畾（ra-i) 60 Stars Astrology Season 6 Mini Appendix Part 4: Angles, Lines, and 7 ② A heptagon is mysterious

60stars astrology

60stars Astrology Season 6

English version

By Tokyo-Tanuki

畾（ra-i) 60 Stars Astrology Season 6 Mini Appendix Part 4: 

Angles, Lines, and 7　 ②　A heptagon is mysterious

1. well, last time I mentioned a little bit about heptagons, and heptagons have two types of diagonals.

Let the length of each side of the heptagon be "a", the shorter diagonal be "b", and the longer diagonal be "c",

1/a= 1/b + 1/c  

.....This is the famous relationship. 
Well, you can find it in textbooks.



2. So if we assume that "a" = 1, then

b + c = b × c
b squared = 1 + ⅽ
c = b squared - 1
c squared = b squared + b
c squared = 1 + （b × c） = 1 + b + c

Thus, we have the strange relation

b=2.246979603717....
c=1.801937735804....
b + c = b × c=4.0489177395....

So, at first glance, the numbers look featureless, but it is strange!

3.  By the way, with the lengths of the sides,　

"b squared=1 + c"　

This reminds me a little of the Pyramid of King Khufu, Φ=√Φ+1, or the Golden Ratio, Φ squared=Φ+1.

So this time, I looked for a cool relationship between "b" and "C".

Well just "b + c = b × c " is pretty cool, though.



...... Now, if we do the math, we find the following relationship between "c" and "b".

c = b squared -1
c squared = b squared +b
c cubed = 3 × squared b +2b -2
c to the 4th power = 6 × squared b +5b -3
c to the 5th power = 14 × squared b +11b -8
c to the 6th power = 31 × squared b +25b -17
c to the 7th power = 70 × squared b +56b -39
c to the 8th power = 157 × squared b +126b -87
c to the 9th power = 353 × squared b +283b -196

This is the relationship between the two items "b" and "c"!

🌟　🌟　🌟　🌟

This is, where the "b-squared terms" are,
For example, 
for the 7th power of ⅽ,
70=31+31+(14-6)
In the case of the 8th power of c
157=70+70+(31-14)
.....and so on.

There is a constant relationship between the coefficients of the terms.

The last "integer term" is,
For example, for the 7th power of c, 
it is -39, 
but it is also 39=17+17+(8-3)
For the 8th power of ⅽ, it is -87, 
but this is 87=39+39+(17-8)

....and there is also a constant relationship between the coefficients of the terms.



The "b term" is,
For example, in the case of the 8th power of c, 
it is 126.

This is the sum of the coefficients of the b squared and b terms in the case of the 7th power of c,
In other words, 126=70+56

....Well, mathematically, the length of diagonal "c", no matter how many times it is squared,

"b squared",  "b" , and  "an integer"   added together.

which is, well, neat....... But it's not so cool.



4.  So, Tanu-chan thought about it a little.

...And since it's New Year's, I searched for a cool relationship that fit the heptagon, using my Japanese raccoon dog power.

...... Hmmm... I see it... I see it. Oh my gosh⚡

"b to the 7th power - 14 (b + c) = b to the 7th power - 14 (b × c)　=　5 "
or 
"b to the 7th power - 14 (b + c +１) = - 9 " 

This would fit the heptagon, since it has both 7 and 14 in it!

....And, let's compare it with the diagonal “Φ” of the pentagon!

⇓

" Φ5 to the 5th power - 5 (Φ) = 3 "

" Φ5 to the 5th power - 5 (Φ + 1) = -2"

Oh wow!　

Maybe Φ's best friend has finally appeared!

Well, it's Tanu-chan's delusion, so everyone do your own calculations to see if the math is correct!

... If it's not correct,

" b squared - 3 = 1/ (b × c) "

or

" b to the 5th power  - 5b = 4c +1 "

........just bear with them. It's not very flashy.

Also, just to be clear, Tanu-chan is a liberal arts student, so I don't know anything about Chebyshev polynomials or anything difficult like that.

I am just looking for a cool formula!

That's all for today!

Tanu-chan💓   TOKYO-TANUKI💛