www.ConcurrentInverse.com

https://www.cut-the-knot.org/htdocs/dcforum/DCForumID4/606.shtml

There will always be a use for large prime numbers, eg 111, when

expressing approximations to irrational numbers. There are 11*6 feet

in a chain, 13 weeks in a quarter, and 2*11/7 = 3.1428 = Pi + 0.0013.

666 - 720 = 54 = 270/5 deg = 3 Pi / 10 radians

pi/5 is related to the golden ratio

616 - 630 = 14 is not likely to be related to the golden ratio

5**1/2 = 2.2360679774997896964091736687313

(5/7)**1/2 = 0.84515425472851657750961832736595

(5/7)**1/3 = 0.89390353509656765128791642511372

1/ Phi = Phi -1

Phi**2 - Phi -1 = 0

2*Phi = 1 +-(1+4)**0.5 = 1 +- (5)**0.5 = 2* 1.6180339887498948482045868343656

Phi and pentagonal symmetry are discussed in depth at

https://www.mcs.surrey.ac.uk/Personal/r.Knott/Fibonacci/phi2DGeomTrig.html#trigmore

1/(5)**1/2 - 0.44721359549995793928183473374626

pentagonal symmetry pi/5 = 36deg

sin36 = 0.58778525229247312916870595463907

cos36 = 0.80901699437494742410229341718282

= Phi /2 = (1+(5)**0.5)/4

tan36 = 0.72654252800536088589546675748062 = (5-2*5**0.5)**0.5

cos72 = ((5**0.5-1)**0.5/2

Thus in pentagonal symmetry the is a correspondence or relationship of

Phi, 5**0.5 and Pi/5 . Why or how can this be derived?

36-54 triangle hypotenuse = 4

side opposite 36 = 2 Phi = 2.35114100 = 2*1.17557050

side opposite 54 = = 3.236067977

ref: https://www.mcs.surrey.ac.uk/Personal/r.Knott/Fibonacci/simpleTrig.html#xactrig

exact trig values

ref: https://www.mcs.surrey.ac.uk/Personal/r.Knott/Fibonacci/phi2DGeomTrig.html#trigmore

I have observed the phenomenon when I calculated the dodecahedral and

icosahedral angles. Consider a pentagonal jello mold with each if

the five sides a cylinder centered at the center of the pentagon base

with unit vectors going from the center of the pentagon and scribing

an arc to the vertical. At some angle the unit vector points are

unity apart; this defines the dodecahedral symmetry. The sides of

the base are then 2*sin(Pi/5) = 1.176 . If the five unit vectors are

raised by an angle alphathen the sides of the elivated pentagon are

2*cos(alpha)*sin(Pi/5) = 1 .

dodecahedral, icosahedral symmetry

In calculating the unit vectors to the vertices I observed

Phi and/or the square root of 5 cropping up.

cos(pi - dihedral) = 1/5**0.5

We have levels of complexity

Pi/2 2**0.5, Pythagoras, square root

Pi/6 3**0.5

Pi/5 Phi, golden, quadratic equation root

Pi/7 cubic equation root

...

My discussion of possible and impossible things is at

https://www.issi1.com/corwin/possible.txt

ref: https://www.issi1.com/corwin/platonic.txt

https://www.issi1.com/corwin/unit_v.txt

https://www.issi1.com/corwin/vertices.txt

ref: www.vb-helper.com/tutorial_platonic_solids.html

cc:golden numerology

cosmos2000@iquebec.com

www.666myth.co.nr/

cosmos2000@iquebec.com?subject=666 Bivalent Myth

www.chez.com/cosmos2000/Numbers/IsomorphousTriplets.html