 1 is equal to its inverse. First of all, youĬould, if you subtract 1 from both sides of this, Ratio, phi is equal to 1 plus 1 over phi. Number, which we're going to call this special Right here over here is phi- so b over a is Inverse of this statement right over here. Well if phi is equal to a overī, which is equal to a plus b over a, we know that So let's see what we can learnĪbout this special ratio phi. And let's just assume that weĬan find a ratio like that. Longer side to the shorter side is equal to the ratio of the Referring to right here, so that the ratio of the So I want to sit and thinkĪbout this a little bit. Of the sum of these two to the longer side? So it's equal to the That the ratio of a to b is equal to the ratio and passing their advantages on to their offspring. What we see now is the result of eons of advantaged plants surviving over disadvantaged plants. so the survivors tend toward that value over time. So plants that tend toward this value have an advantage against plants that don't. It turns out that if a plant grows one leaf, then the next phi (the golden ratio) rotations from the first, then the third phi rotations from the second, and the fourth phi rotations from the third, and so on, that process will result in the longest possible time before the newest leaf is in the shadow of any existing leaf. same problem.Īs this species evolves, the plants whose leaves are most often useful have an advantage and breed more. If the second leaf is opposite the first then that is good, but the third will be in shadow and useless. If it grows a second leaf in the shadow of the first then that leaf is useless. And there are places in the natural world were extreme irrationality is the most efficient solution to a problem, so by natural selection living systems tend toward that value where it works best.Ĭonsider a plant that has grown one leaf. It turns out that the golden ratio is not only an irrational number.
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