# OPEN THREAD 20200306

Basically, all legal free speech is allowed. We will assist the authorities in dealing with illegal speech. You are each other’s moderators. Have fun. And don’t forget to MAGA at nuclear levels.

After going through the elements, we now enjoy a sequence of RANDOM somewhat pseudo-random topics that will be thrown out for investigation and commentary on each open thread. At some point, in a way something like composite numbers, I will accidentally hit a second occurrence of one of them – that’s just normal.

Have fun!

Citizen U

(a.k.a. W on the OTHER site)

Day 121 – The number “phi” (Φ)

## 2 thoughts on “OPEN THREAD 20200306”

1. cthulhu says:

I inherited a 78 record from my maternal grandfather that was titled “Cigareets, Whusky, and Wild Wild Women”.

The song was very popular in its time. It’s even on YouTube.

Seems like there’s a near-universal impulse towards the rawer pleasures of life…..where appreciation of mathematical constants is a more selective taste.

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2. slowcreekno says:

This constant Φ is known as the golden ratio. There is a certain aestetic pleasantness to rectangular objecs (tables, doors, paintings, pieces of paper…) where the two sides have the ratio Φ and 1.

It comes from wanting the ratio of a whole, lengths (a+b)/a equal to the ratio of the parts, a/b.

(a+b)/a = a/b = Φ

which leads us to the equation Φ^2 – Φ – 1 = 0. and the positive root of this Quadratic equation is

Φ = (1 + sqrt(5))/ 2 = 1.618…

Since Φ can be expressed as the value of a closed formula like this, it does not qualify as transcendental, unlike e and π which only can be expressed as limits where some variable goes to infinity. This doesn’t make it less interesting, and like π and e it shows up in some curious places, for example, as the ratio of two adjacent numbers in the Fibonacchi sequence converges to Φ.

And the inverse of Φ 1/Φ = 0.618… – Φ – 1

US Legal 8.5″ by 14″ paper has a height/width ratio of 1.64, slightly more than the golden ratio.
The Letter 8.5″ by 11″ paper has a shorter height/width ratio of 1.29.

And the ISO A-series (A0, A1, etc) have a ratio of 1.414 between height and width (or vice versa). This means that if the sheet is cut in half across its longest dimension, each half will retain the same ratio. We may recognize the ratio as the square root of 2 and this is no coincidence.

The A0 is 0.841m by 1.189m which gives it the area of 1 square meter. Then A1 is half of this, A2 is 1/4 of this etc. thus the area of a sheet of paper size An = 1/2^N m^2 in size.

And a sheet of A4 paper of the very common grade of 80 g/m^2 thus weighs pretty exactly 5 grams. (0.005 kg) Can be useful to know when estimating the weight of a letter to be sent through the mail; or if a fairly precise weight reference is needed in some other situation.

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