  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!