It's PhiDay
A previous Math News proposed the idea that mathematics classrooms should celebrate other numbers besides Pi. If you are interested, Phiday occurs this week and eday is lurking nearby...January 6th and February 7th to be "approximate"!
But, I hear you ask, what can be done on phiday? Eating Pie is of little value, maybe you should switch to artichokes, pineapples, etc (any idea why?).
So, to celebrate Phiday, consider theses ideas gleaned from multiple resources......
To memorize the digits of Phi = 1.61803398874989484820458683436563811772… , I know of no mneumonics where each word has the number of letters equal to the digits like those known for e or pi.... So why not make up one? (Share it with me and I will try to share it via this web site.)
Phi does posess some interesting numerical properties. For example, the first 18 digits of phi total 100 and the first 42 digits total 200. Will any other initial sum of phi's digits sum to a hundredsmultiple?
Calculate the limit of the sequence sqrt(1), sqrt(1+sqrt(1)), sqrt(1+sqrt(1+sqrt(1))),..., sqrt(1+sqrt(1+sqrt(1+.....+sqrt(1)))))...
Regarding the previous, assume that w is this limit number, show that w equals Phi algebraically. That is let w = sqrt(1+sqrt(1+sqrt(1+.....+sqrt(1)))))... Square both sides to get w^{2} = 1 + sqrt(1+sqrt(1+sqrt(1+.....+sqrt(1)))))..., or w^{2} = 1 + w, etc....so, does w = Phi?
Watch the Walt Disney video Donald Duck in Mathemagic Land.
Investigate the idea of Golden Triangles and Golden Ellipses.
Investigate connections between Phi and the Fibonacci Sequence.
Investigate occurences of Phi in art, architecture, nature, music, and the human body.
Interview a dentist about the "common" process of reforming each tooth into a Golden Rectangle.
Investigate the history of the number Phi, starting with Phidias and moving through Euclid, Pacioli, Kepler, Penrose, etc.
Or, investigate this sequence of computations:
(1+1/1)
(1+1/(1+1/1))
(1+1/(1+1/(1+1/1)))
(1+1/(1+1/(1+1/(1+1/1))))
(1+1/(1+1/(1+1/(1+1/(1+1/1)))))
(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1))))))
(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1)))))))
etc.
Whatever you do, have some phiun!
