Remarkable Foursome is a Mathematical Wonder. It presents quite a take on the Fibonacci number series. It develops into what is called the Golden Ratio. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. The Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. What sequence defines this series? Eventually the second of two consecutive numbers divided by the 1st = the Golden ratio: φ is the symbol for this ratio.
In mathematics, the Fibonacci numbers are the numbers in a specific integer sequence. Their defining feature is every number after the first two is the sum of the two preceding ones:}
At 1st glance the “foursome” number square and Fibonacci seem in-congruent. But the remarkable foursome fuses the numbers on the 4 x 4 square as follows: Examine it numbers vertically left to right:
First row: 4 + 9 = 13. Then 5 + 16 = 21.
Second row: 14 + 7 = 21. Then 11 + 2 = 13.
Third row 15 + 6 =21. Then 10 + 3 = 13.
Fourth row: 1 + 12 = 13. Then 7 + 13 = 21.
All total are either 13 or 21. Of course, 13 = 21 = 34. This is just the tip of the iceberg. Fibonacci numbers on the square go further. Again look at the numbers vertically. Consider the four vertical rows again. This time add every other number as follows:
Row one: 4 + 5 = 9. Then 9 + 16 = 25.
Row two: 14 + 11 = 25. Then 7 + 2 = 9.
Row three: 15 + 10 = 25. Then 6 + 3 = 9.
Row four. 1 + 8 = 9. Then 12 + 13 = 25.
I know what my reader is thinking: 25 and 9 are not in the Fibonacci series. But, yes they are. Three and five are both Fibonacci numbers:
3² = 9.
5² = 25.
Mathematician Mark Barr proposed using the first letter in the name of Greek sculptor Phidias, phi, to symbolize the golden ratio. Usually, the lowercase form (φ or φ) is used.
Michael Maestlin, first to publish a decimal approximation of the golden ratio, in 1597.
The golden ratio has been claimed to have held a special fascination for at least 2,400 years, although without reliable evidence. According to Mario Livio.
Number Square of the Remarkable Foursome
Number squares were once the backbone of a lost civilization. This time marked a Golden Age of peace and plenty. Tradition associates the 4 x 4 number square with Jupiter. Each planet had its own set of numbers. Jupiter was the leader of the Olympian Gods; Jupiter means “Jovial King” and/or “Father of Thunder.” Jupiter is associated with luck and good fortune. It is the mentor/Guru /teacher of gods. More on internal link below.
Phi Tie is Like Tied Musical Notes Crossing Many Measures. Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets.
There was once a Golden Age. It was maintained by a mathematical code. The code has been hidden from most for millennium. On appearance, it is strikingly simple. On the featured number square, we have the following:
Any straight row of 3 numbers totals 15.
Any opposite two numbers totals 10.
The 8 perimeter numbers total 40.
The central number is 5.
This featured number square shows the traditional arrangement of the numbers. There are other solutions. The only rules used in diagramming the number square is that; (1) Even numbers must occupy the four corners. (2) Odd numbers must be placed in between the even numbers. (3) The mathematical totals must be the same as those given above.
So Where is the Phi Tie?
For this blog I will just show the obvious one. It is taken as a straight read across the bottom three numbers of the featured picture, from right to left. This code goes back to antiquity. Remember, Hebrew and many ancient languages were read from right to left. These civilizations would prefer the featured picture arrangement of the numbers.That exactly duplicates the first three numbers given as the quotient in the phi formula above: .6180… In antiquity zero was not considered a number on its own. It was a synthetic number. As just mentioned: Any opposite two numbers totals ten. Primary numbers from 1 to 9 were given their own special place of honor on the square.
For those wishing to know about the hidden codes that I discovered and the story that goes with it, go to DSOworks.com and type in “phi”.