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If you draw lines through vertices of a pentagram, and draw lines between the new points you get that way, and so on, you can get an infinity of pentagrams - all rescaled by different powers of the golden ratio! This picture was drawn by James Dolan using Mathematica. (1/n)pic.twitter.com/GCLjMGvwIu
If you run up any arm of the big pentagram you'll see little pentagrams, alternating blue and green, each 1/Φ times as big as the one before. If you *define* Φ this way you can prove Φ = 1 + 1/Φ just by examining the picture. See how? (2/n)
It follows that Φ = 1 + 1/Φ = 1 + 1/(1 + 1/Φ) = 1 + 1/(1 + 1/(1 + 1/Φ)) = 1 + 1/(1 + 1/(1 + 1/(1 + 1/Φ))) and so on. This means that the continued fraction expansion of Φ never ends, so it must be irrational. (3/n)
The Pythagoreans loved the pentagram, and James Dolan speculated that this might be the reason why. It contains within it a proof that there exists an irrational numbers! (4/n)
It's hard to tell. But there's evidence that early Greeks were interested in continued fraction expansions... David Fowler, The Mathematics Of Plato's Academy: A New Reconstruction, 2nd edition, Clarendon Press, Oxford, 1999. Review at http://www.maa.org/reviews/mpa.html … (5/n, n = 5)
Is this how we summon #Cthulhu?
No, you summon it by writing its name after a # sign. I feel sorry for you.
We made this shape from our zometool set a while back and have it hanging on our living room wall :)pic.twitter.com/4x3bczoOUQ
Oh god, not more fodder for the golden ratio bandwagonerspic.twitter.com/ndq2ktqAhf
whatcha got against the golden ratio? o_O
Solar wind speeds during coherent magnetospheric mode at L1 are also scaled by the golden ratio, downward from the magnetopause spreading rate, deterministic at 847 km/s.
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