OT: Fibonacci Numbers Video

[kimosabe]

The Fibonacci ratio shows up a lot in naturally evolved designs.

The length of the bones in the hand and fingers show the Fib ratio in their lengths. Also the length of the hand compared to the length of the forearm. The distance from the floor to a human's naval as compared to the distance from the naval to the top of the head is Fib. The placement of the F holes on a violin are determined using F. numbers.

And, it's not just the ratio, it's the actual numbers that also recur. The number of rows on a seed head of wheat and the number of seeds per row.

Livio's book, "The Golden Ratio," can be dense reading. Here's a quick entertaining read loaded with examples. "Fascination Fibonaccis

- Mystery and Magic in Numbers," by Garland. You can get a used copy for as little as $1.62

[Robert Bonomi]

'Smoother' response --- no reinforcing resonances on two axes.

The 'ideal' ratio is .618:1:1.618.

[Robert Bonomi]

There is a complex set of trade-offs involved.

Peripheral vision is more effective to the sides than up/down. This is in part because, 'historically', threats were more likely to appear from the sides. The 'range of vision', _vertically_, is typically about +/- 60 degrees from the horizontal. However, 'to the sides', it is typically 80+ degrees from 'straight ahead', and in a significant number of people it can range to 90-95 degrees _and_more.

"Portrait" orientation (the long dimension vertical) is optimal -- in terms of 'visually pleasing', that is -- at 1.618:1. This ratio occurs 'naturally' in a bunch of aspects in the human body -- See da Vinci's figure studies.

"Landscape" is more natural, and 'panoramic', at a ratio that is 'wider' and 'flatter'. You don't get much from the extra 'sky' in an exterior shot. Similarly, for interiors, the floor-ceiling dimension tends to limit the usefulness of a greater display height.

Also, realize that 'wide-screen' in the movie theater is typically _1.88_:1.

and the famous 'Cinerama' process from the 1950s, 1960s, and 1970s, was

*really* wide -- at _2.66_:1.

[Bill]

I don't doubt it. IIRC, the trickiest part is showing the limit exists, then the quadratic formula decides the value of the limit. Probably, the more general result can be obtained by using the original one: If you multiply the Fibonacci sequence by some any non-zero value c, then when you take the ratio of successive elements the c cancels out. So the sequuence 0, c, c, 2c, ... will yield the Golden Ratio too. Beginning with -1, 0 instead we get -1, 0, -1, -1, -2,.. and this also yields the Golden Ratio (discard the first term to see this). Taking the sum of the 2 sequences

0, 1, 1, 2, 3, ... and 1, 0, 1, 1, 2, 3, ..., or equivalently any zon-zero multiples of them 0, a, a, 2a, 3a, ... b, 0, b, 2b, 3b ... basically helps establish your result. Minor technicalities omitted. If a or b is zero, but not both, the conclusion is unaffected. So yes, I agree. Score another one for the Golden Ratio!

Bill