## Saturday, June 2, 2012

### Fibonacci Day: Some Additional Fibonacci Fun

I don't know if you noticed, but today is a Fibonacci day. It is the second of June, and 2 is the third Fibonacci number.

Something mentioned frequently in the mathematical world is that the Fibonacci numbers often appear around nature. It is also in science, architecture, we even found it in literature.

Rather than explaining some applications, I thought I would show you a neat video I found instead.

I find all of the applications fascinating, as well as the fact that we can easily draw our own golden rectangle. You don't even have to be an artist to do it.

Bonus: In the video, they mentioned that if you take the square out of the golden rectangle, the remaining rectangle is a golden rectangle. You can prove this 2 ways.

First off, the sides of a golden rectangle can be two consecutive Fibonacci numbers. Say they are 55 and 89.

If you cut off a 55x55 square, you are left with a 34x55 rectangle. Since these are two consecutive Fibonacci numbers as well, it is a golden rectangle.

The more interesting one, however, is to look at the golden ratio itself. If you do 1/1.618034..., you get 0.618034...

In other words, phi:1 = 1:phi-1. So, if we cut off a 1x1 square from the rectangle with side ratio phi:1, we are left with a side ratio of 1:phi-1, which is the same as before. I found this really cool about Fibonacci numbers.

1. The video mentioned the Egyptian Pyramids:

0thly, I do not get the fascination with huge piles of stone. the usable volume of a pyramid is less than < .1%. The nonstructural volume of modern skyscrapers is about ~90%. After subtracting utility-space (airducts, crawlspaces, et cetera) is still about ~ 80%. The enclosed volume of geodesic domes* and monolithic domes*(which honestly, because of extremely high ceilings, may be hard to fully utilize from the ground all of the way to the ceiling) is greater than > 99.9%.

* http://wikipedia.org/wiki/geodesic_dome
* http://wikipedia.org/wiki/monolithic_dome

As for τ/4 or ϕ appearing in the Egyptian Pyramids, that is an accident:

The Egyptian Cubit, believed to be the length of the forearm of an early Egyptian Pharaoh, was the unit the Egyptians used to measure length. The Egyptians had only a vague idea about τ.

One defines the standard Egyptian Pyramid has a base x rolls of a wheel with a diameter of 1 Egyptian Cubit and 2x number of rolls in cubits high. Initially, the ration of base to height was τ/4.

Because of erosion, the ratio of base to height has often closer or even gone through ϕ. The Egyptian Pyramids erode faster at the top than the base because the Egyptians built them over > 4,000 BBF (Before Benjamin Franklin):

Benjamin Franklin invented the lighting rob along with bifocals, the cast iron stove, et cetera. Lightning strikes the tops of the Egyptian Pyramids. Since stone is a poor conductor, much of the energy becomes heat. Sometimes the stone explodes from the heat. Thus, the Egyptian Pyramids erode faster from the top.

2. If you like Golden Rectangles, you will love ISO 216:

http://www.cl.cam.ac.uk/~mgk25/iso-paper.html
http://wikipedia.org/wiki/ISO_216

This leads to grammage:

http://wikipedia.org/wiki/paper_density

Since we get into measures and standards, we might as well get into Planck-Units:

http://wikipedia.org/wiki/Planck_Units