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.
Numerical Kin at the Great Pyramid & Piano Keys. The numerical kin are numbers 8 and 5. Animating numbers is an ancient concept. Today, many find animating numbers as archaic– in a negative sense. I almost hate to think about how the ancients would think of our “civilization.” Here is the ancient view of how 8 and 5 are numerical kin:
Both 5 and 8 are in the Fibonacci numbers. They are actually “kissin’ kouzins”. They are adjacent numbers in the series: 1,1,2,3,5,8,13,21,34.. etc. Each number, after 1,1, is the sum of the previous two numbers. From large to small, nature uses this series. See Fascinating Fibonaccis: Mystery and Magic in Numbers. It is written by Trudi Hammel Garland.
Ratio of the height of the Great Pyramid to the base is 5 to 8, respectively. The height, in shorter Egyptian cubits (1.71818…ft) is 275. The length of any side of the square base is 440 . See John Michell’s The New View Over Atlantis.
The Great Cubit of 55 cubits also measures the Great Pyramid: 440/275 = 8/5.
5 to 8 as Numerical Kin on the Piano Keyboard
When you play a keyboard, you activate the power of the Great Pyramid. Here’s how:
Its 8 to 5 ratio is found in the ratio of white keys to black in the octave.
As “5” represents the height of the Great Pyramid, there are 5 elevated black keys within the c to c’ octave.
As 8 is the number of the base; the 8 white keys are the foundation from which the 5 black rise.
The side of a square base of the Great pyramid measures 440 cubits. We, in the U.S. and England, tune to “A” 440. It is one of the white keys.
Please keep checking the blogs. Very soon I will reveal one of the fundamental forgotten principles of the Great Pyramid. It has been has been long buried.
Finally, I have just finished my “labor of love” as the pianist on the isle of Boca Grande at the Gasparilla Inn. I’ll be available for a while as a piano teacher in Sarasota.