PHI
Phi has a value equal to the
(square root of 5) + 1
2
The decimal equivalent = 1.61803 39887 49894 84821...
Phi may also be seen as the length of the hypotenuse in the right triangle having the other two sides equal to 1, and the square root of Phi.
Using a
^{2}
+ b
^{2}
= c
^{2}
, Phi angle, I would suggest, could be seen as,
51 degrees 49'38.23"
From the trig tables...
angle
sin
cos
tan
ctg
sec
csc
51 degrees 49'
.61818
.78601
.78646
1.27153
1.27225
1.61765
51 degrees 50'
.61795
.78622
.78598
1.27250
1.27191
1.61825
Phi
.61803
.78615
.78615
1.27202
1.27202
1.61803
Phi -1 ***
Phi
^{-2}
Phi
^{-2}
Phi
^{1/2}
Phi
^{1/2}
Phi
this Phi angle of 51 degrees 9'38.23"or 51.82729 degrees
is the only angle in trig chart where
cos = tan
ctg x sin = cos
cos
^{2}
= sin
cos x ctg = 1
ctg
^{2}
= Phi
SOME PHI RELATED ANGLES
ANGLE
sin
cos
tan
ctg
tan+ctg
ctg/tan
20 degrees 54' 18.585" *
Phi
^{-2}
Phi
^{2}
3
Phi
^{4}
31 degrees 43' 02.908" **
cos(ctg)
Phi(sin)
Phi -1 ***
Phi
root 5
Phi
^{2}
58 degrees 16' 57.092"
Phi(cos)
ctg(sin)
Phi
Phi -1 ***
root 5
Phi
^{-2}
69 degrees 05' 41.415"
Phi
^{2}
Phi
^{-2}
3
Phi
^{4}
* ½ angle from dodecahedron (10 zone diamond)
** ½ angle from icosahedron (6 zone diamond)
Diamonds are made by taking an edge and drawing radii from the end points. This gives you ½ of the diamond.
*** noting that Phi - 1 = 1/Phi = Phi
^{-1}
Phi exponent
decimal value
-3
.23606 79775
-5/2
.30028 30967
-2
.38196 60113
-3/2
.48586 82568
-1
.61803 39887
-1/2
.78615 13535
1/2
1.27201 96888
1
1.61803 39887
3/2
2.05817 10908
2
2.61803 39887
5/2
3.33019 21120
3
4.23606 79775
4
6.85410 19662
Phi angles are seen in the basic structures of a pentagon and in the penrose tiles. Phi appears in all 5 fold polyhedra...
dodechedron, icosahedron, triacontahedron, enneacontahedron, truncated cuboctahedron, great rhombicuboctahedron, truncated icosadodecahedron, great rhombicosadodecahedron, pentagonal hexacontahedron among others.
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