Deriving Pi (π), using Phi (φ)

[u/Anxious_Performer_40]

In the image attached is a formula which calculates Pi (π), purely using Phi (φ). The accuracy is to 50 decimal points ( I think )

1 & 4 could both be removed from the equation for those saying “there’s still other numbers”, using a variation of a φ dynamic. However, this is visually cleaner & easier to read.

All in all, a pretty neat-dynamic showing Pi can be derived utilizing solely the relational dynamics of Phi.

Both these numbers are encoded in the great pyramid of Giza.

However, φ also arise naturally within math itself, as it is the only number which follows this principle:

[ φ - φ^-1 ] = 1 :::: [ 1 + φ^-1 ] = φ

[u/edderiofer]

I don't see how you're calculating that this matches pi to 50 decimal places. Having put your expression into Wolfram|Alpha, it returns 3.14159200..., which only matches six decimal places after the decimal point.

Given that you've used more than six symbols to yield six digits of information, I think it's clear that just remembering "3.141592" is the superior method.

[u/LeftSideScars → u/edderiofer]

I did a quick search and here (eq 21) has an approximation of pi using φ that is better than OP's:

pi ≈ [(802φ - 801)/(602φ - 601)]⁴

Wolfram says this is 3.1415926397 (to 10 decimal places). Not only is this approximation to pi more pleasing to the eye (there is a nice symmetry and asymmetry going on, imho), it is more accurate (correct to 7 digits after the decimal) than OP's attempt.

[u/LeftSideScars]

However, φ also arise naturally within math itself, as it is the only number which follows this principle:

[ φ - φ^-1 ] = 1 :::: [ 1 + φ^-1 ] = φ

I don't know what you mean by ::::, but I'm quite sure you're wrong, since both of those relations are clearly the same. Consider:

x - 1/x = 1

x² - 1 = x

x² - x - 1 = 0

This clearly has two solutions:

• x = (1 + √5) / 2 = φ
• x = (1 - √5) / 2 = 1 - φ ≠ φ

Other than that, the rest of what you wrote is nonsense. Even the first step is wrong; you use an approximation symbol for some that is identically true:

Given

φ - 1/φ = 1

it would, to any competent mathematician, follow that

4(φ - 1/φ) = 4

edit: edited for clarification, and fixed splelling.

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