Golden Ruler
Golden Ruler

God Preserves Kepler’s Great Treasure of Gold!

"It is impossible for a cube to be the sum of two [positive integer] cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain." 

 Pierre de Fermat -- conjectured in 1637 and proved in 1994 by Andrew Wiles

  

 Proof of Fermat's “last theorem” purifies Kepler’s “measure of gold” so that the special place in God’s Mind for the 3-4-5 Pythagorean “parent of all triples” might shine forth delightfully in conjunction with Kepler’s other “great treasure” — the Golden Ratio! 

  Inter-relationship of the 3-4-5 Pythagorean Triple and the Golden Ratio:

 

Consider the following proof from H. E. Huntley's The Divine Proportion, Dover, 1970

PQ / BP = φ 

 

Let ABC be such a triangle with BC = 3, AC = 4 and AB = 5.

Let O be the foot of the angle bisector at B.

Draw a circle with center O and radius CO.

Extend BO to meet the circle at Q

Let P be the other point of intersection of BO with the circle.

Then PQ / BP = φ.

AO = 5/2 and CO = 3/2.

Thus the circle's radius r is 3/2.

By the Power of a Point Theorem,

BP·BQ = BC2.

In other words, (BO - 3/2)·(BO + 3/2) = 32.

From which, BO = 35/2. We thus find BP = 3(5 - 1)/2.

And finally,

PQ / BP = 2·r / [3(√5 - 1)/2]

               = 2 / (√5 - 1)

               = 2 · (√5 + 1) / 4

               = (√5 + 1) / 2 = φ

 

Thus:  PQ / BP = (√5 + 1) / 2 = φ

           The Tree of Primitive Pythagorean Triples                         

•••                                                                        •••                                                                      ••• 

91

60

109

187

84

205

117

44

125

 

299

180

349

459

220

509

165

52

173

 

209

120

241

273

136

305

63

16

65

105

88

137

297

304

425

207

224

305

 

377

336

505

697

696

985

319

360

481

 

275

252

373

403

396

565

133

156

205

9

40

41

105

208

233

95

168

193

 

57

176

185

217

456

505

175

288

337

 

51

140

149

115

252

277

85

132

157

     

Generation 3

             

Generation 3

             

Generation 3

     
      Matrix

 

                 

Matrix

                 

 

Matrix      
                                                         

7

24

25

55

48

73

45

28

53

 

39

80

89

119

120

169

77

36

85

 

33

56

65

65

72

97

65

72

97

     

Generation 2

             

Generation 2

             

Generation 2

     
     

Matrix

 

                 

Matrix

                 

 

Matrix

     
                                                         
     

5

12

13

             

21

20

29

             

15

8

17

     
     

Generation 1

             

Generation 1

             

Generation 1

     
                   

1

-2

2

1

2

2

-1

2

2

                   
                   

2

-1

2

2

1

2

-2

1

2

                   
                   

2

-2

3

2

2

3

-2

2

3

                   
                         

Matrices

                         
                         

3

4

5

                         
                       

Original Parents

                       

 

In mathematics, a Pythagorean triple is a set of three positive integers a, b, and c having the property that they can be respectively the two legs and the hypotenuse of a right triangle, thus satisfying the equation a2 + b2 = c2;  the triple is said to be primitive if and only if a, b, and c share no common divisor. The set of all primitive Pythagorean triples has the structure of a rooted tree, specifically a ternary tree, in a natural way. 

Each child is itself the parent of 3 more children, and so on. If one begins with primitive triple [3, 4, 5], all primitive triples will eventually be produced by application of [above] matrices. The result can be graphically represented as an infinite ternary tree.  

                                                                                                                                                                                                                              Wikipedia  

 

"Wisdom is proved right by her children"    Luke  7:35

 

The numbers  5 12, and  60  are the 'mitochondrial DNA organelles'

which trace the ancestry of all Pythagorean triples back to a first parent —

     The   3-4-5   Triangle!

 

One side of every Pythagorean triple is divisible by  5.

The product of the two non-hypotenuse legs is always divisible by  12.

The largest number that always divides the product of all three sides is  60.

 

The age our universe can be determined utilizing the numbers:

 

5- 12- 60,

 the Golden Ratio  Ø,

  1 Plank time = 5.391x10-44 secs,

and the literal truth of Isaiah 40:12.

 

5 x 10 12) 5(tp) x Plank time factor (sec/tp)  ÷  Julian year (sec/yr)    Age of Universe(yrs)   

 5 Ø (10)60 x 5.39106(32)x10 -44  ÷ (60x60x24x365.25) =  13.82065(85) x 109

 

Conclusion:

  God's Hand precisely   measured off   our Space-time heaven!

 

 

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