壘(To-riｰde) 60-Stars Astrology Season ▲ What Is This "3–4–7" Anyway?

60 Stars Astrology Season ▲

English　Version

By　TOKYOｰTANUKI

壘(To-riｰde)　60-Stars Astrology  Season ▲　

What Is This "3–4–7" Anyway?

1.　Well... last time I wrote that Menkaure's Pyramid has an angle ratio of 3:4:7, 
and that I think it may represent the Lucas sequence.

But I'm sure some people are thinking:

"Why are you so confident when you have no proof?"

or

"Don't treat your own daydreams as if they were facts."

or

"The other two pyramids were explained by length ratios, so why did you suddenly switch to an angle ratio?"

...Ah yes...I can already hear everyone's criticism...and their laughter.




2.　So I escaped by saying,

"Maybe I'll explain it next time."

But honestly, I don't really have a clear answer yet.

...Well, this has always been a blog of little delusions, not a science blog.

So I'd simply like to write down what I'm thinking right now.

To be honest, I probably won't be able to keep updating this blog at the current pace after around next spring.
I may still write occasionally, but probably not very often.

So I'd like to leave these thoughts here while I can.


What I'm imagining is something like this:

Φ¹      1       (Φ−1)¹
Φ²      3       (Φ−1)²
Φ³      4       (Φ−1)³
Φ⁴      7       (Φ−1)⁴
Φ⁵     11       (Φ−1)⁵

....As many people already know,

if we calculate
[Φ^n ± (Φ-1)^n,]
using subtraction when "n" is odd and addition when "n" is even, the Lucas sequence appears.

(Strictly speaking, we only need to consider the absolute value.)

n = 1 (odd)
Power of the curve (Φ): **1.6180**
Seed of the empty space: **0.6180**
Calculation (subtract because n is odd):
**1.6180 − 0.6180**
Point appearing on the number line (Lucas number):
**1.0000** (integer **1**)

n = 2 (even)
Power of the curve (Φ): **2.6180**
Seed of the empty space: **0.3820**
Calculation (add because n is even):
**2.6180 + 0.3820**
Point appearing on the number line (Lucas number):
**3.0000** (integer **3**)

n = 3 (odd)
Power of the curve (Φ): **4.2361**
Seed of the empty space: **0.2361**
Calculation:
**4.2361 − 0.2361**
Lucas number:
**4.0000**

n = 4 (even)
Power of the curve (Φ): **6.8541**
Seed of the empty space: **0.1459**
Calculation:
**6.8541 + 0.1459**
Lucas number:
**7.0000**

n = 5 (odd)
Power of the curve (Φ): **11.0902**
Seed of the empty space: **0.0902**
Calculation:
**11.0902 − 0.0902**
Lucas number:
**11.0000**

n = 6 (even)
Power of the curve (Φ): **17.944**
Seed of the empty space: **0.056**
Calculation:
**17.944 + 0.056**
Lucas number:
**18.000**

n = 7 (odd)
Power of the curve (Φ): **29.0344**
Seed of the empty space: **0.0344**
Calculation:
**29.0344 − 0.0344**
Lucas number:
**29.0000**
---

...Well...this is what I think may be happening.

In short, the integers nearest to the powers of Φ form the Lucas sequence, and adding or subtracting ((Φ-1)^n) adjusts them into exact integers.

This seems to suggest that when we compare the ordinary arithmetic progression with the powers of Φ,the sequence that quietly emerges is the Lucas sequence.

In other words, the arithmetic progression and the golden ratio are what appear on the stage, but the framework that quietly keeps everything aligned may actually be the Lucas sequence.


...If we think in terms of geometry,

the curve (Φ) and the straight line (1) are visible, 

but perhaps something else is quietly hiding behind them.

...That might explain why the visible curves and straight lines are represented by the length ratios

1 – √Φ – Φ(Khufu)
3 – 4 – 5 (Khafre)

while the hidden Lucas sequence is represented by the 3–4–7 "angle ratio" of Menkaure's Pyramid.



Of course...this is only one of my little delusions.

"What do you even mean by something hiding behind them?"

"You're just a tanuki!"

...Ah...I can hear their whispering again!



3. Or perhaps the Lucas sequence has been the main character all along.

Anyway, I really don't know.

Well... that's just another one of Tanu-chan's little delusions.

So, I'll leave the story here for today.

See you next time!




Tanu-chan 💓 TOKYO-TANUKI 💛