Fibonacci Numbers - Modified Fibonacci Series - Sacred Geometry

Modified Fibonacci Series


Modified Fibonacci Series

Golden ratio – 3d rendering

A “modified” series of Fibonacci Numbers can be easily had by using starting numbers other than 0 and 1.  For example, we can write a whole series of modified Fibonacci series by using as the first numbers, 1 and another integer.  This is shown in Table 1.  In fact, we can also use non-integer numbers (as in the so-called “crossing sequence” in Golden Mean Mathematics, where we used 1 and Ö5).  In terms of the ratio of adjacent numbers always approaching the Golden Mean (F or f), everything seems to work!

Table 1

        1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349…

        1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665, 1076, 1741, 2817, 4558, 7375, 11,933…

        1, 5, 6, 11, 17, 28, 45, 73, 118, 191, 309, 500, 809, 1309, 2118, 3427, 5545, 8972…

        1, 6, 7, 13, 20, 33, 53, 86, 139, 225, 364, 589, 953, 1542, 2495, 4037, 6532, 10,569…

        1, 7, 8, 15, 23, 38, 61, 99, 160, 259, 419, 678, 1097, 1775, 2872, 4647, 7519, 12,166…

A significant result is: Any series which uses the methods of adding the last two numbers to obtain the next in the series, will always have as a limit of the ratio of those last two numbers, F and f.  Obviously, if the numbers are squared, cubed and so forth — i.e. any of the numbers taken to the nth power — we will have as limits, Fn and fn.  More importantly, if there is a separation between the numbers (where the numbers between the ratio numbers total n), we obtain the equivalent limits of  Fn+1 and fn+1.  For example, dividing 2817 by 11,933 yields 0.2361, which in turn equals 0.618… to the 3rd (2+1) power — the 2 representing the separation of the two numbers, 4558 and 7375.

You may well ask: Is there no end to this madness?


For in continuing the alexandrian tradition of observing strange phenomena, we might note that the 12th number of each sequence (not counting the zero in the original Fibonacci Series and noted in bold in Table 1), i.e. 233, 322, 411, 500, 589, 678… all differ by 89 between adjacent sequences.  89 is, of course, the 10th number of the original Fibonacci series.  Meanwhile, the 17th numbers, 2584, 3571, 4558, 5545, 6532, 7519… all differ by 987, the 15th number of the original Fibonacci series.  Not surprisingly, the general rule is that the nth number of all of the different, modified Fibonacci series, all differ between the adjacent series by the (n-2)th number of the original Fibonacci series.

We can also match a Fibonacci series with its cumulative sequence.  Table 2 shows two examples: the traditional Fibonacci series and its cumulative totals on the second line, as well as the 1-3 sequences matched to its cumulative sequence.

Table 2

        1, 2, 3,  5,   8, 13, 21, 34,   55,   89, 144, 233, 377,   610,   987, 1397, 2584, 4181…

        1, 3, 6,11, 19, 32, 53, 87, 142, 231, 375, 608, 985, 1595, 2582, 3979, 6563, 10,744…

        1, 3, 4,   7, 11, 18, 29,   47,  76, 123, 199, 322,   521,  843, 1364, 2207, 3571, 5778…

        1, 4, 8, 15, 26, 44, 73, 120, 196, 319, 518, 840, 1361, 2204, 3568, 5775, 9346,15,124…


The cumulative sequences also approach F and f!  But you probably guessed that, right?

What you might not have guessed is the manner in which the nth number in the cumulative sequence is less than the n+2 number in the regular sequence by the second number in the regular sequence.  Really.

Returning to the “crossing” Fibonacci series defined in Golden Mean Mathematics by Fn + fn and Fn – fn, we note that the whole numbers of the opposing cross sequence of numbers are also exact square roots.  Comparing the two series…

         5, 20, 45, 125, 320, 845, 2205, 5780, 15125, 39605, 103680, 271445, 710645, 1860500

         9,16, 49, 121, 324, 841, 2209, 5776, 15129, 39601, 103684, 271441, 710,649, 1860496

We note that each term differs from the corresponding number in the other series by a factor of 4 (or 22), with the two series interweaving between themselves.  The two series can be said to be based on Ö5 = F + f and Ö9 = F – f + 2.  If we then compare any two adjacent numbers in the sequence, we see that the quotients of adjacent numbers approach fand F2, and the quotients of two non-adjacent numbers approach fn+1 and Fn+1 where n is the count of numbers in the series between the numbers being divided.  Amazing!

In summary, all of the Fibonacci numbers, taken in a host of different combinations and different sequences, all connect directly through the Golden Means, turning into and connecting with themselves like fractals in Chaos Theory.

Mathematics can truly be philosophical — as in Sacred GeometryTranscendental Numbers, and F-lo-sophia (to name just a few).

Accordingly, you can always return to Sacred Mathematics,

Or press on to Sacred GeometryMusicConnective Physics, and Creating Reality.

But whatever you do, don’t stop now!

Fibonacci Numbers

We begin our whirlwind tour of F Lo Sophia and Sacred Geometry by first stopping in Pisa, Italy, where in the year 1202 A.D. (or as currently written, C.E. for “Current Era”), a mathematician and merchant, Leonardo da Pisa wrote a book, Liber Abaci (The Book of Computation).  Born in 1179, Leo had traveled during the last years of the 12th century to Algiers with his father, who happened to be acting as consul for Pisan merchants.  From the Arabs the young Leonardo Bigollo discovered the Hindu system of numerals from 1 to 9, and from the Egyptians an additive series of profound dimensions.  Leo promptly shared his illumination with Europeans by writing his book and offering to the intelligentsia (the small minority who could read) an alternative to the reigning, clumsy system of Roman numerals and Greek letters.

Books on mathematics are not normally among the best sellers of any era.  Leo’s book, nevertheless, had the effect of convincing Europe to convert its unromantic, Romanized numeral system to the one known today as the Hindu-Arabic numeral system.  Leo also introduced to the Western World what has become known as the Fibonacci Series.  The term derives from the fact that Leonardo’s father was nicknamed Bonaccio (“man of good cheer”), and thus Leonardo was known in Latin as the son of Bonaccio, or “filius Bonaccio”.  This latter moniker has been contracted, for the benefit of non-Latin scholars, to “Fibonacci” (fib-oh-NAH-chee) — and the name we will use hereafter.

Clearly our society owes a great debt of gratitude to Fibonacci — as well as the Arab scholars who kept the knowledge alive, and the Egyptians for holding the mysteries intact.  If you question this statement (as you should question all such statements), try multiplying XCIV by LXXXIII.  Better yet, try your hand at long division using these same numbers (and in whichever ratio you prefer).  Or take the historical route and try to imagine how European commerce, banking and measurement (science) managed to progress from the first to the twelfth century using Roman Numerals!  Scary, isn’t it?  There’s a reason for that period of time to which historians have referred to as the Dark Ages.  Therefore, after these exercises, you might consider offering a heartfelt word of thanks to the Hindu mathematicians and their intermediaries, the Arab scholars who preserved the knowledge, and our Italian friend, Fibonacci.

History has decreed our Italian hero’s most famous mathematical contribution to be the series of numbers named after him.  The original series is constructed  from the numbers, 0 and then 1, and then adding the last two numbers in the series to obtain the next number.  For example, 0+1=1, 1+1=2, 1+2=3, 2+3=5, 3+5=8… (and so forth). [The three dots at the end, “…”, denotes the fact the sequence continues ad infinitum, and is a mathematical shorthand for “and so forth”].  The resulting Fibonacci sequence becomes:

0  1  1  2  3  5  8  13  21  34  55  89  144  233  377  610  987  1597  2584  4181

6765  10,946  17,711  28,657  46,368  75,025  121,393  196,418  317,811…

For the mathematician, the Fibonacci numbers can be calculated from:

F(n) = (2/Ö5) {- [-2/(1-Ö5)]n / [1 – Ö5] + [-2/(1+Ö5)]n/ [1 + Ö5]}

where the number 5 and Ö5 figure prominently — as well as the Golden Mean, F.

These Fibonacci numbers might be merely an Italian mathematical curiosity except for the fact Mother Nature has an apparently decided fondness for this strange sequence of Hindu-Arabic numbers!  The most notorious of the “natural” examples, and the one of which Fibonacci is credited in bringing from Egypt to Europe, is known as “The Rabbit Riddle”.  This puzzlement makes the initial assumptions of a pair of newborn rabbits (one male and one female), who take precisely one month to mature, after which they immediately mate (typical!).  The female then gestates for one month, gives birth to another pair like the first two, and mates every month thereafter.  Every newborn pair repeats this pattern of monthly maturing, mating, gestating, and breeding other identical pairs, all of whom continue the family tradition and do likewise.  Then, assuming that no pair dies or deviates from the pattern — e.g., none come out of the closet and announce they’re gay — how many pairs of rabbits will there be after any given number of months?

As it turns out, a count of the newborn, mature, and total rabbit pairs each month produces a pattern, which is nothing more than three versions of the Fibonacci Series (all the same numbers, but beginning on different months).  Thus at the outset, the total number of rabbit pairs is 1, and each succeeding month there are: 1, 2, 3, 5, 8, 13, 21… and so forth.  Isn’t it amazing what mathematics and/or rampant incest can accomplish!?

Curiously, this same pattern occurs in the case of spreading rumors in a crowd — an apparently “natural” process, judging by its popularity.  In this case, we assume each person passing on the rumor does so after a specified time of thinking about it (say half-a-minute), and then tells another person (who hasn’t already heard it) every half-minute thereafter.  When everyone else gets into the same spirit of uncontrolled gossip, the numbers of knowers, tellers, and hearers, follow the same Fibonacci sequence of numbers.

In other areas of nature, Fibonacci-inspired, growth patterns arise in honeybees, the branches of the sneezewort (Achillea ptarmica) plant (which has the appearance of a Jewish Minora run amuck), and any process which grows from within itself.  The number of flower petals for different types of plants, for example — such as those given in Table 1 (below) — may be Fibonacci inspired:

Table 1

  Number of Petals                                Flower(s)

2                    Enchanter’s nightshade

3                    Iris, Lilies, trillium

                   All edible fruits, some delphinium, larkspurs, buttercups, columbines, milkwort

8                    Other delphiniums, lesser celandine, some daisies, field senecio

13                 Globe flower, ragwort, “double” delphiniums, mayweed, corn marigold, chamomile

21                  Heleniums, asters, chicory, doronicum, some hawkbits, many wildflowers

34                 Common daisies, plantains, gaillardias, hawkbits, pyrethrums, hawkweeds

55                 Michaelmas daisies

89                 Michaelmas daisies

Humans have caught on to this fad-setting trend by composing the musical scale, where 1 piano has 1 keyboard with black keys arranged in groups of 2 and 3, consisting of 5 black keys (sharps and flats) and 8 white keys (whole tones) for a 13 note chromatic musical octave.  Musically, we have thus accounted for seven consecutive Fibonacci numbers.  And from the viewpoint of Nature, this form of music becomes harmonious with our physical, emotional, mental, and undoubtedly spiritual bodies.

Why is Nature so intrigued by the Fibonacci series?  Perhaps, because of the manner in which it originates, beginning with only two terms, zero and unity.  These two numbers may be considered to be the Unknowable and the manifest Monad.  Curiously, this quickly yields another Monad (representing the duality or male-female aspects of the creators or possibly the first holy offspring, as per a Vesica Pisces).

But the real intrigue begins with relating the Fibonacci Sequence to the Golden Mean and thereafter to Sacred Geometry.  In order to initiate this relationship, we divide each of the numbers in the sequence by the previous number, to yield the series:

¥ (infinity), 1, 2, 1.5, 1.6667, 1.6000, 1.6250, 1.6154, 1.6190, 1.6176…

Slowly but surely, each subsequent quotient approaches, ever closer, the number: 1.61803398875… — the mathematical ratio which has become known as the Golden Mean.  For our purposes, we will distinguish between two forms:

F  =  1.61803398875…       and          f =  0.61803398875…

Both of these numbers are considered by different authors as the Golden Mean (although some select one, some the other).  Fortunately, there is no real problem with this variation in opinion in that one can quickly discover that:

1 / F  =  1 / 1.61803398875…  =  0.61803398875…  =  f

This relationship is accurate to however many decimal places one cares to carry it.

Thus, we can conclude: Life is good.  Not to mention, something you can count on.

And for further enlightenment, we can consider those Modified Fibonacci Series which are initiated by numbers other than 0 and 1 — and yet achieve the same results (albeit even more interesting).

There is also, and excellent website for the more sophisticated mathematician who may want to know even more about Fibonacci Numbers.  For the moment, we won’t go there.

(6/6/05) Where we might venture to go is Pascal’s Triangle, where diagonals of this famous mathematical delight add to the Fibonacci Numbers. (Actually, Pascal’s Triangle looks a lot like the above mentioned rabbit’s family tree.)

But for everyone, with just this brief introduction, it may now be time to delve into the world of Sacred Geometry, the Golden Mean, and the Philosophupon which this treatise on Sacred Mathematics is inexorably leading.

And from there???

From The Great Pyramids to the Harmony of the Spheres, to the Satellites of Jupiter, There is just no limit to the possible delights and entertaining madness.  Which is the good news!  For ultimately with stops in the Quantum World and Connective Physics, brief forays into other branches of the Tree of Life and Creating Reality, the Crown of Kether can not be far from easy reach.


Dan Sewell Ward

The Library of Halexandria 



Independent Researcher.