-
Golden Ratio, FibonacciNumbers, and the
Golden Spiral
by
Troy Henderson
University of Mobile
-
Creation Reveals GodGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
The heavens are telling of the glory of God; and their expanseis declaring the work of His hands. Day to day pours forthspeech, and night to night reveals knowledge. There is nospeech, nor are there words; Their voice is not heard. Theirline has gone out through all the earth, and their utterances tothe end of the world.
Psalm 19:1- 4a (NAS)
-
Golden RectangleGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden RectangleGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden RectangleGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
1x 1
1 1
x
-
Golden RectangleGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
1x 1
1 1
x
x1=
1x 1
-
Golden RatioGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
x1=
1x 1
x(x 1) = 1x2 x = 1
x2 x 1 = 0
x =1 +
5
2,1
5
2
=x1= x =
1 +
52
1.618
-
Golden RatioGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
x1=
1x 1
x(x 1) = 1x2 x = 1
x2 x 1 = 0
x =1 +
5
2,1
5
2
=x1= x =
1 +
52
1.618
-
Mona LisaGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Mona LisaGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
ParthenonGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
ParthenonGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
ParthenonGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Great Pyramid of GizaGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Great Pyramid of GizaGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Great Pyramid of GizaGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Great Pyramid of GizaGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Great Pyramid of GizaGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
a = 378
b = 481
-
Great Pyramid of GizaGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
a = 378
b = 481c 611.75
-
Great Pyramid of GizaGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
a = 378
b = 481c 611.75
ca=
611.75378
1.618
-
Human ArmGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Human ArmGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
67
8
111
8
-
Human ArmGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
67
8
111
8
111
8
67
8
=
89
55 1.618
-
Earth / Moon RadiiGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Radius of Earth
a = 3,959 miles
Radius of Earth + Moonb = 5,039 miles
a2 + b2 = c2
c = 6,408 miles
ca=
6,4083,959
1.618
a
b c
-
Earth / Moon RadiiGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Radius of Earth
a = 3,959 miles
Radius of Earth + Moonb = 5,039 miles
a2 + b2 = c2
c = 6,408 miles
ca=
6,4083,959
1.618
a
a
b c
-
Earth / Moon RadiiGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Radius of Earth
a = 3,959 miles
Radius of Earth + Moonb = 5,039 miles
a2 + b2 = c2
c = 6,408 miles
ca=
6,4083,959
1.618
a
b
a
b c
-
Earth / Moon RadiiGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Radius of Earth
a = 3,959 miles
Radius of Earth + Moonb = 5,039 miles
a2 + b2 = c2
c = 6,408 miles
ca=
6,4083,959
1.618
a
b c
a
b c
-
Earth / Moon RadiiGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Radius of Earth
a = 3,959 miles
Radius of Earth + Moonb = 5,039 miles
a2 + b2 = c2
c = 6,408 miles
ca=
6,4083,959
1.618
a
b c
-
Relative Mean Orbital DistancesGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Planet Mean distance (million km) Relative mean (Mercury=1)Mercury 57.91 1.00000Venus 108.21 1.86859Earth 149.60 1.38250Mars 227.92 1.52353Ceres 413.79 1.81552Jupiter 778.57 1.88154Saturn 1,433.53 1.84123Uranus 2,872.46 2.00377Neptune 4,495.06 1.56488Pluto 5,869.66 1.30580
Average 1,640.67 1.61874
-
Relative Mean Orbital DistancesGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Planet Mean distance (million km) Relative mean (Mercury=1)Mercury 57.91 1.00000Venus 108.21 1.86859Earth 149.60 1.38250Mars 227.92 1.52353Ceres 413.79 1.81552Jupiter 778.57 1.88154Saturn 1,433.53 1.84123Uranus 2,872.46 2.00377Neptune 4,495.06 1.56488Pluto 5,869.66 1.30580Average 1,640.67 1.61874
-
Fibonacci SquaresGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
1
-
Fibonacci SquaresGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
11
-
Fibonacci SquaresGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
112
-
Fibonacci SquaresGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
112
3
-
Fibonacci SquaresGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
112
3 5
-
Fibonacci SquaresGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
112
3 5
8
-
Fibonacci SquaresGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
112
3 5
813
-
Fibonacci SquaresGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
112
3 5
813
21
-
Fibonacci SquaresGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
112
3 5
813
21 34
-
Fibonacci SequenceGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Fn = {1,1,2,3,5,8,13,21,34,55,89,144,233, . . .}
Fn+1Fn
={
11,21,32,53,85,138,2113
,3421
,5534
,8955
,14489
,233144
, . . .}
limn
Fn+1Fn
= =1 +
5
2 1.618
-
Fibonacci SequenceGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Fn = {1,1,2,3,5,8,13,21,34,55,89,144,233, . . .}
Fn+1Fn
={
11,21,32,53,85,138,2113
,3421
,5534
,8955
,14489
,233144
, . . .}
limn
Fn+1Fn
= =1 +
5
2 1.618
-
Fibonacci SequenceGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Fn = {1,1,2,3,5,8,13,21,34,55,89,144,233, . . .}
Fn+1Fn
={
11,21,32,53,85,138,2113
,3421
,5534
,8955
,14489
,233144
, . . .}
limn
Fn+1Fn
= =1 +
5
2 1.618
-
Finger BonesGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
8
5
32
-
Finger BonesGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
8
5
32
-
Finger BonesGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
8
5
32
-
Human ArmGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
67
8
111
8
111
8
67
8
=
89
55
-
Noahs ArkGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Make for yourself an ark of gopher wood; you shall make theark with rooms, and shall cover it inside and out with pitch. Thisis how you shall make it: the length of the ark three hundredcubits, its breadth fifty cubits, and its height thirty cubits.
Genesis 6:14-15 (NAS)
5030
=53 1.66
-
Noahs ArkGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Make for yourself an ark of gopher wood; you shall make theark with rooms, and shall cover it inside and out with pitch. Thisis how you shall make it: the length of the ark three hundredcubits, its breadth fifty cubits, and its height thirty cubits.
Genesis 6:14-15 (NAS)
5030
=53 1.66
-
Ark of the CovenantGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
They shall construct an ark of acacia wood two and a half cubitslong, and one and a half cubits wide, and one and a half cubitshigh. You shall overlay it with pure gold, inside and out you shalloverlay it, and you shall make a gold molding around it. You shallcast four gold rings for it and fasten them on its four feet, and tworings shall be on one side of it and two rings on the other side ofit. You shall make poles of acacia wood and overlay them with gold.You shall put the poles into the rings on the sides of the ark, to carrythe ark with them. The poles shall remain in the rings of the ark; theyshall not be removed from it. You shall put into the ark the testimonywhich I shall give you.
Exodus 25:10-16 (NAS)
-
Ark of the CovenantGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Ark of the CovenantGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Ark of the CovenantGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
212
112
212112
=53
-
Bronze AlterGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
And you shall make the altar of acacia wood, five cubits longand five cubits wide; the altar shall be square, and its heightshall be three cubits. You shall make its horns on its fourcorners; its horns shall be of one piece with it, and you shalloverlay it with bronze.
Exodus 27:1-2 (NAS)
-
Flower PetalsGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Calla Lily
1 Petal
-
Flower PetalsGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Euphorbia
2 Petals
-
Flower PetalsGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Iris
3 Petals
-
Flower PetalsGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Columbine
5 Petals
-
Flower PetalsGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Bloodroot
8 Petals
-
Flower PetalsGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Black-eyed Susan
13 Petals
-
Flower PetalsGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
African Daisy
21 Petals
-
Flower PetalsGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Shasta Daisy
34 Petals
-
Flower PetalsGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
Michaelmas Daisy
55 Petals
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Golden SpiralGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Nautilus ShellGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Nautilus ShellGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Bighorn SheepGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Bighorn SheepGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
SunflowerGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
SunflowerGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
SunflowerGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
SunflowerGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
SunflowerGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
1 2 34
56
78910
11
1213
1415
161718192021
2223
2425
26
27
2829
3031
3233 34
34 Counterclockwise Spirals
-
SunflowerGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
SunflowerGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
SunflowerGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
1 2 3
4
5
6
7
8
91011
12
1314
15
16
17
1819
2021
21 Clockwise Spirals
-
Human EarGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Human EarGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Human EarGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Hurricane IsaacGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Andromeda GalaxyGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Andromeda GalaxyGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
-
Creation Reveals GodGolden Ratio, Fibonacci Numbers, and the Golden Spiral Troy Henderson
For since the creation of the world His invisible attributes, Hiseternal power and divine nature, have been clearly seen,being understood through what has been made, so that theyare without excuse.
Romans 1:20