Kepler Triangle

KEPLER TRIANGLE: Plato and The Most Beautiful Triangle!

Based on Phi and its Golden Root Triangle 1.272… 
Construction of the Kepler Triangle.

This Article/Newsletter is based on 3 parts:
Part 1 of 3: Investigation of 5 Unique Triangles, and
Part 2 of 3: The Ultimate Triangle: Based on Phi and the Golden Root 1.272…
Part 3 of 3: Construction of the Kepler Triangle from the Golden Rectangle.

PART 1 of 3: 
INVESTIGATION OF 5 UNIQUE TRIANGLES.

This lesson is asking the Question: “Do You Know Your Triangles!” And determining which one has supreme or divine qualities above the rest.
This was asked 2,500 years ago by the Pythagorean community. 200 years later, Pythagoras’ main student was Plato, circa 370BC, and he was obsessed by what he considered to be “The Most Beautiful Triangle”. In his classic book, The Timaeus, Plato is in dialogue: “One, Two, Three, but where is the Fourth, my dear Timaeus?”

Triangle 1: The Equilateral Triangle having 60 degree angles 

Triangle 2: The Root 3 Triangle having the 30 degrees and 60 degree angles.

Triangle 3: The Root 2 Triangle having 45 degree angles.

Triangle 4: The Pythagorean 3-4-5 Triangle having 53.13 degree angles.
Triangle 5: The 10 Right-Angled Triangles Within The Pentagon having 54 degree angles.

PART 2 of 3:

Triangle 6: The Ultimate Triangle: The Phi Right-Angled Triangle aka Kepler’s Triangle based on the Golden Root 1.272…. being the critical height of the Great Cheop’s Pyramid is known as the Square Root of Phi because 1.272… x 1.272… = 1.618… or Phi. It has a base angle of 51 degrees and 51 minutes but when expressed as a decimal it is 51.84 degrees. This is the key to literally Squaring The Circle and to determining the True Value of Pi which is 4/Root Phi = 4 divided by 1.272… = 3.144605511029693144…

PART 3 of 3: 

CONSTRUCTION OF THE KEPLER TRIANGLE FROM THE GOLDEN RECTANGLE.
We can geometrically calculate the Golden Root with 2 arcs. The 1^st Arc, having the length of the diagonal of half the unit square, generates the Golden Rectangle. The 2^nd Arc is the length of Phi 1.618… which arcs down to hit an opposing side at the value of 1.272… completing the magical process of isolating golden harmonic proportions that sing to the Soul.

Plato’s Triangle, that incorporated both the Golden Ratio and the Pythagorean Triangle, inspired his now famous quote:

“Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel”. 

Jain 108