* The 12 Pillars of Phi *
There are 12 tribes of Israel and 12 Apostles, so let's take a look @
[(1+√5)/2]^12 ... OR ... Φ^12 = (Φ^3)(Φ^3)(Φ^3)(Φ^3) ... see graphic
What do you know about the Golden Ratio (Phi Φ)?
(Φ+1)/Φ = Φ ==> Φ^2 = Φ+1
Φ = (1+√5)/2 = 1.618033988749894848204586834365638117720...
Φ ≈ 1.618 which reminds us of 1618 AD.
John Napier, who died in 1617, discovered logarithms and invented the forerunner of the slide rule (Napier's rods or "bones":
Since ln(1)=0, I will add one to 1617 to get 1618. ☺☺
Napier also wrote a treatise on the book of Revelation (1593):
The Synod of Dort met in Holland from 1618 to 1619. Here are their conclusions regarding the Sovereignty of God in salvation:
Now, back to our problem (recall: Φ^2 = Φ+1) ...
Φ^12 = (Φ^2)^6 = (Φ+1)^6 = [(Φ+1)^2]^3 = (Φ^2+2Φ+1)^3
= (Φ+1+2Φ+1)^3 = (3Φ+2)^3 = [(3Φ+2)^2](3Φ+2)
= [9(Φ+1)+12Φ+4)(3Φ+2) = 63Φ^2 + 81Φ + 26 = 63(Φ+1) + 81Φ + 26
So, Φ^12 = 144Φ + 89
We happily come back to twelve squared which is gross (12 doz)‼
My inspiration for this #rant ...
my video on Calc is EZ (1.5K+ views and rising):
my book *Calculus is Easy* >>
FYI: Φ = 2cos(π/5)
