The Golden Ratio

The Fibonacci Sequence

Mathematician Leonardo Pisano Bogollo, better known by his nickname Fibonacci, discovered a number sequence whilst contemplating the rate of increase of a population of rabbits. Despite being known in India hundreds of years beforehand, the sequence came to bear his name, and begins as follows:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ...

And so on into infinity, with each new number being the sum of the two previous numbers. The next number in the series is therefore 89 + 144 = 233, and so forth. In more recent times, both mathematicians and non-mathematicians alike have become increasingly fascinated with the Fibonacci Sequence. The source of this fascination is the golden ratio, which is inherent in our universe in everything including nature, astronomy, biology, art and architecture.

1999 postage stamp from Dominica (the nature island) commemorating Leonardo da Pisa, known as Fibonacci (c. 1170 – c. 1250) whose number sequence defines nature.

Table 1. Phi approximations obtained by dividing any Fibonacci number by it's predecessor. The greater the numbers, the more accurate the approximation.

This golden ratio, also known as phi and represented by the Greek symbol φ, is an irrational number precisely (1 +√5) / 2, or:

1.61803398874989484820458683...

but can be approximated by dividing any number in the Fibonacci Sequence by it's predecessor. The larger the number, the more precise the approximation. The increasingly accurate approximations can be observed in Table 1 at left. Geographically, a very good approximation of phi is obtained by measuring the ratio of the distance from Kaaba at Mecca to the South Pole as a ratio of the distance between the North and South Poles. This gives an approximation with an accuracy similar to that provided by 34 / 21 from Table 1.

There is a plethora of information on the golden ratio and where it can be observed, including these two short introductory videos:

Given the ubiquitous nature of the golden ratio in all of creation, it behooves one to seek it in the words of the Creator, the holy Bible.

The Knowledge of Good and Evil

A cousin of the Fibonacci numbers is a sequence obtained by a similar technique, but beginning with the numbers 2, 1, 3, 4, 7... etc. Numbers in this series are called Lucas numbers (named after another mathematician) which also display interesting properties.

Using the same method outlined above, Lucas numbers can also be used to approximate phi. It's true that the golden ratio can be approximated using many more number sequences, but the most accurate come from the Fibonacci and Lucas number series. For example, the number 5778 approximates phi to within 0.000000175% (5778 / 3571 = 1.1.61803416...).

5778 x 0.61803398875 = 3571.000387 * to ensure sufficient accuracy, use a scientific calculator
5778 x 1.61803398875 = 9349.000387

"For God doth know that in the day ye eat thereof, then your eyes shall be opened, and ye shall be as gods, knowing good and evil." Gen 3:5

"knowing good and evil" (ידעי טוב ורע) has a gematria value of 387
the word "good" appears in 655 Bible verses (KJV)
the word "evil" appears in 569 Bible verses (KJV)
655 - 569 = 86
the gematria value of Elohim (אלהים), translated in the Bible as "God" = 86

Adam and Eve, ceiling fresco from the Stanza della Segnatura. (detail) by Raphael. The day Adam and Eve ate therof, and came to know good and evil.

The Seven Angels of the Apocalypse by Giusto di Giovanni de' Menabuoi (1360-70)

the term "good and evil" appears in 7 Bible verses
the word "seven" appears in 391 Bible verses (KJV)

"Who hath ascended up into heaven, or descended? who hath gathered the wind in his fists? who hath bound the waters in a garment? who hath established all the ends of the earth? what is his name, and what is his son's name, if thou canst tell?" Pro 30:4

"what is his name" (מה שמו) has a gematria value of 391

In Hebrew, Jesus' name can be spelled either יהושע (Yehoshua, which means "Jehovah is salvation") or ישועה (Yeshua, which means "salvation").

Yehoshua (יהושע) has a gematria value of 391 - Strong's H3091
Yeshua (ישועה) has a gematria value of 391 - Strong's H3444

"I am come in my Father's name, and ye receive me not: if another shall come in his own name, him ye will receive." Joh 5:43

"in my Father's name" (εν τωι ονοματι του πατρος) has an isopsephia value of 3227
3227 = 31 x 233 (the 12th Fibonacci number)

When God drove out the man from the Garden of Eden, he placed Cherubims and a flaming sword to keep the way of the tree of life.

"tree of life" (עץ החיים) has a gematria value of 233
the words "tree" and "life" appear together in 13 Bible verses (KJV)
13 and 31 are emirp primes

When the prophet Malachi asked "Have we not all one father? hath not one God created us?", "one" (אחד) has a gematria value of 13 and "God" (אל) has a gematria value of 31. (Mal 2:10)

The Fall of Man and the Expulsion from the Garden of Eden by Michaelangelo. Fresco, ceiling. Sistine Chapel, Vatican. 1508 - 1512

Jeremiah's Lament

Israel Encamped Roundabout the Tabernacle in the Wilderness of Sinai John W. Kelchner

In the Bible, the Tabernacle was the portable dwelling place for the divine presence from the time of the Exodus from Egypt until the conquering of the land of Canaan. Built according to specifications revealed by God to Moses at Mount Sinai, it was transported by the Israelites on their journey through the wilderness and into the Promised Land. It was replaced as the dwelling place of God by the First Temple in Jerusalem. Within the Tabernacle was a shrine (the Holy of Holies) and the Ark of the Covenant. Inside the Tabernacle, the Lord talked with Moses, and the children of Israel encamped round about.

In the book of Numbers, God appointed the Levites to perform guard duties concerning the Tabernacle and it's contents, to protect the Israelites from the consequences of encroachment. The golden ratio of 5778 is used to pinpoint the verse relating to this sacred duty:

5778 x 0.61803398875 = 3571

"And they shall be joined unto thee, and keep the charge of the tabernacle of the congregation, for all the service of the tabernacle: and a stranger shall not come nigh unto you." Num 18:4
ונלוו עליך ושמרו את משמרת אהל מועד לכל עבדת האהל וזר לא יקרב אליכם
101 312 31 213 41 476 80 120 36 980 401 552 130 98 = 3571

5778 = φ¹⁸
5778 is the 18th Lucas number
in Hebrew, the word tabernacle (אהל) has a gematria value of 36 = 2 x 18
5778 is the 54th hexagonal number = 3 x 18

There are a total of 31,102 verses in the Bible. The golden ratio is applied to determine the "golden" verse, and it also relates to the Tabernacle. The prophet Jeremiah lived during one of the most devastating periods in Jewish history; the destruction of Jerusalem and Solomon's Temple which had replaced the Tabernacle as God's dwelling place. This destruction was followed by the beginning of the Babylonian exile.

Jeremiah became a key figure in laying the foundations for the restoration of Jerusalem and the Second Temple. He firmly believed that once the period of punishment was over, the Israelites would return to their land and rebuild Jerusalem.

31,102 x 0.61803398875 = 19,222

In verse number 19,222 in the Bible, Jeremiah laments the parlous state of the Jewish nation:

Jeremiah Lamenting the Destruction of Jerusalem by Rembrandt (1630).

"My tabernacle is spoiled, and all my cords are broken: my children are gone forth of me, and they are not: there is none to stretch forth my tent any more, and to set up my curtains." Jer 10:20 (verse number 19,022 in the Bible)
אהלי שדד וכל מיתרי נתקו בני יצאני ואינם אין נטה עוד אהלי ומקים יריעותי

The words "tabernacle" and "tent" are alternative translations of the same Hebrew word (אהל) Strong's H168.

in total, Strong's H168 is translated 345 times in the KJV
the gematria value of משה ("Moses"), who tabernacled with God, is 345
the word "tabernacle" appears in 297 Bible verses (KJV)
the word "tent" appears in 88 Bible verses (KJV)
88/297 = 0.296296…
296 is the gematria of הארץ ("the earth") from Gen 1:1
296 x 3 = 888 = the isopsephia value of Ιησους ("Jesus")

                                    "O earth, earth, earth, hear the word of the LORD." Jer 22:29

The gematria of Jer 22:29 is 1525, a value shared with two affirmative verses from the book of Psalms:

"The LORD hath prepared his throne in the heavens; and his kingdom ruleth over all." Psa 103:19
                                            יהוה בשמים הכין כסאו ומלכותו בכל משלה
                                                375    52    508 87    85    392       26   =   1525

                     "My help cometh from the LORD, which made heaven and earth." Psa 121:2
                                                  עזרי מעם יהוה עשה שמים וארץ
                                                        297    390      375     26      150    287   =   1525

Apocalypse 7, the 144,000 elect by Martinus, from the Beatus d'Osma manuscript (11th century). Archives de la Cathédrale

The prophet Ezekiel also speaks of the Tabernacle:

"My tabernacle also shall be with them: yea, I will be their God, and they shall be my people." Eze 37:27
והיה משכני עליהם והייתי להם לאלהים והמה יהיו לי לעם
140 40 31 56 116 75 441 155 420 26 = 1500

Eze 37:27 gives 37 x 27 = 999
the gematria of "the meek will inherit the earth" (וענוים יירשו ארץ) is 999 (Psa 37:11)
the length, breadth and height of the New Jerusalem = 1500 miles ( Rev 21:16)

In the book of Revelation, the New Jerusalem is written as a cube 12,000 furlongs, and the number of the elect with his father's name written in their foreheads is 144,000.

the 12th Fibonacci number is 144

The book of Revelation, like most other prophetic books in the Bible, speaks about both the present and future times. The double Fibonacci number 5168 (from the 18th row in Table 2) provides insights:

5168 x 0.61803398875 = 3194

"Write the things which thou hast seen, and the things which are, and the things which shall be hereafter;" Rev 1:19
γραψον α ειδες και α εισι και α μελλει γινεσθαι μετα ταυτα
924 1 224 31 1 225 31 1 120 288 346 1002 = 3194

to be more precise: 5168 x 0.61803398875 = 3194 - 0.000346

5168 x 1.61803398875 = 8362 + 0.000346

hereafter (μετα) has an isopsephia value of 346

Pascal's Triangle

The first ten rows of Pascas's triangle. The sides are all "1's". All the other numbers are the sum of the two numbers above.

French postage stamp commemorating Blaise Pascal.

Another interesting number pattern is Pascal's Triangle, named after Blaise Pascal, a famous French mathematician, physicist and religious philosopher. Pascal's triangle starts with "1" at the top, with numbers placed below in a triangular pattern. Each number is the sum of the two numbers above (except for the sides, which are all "1"). The numbers represented in Pascal's triangle are termed binomial coefficients. Triangular numbers are a subset of binomial coefficients. All triangular numbers are binomial coefficients, but not all binomial coefficients are triangular numbers.

In the illustration, numbers that are not triangular are colored red. You can see patterns beginning to emerge (triangles within triangles. The notation for binomial numbers is based on a simple address matrix, according to row and column numbers (both starting with zero). For example, the number 20 is located at row 6, column 3. It's notation is therefore written as C(6,3) or ₆C₃ (C is for coefficient), whilst number 28 is written as C(8,2).

expressed as a binomial coefficient, 5778 = C(108,2)
108 x 2 = 216

By drawing a series of diagonal lines, the Fibonacci number series can be calculated from Pascal's triangle. The numbers dissected by the lines add to the relevant Fibonacci number. The first diagonal dissects 1, the second diagonal also 1, the third diagonal 1 + 1 = 2 and so forth through the Fibonacci series. In another interesting pattern, the triangular numbers form a diagonal starting from Row 2 (the numbers colored red are the triangular number series).

By arranging the binomial coefficients in a simple matrix explained in this video, the distribution of prime numbers form a spiral pattern (called a Ulam spiral), as seen in the diagram using numbers up to 256². The spiral arms are formed by triangular numbers, and the "noise" by non triangular numbers. The accompanying diagram uses the triangular numbers alone. The mathematical elegance of the triangular numbers is pertinent considering the triangular structure of the Bible's very first verse, and the reference to other triangular numbers in Scripture (see examples in John 21:11, Acts 27:37, Rev 13:18). Most of these numbers are encompassed in the 5778 triangle.

Ulam Prime Spiral of binomial coefficients for numbers up to 256²

Ulam Prime Spiral of triangular numbers for numbers up to 256²

Golden Spirals

Fibonacci squares of side length 1, 1, 2, 3, 5, 8,13 ...

A grid of abutted squares can be constructed based on the numbers from the Fibonacci sequence (1², 1², 2², 3², 5²..., you know the drill). By joining the dots, a spiral is obtained which is frequently observed in nature, as it is in art.

Take a tour through the slide show gallery to appreciate the beauty that surrounds us every day, courtesy of the Creator.

Mona Lisa by Leonardo da Vinci (1503 - 1517) with superimposed Golden Spiral.

Hurricane Floyd near peak intensity on September 14 at 2030 UTC. This image was produced from data from NOAA-14, provided by NOAA.

One of the most spectacular natural examples of the golden spiral is in the patterns described by hurricanes (from urikan, or "big wind" as the Carib Indians knew them). The 1999 Atlantic hurricane season had five Category 4 hurricanes - a record number for a single season. Scientists believe that although global warming may not be increasing the number of hurricanes per season, it may be increasing their strength. The third (and strongest) of the five major hurricanes in 1999 was named Floyd, a spectacular example of the golden spiral (photo at left and also featured in the slide show gallery).

Floyd, which caused the third largest evacuation in US history, had it's beginning on September 7th, when it was designated as Tropical Depression Eight. It was upgraded to Tropical Storm Floyd on September 8th before reaching hurricane status on September 10th.

In the Jewish tradition, Adam and and Eve were created on the 1st day of Tishrei (Rosh Hashanah). The first day of creation was six days earlier on 25th Elul. On the second day, God divided the waters from the waters, vapour above, water below Gen 1:6-8), and on the third day, He created land and sea (Gen 1:9-10). Then on the fifth day, God created animal life, including the sea creatures (Gen 1:20-23).

2, 3 and 5 are all Fibonacci numbers
Floyd became a Tropical Depression on 26th Elul 5759 on the Hebrew calendar (5778 - 18 years after the second day of creation)
Floyd became a Tropical Storm on 27th Elul (5778 - 18 years after the third day of creation)
Floyd became a Hurricane on 29th Elul (5778 - 18 years after the fifth day of creation)
5778 = φ¹⁸

God moves in a mysterious way
His wonders to perform;
He plants His footsteps in the sea
And rides upon the storm.
William Cowper (1773)

On a much smaller scale, the cross section of the Nautilus shell is probably the most precise and elegant expression of the golden spiral in all of nature, as evidenced in the images below. The chambered Nautilus is an odd yet beautiful creature that's been swimming in the sea for half a billion years without evolving much in all that time.

Cross section of a Nautilus shell.

A real live Nautilus.

Sadly, this beautiful creature may be headed for extinction, according to marine biologists. They are being fished out, due to their increasing popularity in the worldwide jewelry trade. There is great demand for its delicate shell with near-perfect spirals when cut in cross-section. Download a video of the Nautilus here, before it's too late (5.2 Mb).

Thanks for the heavenly message brought by thee,
Child of the wandering sea.
From "The Chambered Nautilus" by Oliver Wendell Holmes (1809-1894)

The Golden Ratio (Phi) in Pi

Oftentimes the most significant gematria in the Bible is punctuated with repdigits. Consider the Bible's famous first verse, for example. In this verse, consisting of seven Hebrew words, repdigits are abundantly evident:

                        "In the beginning God created the heaven and the earth."
                                                 בראשית ברא אלהים את השמים ואת הארץ
                                                    296      407        395    401    86         203    913   =   2701

Remembering that Hebrew is read from right to left:

the gematria value of words 1 and 3 = 999
the gematria value of words 2, 4 and 5 = 999
the gematria value of words 3, 5 and 6 = 888
the gematria value of words 3, 5 and 7 = 777

All 3 digit repdigits are divisible by 37, as also is the total gematria of this verse (2701 = 37 x 73). This repdigit design is also evident 13 verses later, when God said "Let there be lights in the firmament of heaven" (יהי מארת ברקיע השמים).

the gematria value of "Let there be lights" (יהי מארת) is 666
the gematria value of "in the firmament of heaven" (ברקיע השמים) is 777

the total gematria value of Gen 1:14 = 3744 = 13 x 288 (holy sparks)
in Hebrew, the number six is spelled וו (vav) which appears in 13 Bible verses (KJV) Strongs H2053
in Greek the word six appears in 13 NT Bible verses (KJV)

Remarkably, in a search for the golden ratio in pi, the results were similarly punctuated with repdigits. The first occurrence of the 8-digit string 16180339 begins at position 19,445,230 (after the decimal point) in pi.

the next three digits following this string are 999
the totient (φ) of 19,445,230 = 6666912
19,445,230 = 7 x 2777890
φ(2777890) = 1111152
19,445,230 = 35 x 555578
φ(555578) = 277788

There are two occurrences of 16180339 in the first 100 million digits of pi. The second occurrence begins at position 21118044 after the decimal point. This number not only includes repdigit 111, but also the gematria value of the Shema (Deu 6:4). Shema (שמע) is the Hebrew word that begins the most important prayer in Judaism.

                                       "Hear, O Israel: The LORD our God is one LORD:"
                                                          שמע ישראל יהוה אלהינו יהוה אחד
                                                            13        26         102        26           541        410    =   1118

We know that:

1 / 1.618033988 = 0.618033988
1 / 0.1618033988 = 1.618033988

So it's ominous to note that the first occurrence of the 8-digit string 61803398 begins at position 62,078,146 in pi. The next four digits following this string are 5777, which year on the Hebrew calendar corresponds with 2016/2017 on the Gregorian calendar.

62 + 078 + 146 = 286
the Greek word for tabernacle (σκηνη) has an isopsephia value of 286 (Strong's G4633)

The odds of an eight digit string appearing in the first 100 million digits of pi are 63%, so it's interesting to note that 16180339 and 61803398 each occur twice within this number set. No further occurrence of either 8-digit string is found within the first 300 million digits of pi.

"And for that the dream was doubled unto Pharaoh twice; it is because the thing is established by God, and God will shortly bring it to pass." Gen 41:32

Pharaoh's Dream. Illustration by Owen Jones from "The History of Joseph and His Brethren" (Day & Son, 1869).

The repdigit coincidences continue:

the first occurrence of 111 in pi begins at position 153 after the decimal point (the number of fish in the net)
the first occurrence of 111 in phi begins at postion 441 (the gematria value of the Hebrew word for truth (אמת))
the word "truth" appears in 222 Bible verses (KJV)
after the first occurrence of 111 in phi, the next 3 digits are 777

Benjamin was the last born of Jacob's twelve sons, and the only founder of a tribe to be born in Israel. Benjamin was born on 11 Cheshvan, 2208 on the Hebrew calendar, and also died on 11th Cheshvan, 111 years later.

Benjamin (בנימין) H1144 has a gematria value of 162
in Chapter 7 of the Book of Revelation, the elect of the tribes of Israel number 144,000
144,000 / 162 = 888.888 888 ...

the first occurrence of 888888 in pi begins at position 222299

The last biblical mention of Benjamin is in Rev 7:8, where "of the tribe of Benjamin" (εκ φυλης βενιαμιν) has an isopsephia value of 1331.

1331 = 11³ and is written as 1000 in base 11

The only 9-string repdigits that appear within the first 2 billion digits of pi are 666666666 (5 times), 777777777 (4 times) and 888888888 (once). Given that the isopsephia value of Jesus (Ιησους) is 888, it is therefore interesting to note:

the square root of 666666666 = 25819.88896
the square root of 777777777 = 27888.66753

888888888 = 3 x 296 296 296
the gematria value of the earth (הארץ) from Gen 1:1 = 296

"O earth, earth, earth, hear the word of the LORD." Jer 22:29

the words "Jesus" and "earth" appear together in 6 Bible verses (KJV)
the only 10-string repdigit to appear within the first 2 billion digits of pi is 6666666666
in base 11 (which is a repdigit) 6666666666 = 2911183380
1118 = the gematria value of the Shema (Deu 6:4)
the string 6666666666 begins at position 386,980,412 after the decimal point in pi
386 + 980 + 412 = 1778

"Ask thee a sign of the LORD thy God; ask it either in the depth, or in the height above." Isa 7:11
שאל לך אות מעם יהוה אלהיך העמק שאלה או הגבה למעלה
175 15 7 336 215 66 26 150 407 50 331 = 1778

Isa 7:11 gives 7 x 11 = 77
77 = 4² + 5² + 6²

The first appearance of the four digit string 2368 (Jesus Christ) begins at position 7481 after the decimal point in pi.

2 x 3 x 6 x 8 = 288
after this first appearance of 2368 in pi, the following three digits are 288
7481 is the 947th prime number
947 is also prime

         "God judgeth the righteous, and God is angry with the wicked every day." Psa 7:11
                                            אלהים שופט צדיק ואל זעם בכל יום
                                                           56    52    117    37 204    395    86   =   947

Beginning at position 2368 after the decimal point in pi is the four digit string 3534.

"Only fear the LORD, and serve him in truth with all your heart: for consider how great things he hath done for you." 1 Sam 12:24
                              אך יראו את יהוה ועבדתם אתו באמת בכל לבבכם כי ראו את אשר הגדל עמכם
                               170    42 501 401 207 30    94         52      443     407    522         26    401    217   21   =   3534

I Sam 12:24 gives 12 x 24 = 288

The only 9-string repdigits that appear within the first 500 million digits of phi are 222222222 (twice), 333333333 (once) and 888888888 (once).

the string 888888888 begins at position 433,356,674 after the decimal point in phi
433 + 356 + 674 = 1463
the Greek word for prayer (προσευχή) has an isopsehia value of 1463 Strongs G4335
Strongs G4335 is translated 37 times in the Bible (KJV)
all 3 digit repdigits are divisible by 37

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