SWEETSWORDS 6 [ Fibonacci Fitna ] Sacred geometry in nature

3 - 2 - 1 - 1 + 2 + 3 + 5 + 8

BISMILLAH

By Poga Humayun Dundiwala

Heavy heart and weight of gratitude
Counting the blessing with extravagant attitude
Number of blessing and one too many
The IMAM of want and chant and Psalm of content phony

Heavy heart and weight of gratitude
Counting the cost under apparent magnitude
Number of blessing and needful suffice
Weight of beholden and audit office

Poga Say’s in a fooligton maze

Weight of a being and measure of existence
A Eternal machine refueling with instance
A overflowing fullness that remain empty
Occasional outrage and permanent serenity

Heavy heart and weight of gratitude
Counting the beads with blessed blind attitude
Number of chant and want of man
Lightness of mark and weight of the pen

Mr Barzakh Fitrath Ullah: The dark square niche with glowing cube crystal
The dark square footstool with the glowing circle
The four legged dark KURSI and the eight headed glowing ARSH
The first dimensional lines of the audible pen and the second dimensional planes of the visual brush
These are the silent blessing of watery niche with the sounding clay crystal
Square up of all SAWAB with rolling down KHOAB circle
These are payment of silent blessing with the grateful AWAZ
These are payment to silent water from vibrating QUARTZ

Mrs Be Aql Khan Usta : This life is not some unreal Khawab
But this life is the place to gain the real Sawab
This life is where
I will my awareness to submit in sincere
To will of GOD
This life is where i sing the DURUD OF MUHAMMAD
Sallel La Hu Alahi Wa Sallaim
This life is blessing this life where we drink the living breathing Tasnim

Mr Barzakh Fitrath Ullah : ALLAH is manifest ALLAH is hidden and he have no partner
Therefore this life is not real nor unreal but reflectional SANSAR
Yet this life not the blind mutation of two
But the clear water with the NURANI view
This life is like this living breathing vibration
The WORD of GOD that occupies the whole without GOD being present in part portion or section

Mrs Be Aql Khan Usta : The best business is religious SAWDAH
The best of innovation is religious BIDAH
It is the business where profits multiplies by SUNNAH sanction
It is the TIJARAH where SAWAB multiplies by Fibonacci function
Therefore count this blessing where one is many
It is one present JULM where punishment multiplies without losing its previous tyranny

Barzakh : He gave Fibbonacci formula in the HADITH of BIDAH E HASANAH and the sequence in the DEEN of ISLAM
In the First KALAMA of ALLAH chose from his infinite names and infinite prophets he chose ALLAH 1 and MUHAMMAD 1 SALLEL LA HU ALAHI WA SALLAM
Now he who chooses the first sequence 1 + 1 may continue toward the two
With the QURAN and SUNNAH by his body and RUH 1 + 1 = 2
Then toward the third run
With IMAN AMAL and EHSAAN 1 + 1 + 1 = 3
He continues to ascend toward the infinite area
He continues to grow By five KALAMA E SHAHADA and eight AHKAM E SHARRIAH 5 + 8 = 1 + 3 + 4 + 7 + 1 + 1 + 2 + 3 + 5
This is the ASL asset of Islamic Bidathi innovation
And the inheritance SAWAB of its NAQL repetition
This is the real ISLAMIC enigmatic BIDAH and the ASL concept of ISLAMIC AWL
The RUHANI solution to all mental BE AQL

Be Aql : Prophet MUHAMMAD SALLEL LA HU ALAHI WA SALLAM said i warn you of the newly invented matter
And every newly invented matter is an innovation of its creator
And every innovation is misguidance
And every misguidance is in the hell fire residence
According to HADITH An Nasaee
All Ulema Say’s this HADITH is Kullu shaee
Now doe’s all newly invented matters leads to hell
Ancient water and modern well
I don’t know that but i know this
When you neglect 13 Arkan and Ahkam and 21 SALAH SUNNAH your AHZAB increases while your SAWAB decrease 13 + 21

Barzakh : Now who else but you to tell me about 13 SALATH E MUSTAHAB and 21 SALATH SUNNAH SAWAB 13 + 21 = 34
Who else but you to tell me about 13 SALATH E MUSTAHAB and 21 SALATH E SUNNAH what can nullify SALATH by 21 acts MOQRUH AHZAB 13 + 21 + 21 = 55
Who else but you to tell me about 40 SUNNATH E MUHAQADA and 44 SUNNAH E GAIR E MUHAQADA in our 5 daily SALAH 40 + 44 + 5 = 89
Who else but you to tell me about SUNNAH OF MUHAMMAD UR RASUL ALLAH
Yes i agree i know very little about AL QURAN and AL HADIS
But i know lot about every other which what that and this
I know he is under ALLAH but above AL QURAN
I know he is the spiral stair case through what AL QURAN descended to the creation by Fibonacci function
I know he is above AL QURAN but under SUNNAH
I know he is the highest Father of every lowly Munnah
I know he is under ALLAH but above AL QURAN
I know he is above AL FURQAN but under its FORMAN 8 + 13 + 21 + 34 + 55 + 89

General | Scientific Miracles in the Quran


Fibonacci Numbers: A Measure of Beauty

By Harun Yahya
March 02, 2005

The Golden Number is not mathematical imagination, but a natural principle related to the laws of equilibrium
[Allah has appointed a measure for all things.] (Qur’an, 65: 3)

The Wikipedia encyclopaedia describes beauty as ‘the phenomenon of the experience of pleasure, through the perception of balance’. Everyone admires beauty in nature and the unique balance found in it. Some say that this balance and perception of beauty is due to the Golden number or the ratio that gives certain things their exquisitness.

If a pleasing or exceedingly balanced form is achieved in terms of elements of application or function, it is there that we may look for a function of the Golden Number. The Golden Number is a product not of mathematical imagination, but of a natural principle related to the laws of equilibrium.[1]

What do the pyramids in Egypt, Leonardo da Vinci’s portrait of the Mona Lisa, sunflowers, the snail, the pine cone and your fingers all have in common?

The answer to this question lies hidden in a sequence of numbers discovered by the Italian mathematician Fibonacci. The characteristic of these numbers, known as the Fibonacci numbers, is that each one consists of the sum of the two numbers before it.[2]

Fibonacci numbers

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

Fibonacci numbers have an interesting property. When you divide one number in the sequence by the number before it, you obtain numbers very close to one another. In fact, this number is fixed after the 13th number in the series. This number is known as the “golden ratio.”

GOLDEN RATIO = 1.618

233 / 144 = 1.618

377 / 233 = 1.618

610 / 377 = 1.618

987 / 610 = 1.618

1597 / 987 = 1.618

2584 / 1597 = 1.618

The Golden Ratio

When conducting their researches or setting out their products, artists, scientists and designers take the human body, the proportions of which are set out according to the golden ratio, as their measure. Leonardo da Vinci and Le Corbusier took the human body, proportioned according to the golden ratio, as their measure when producing their designs. The human body, proportioned according to the golden ratio, is taken as the basis also in the Neufert, one of the most important reference books of modern-day architects.

The Golden Ratio in the Human Body

Click to Enlarge

The “ideal” proportional relations that are suggested as existing among various parts of the average human body and that approximately meet the golden ratio values can be set out in a general plan as follows:[3]

The M/m level (shown in the table to the right) is always equivalent to the golden ratio. M/m = 1.618

The first example of the golden ratio in the average human body is that when the distance between the navel and the foot is taken as 1 unit, the height of a human being is equivalent to 1.618. Some other golden proportions in the average human body are:

The distance between the finger tip and the elbow / distance between the wrist and the elbow

The distance between the shoulder line and the top of the head / head length

The distance between the navel and the top of the head / the distance between the shoulder line and the top of the head

The distance between the navel and knee / distance between the knee and the end of the foot

The Human Hand

Lift your hand from the computer mouse and look at the shape of your index finger. You will in all likelihood witness a golden proportion there.

Our fingers have three sections. The proportion of the first two to the full length of the finger gives the golden ratio (with the exception of the thumbs). You can also see that the proportion of the middle finger to the little finger is also a golden ratio.[4]

You have two hands, and the fingers on them consist of three sections. There are five fingers on each hand, and only eight of these are articulated according to the golden number: 2, 3, 5, and 8 fit the Fibonacci numbers.

The Golden Ratio in the Human Face

There are several golden ratios in the human face. Do not pick up a ruler and try to measure people’s faces, however, because this refers to the “ideal human face” determined by scientists and artists.

For example, the total width of the two front teeth in the upper jaw over their height gives a golden ratio. The width of the first tooth from the centre to the second tooth also yields a golden ratio. These are the ideal proportions that a dentist may consider. Some other golden ratios in the human face are:

Length of face / width of face

Distance between the lips and where the eyebrows meet / length of nose

Length of face / distance between tip of jaw and where the eyebrows meet

Length of mouth / width of nose

Width of nose / distance between nostrils

Distance between pupils / distance between eyebrows

Golden Proportion in the Lungs

In a study carried out between 1985 and 1987,[5] the American physicist B. J. West and Dr. A. L. Goldberger revealed the existence of the golden ratio in the structure of the lung. One feature of the network of the bronchi that constitutes the lung is that it is asymmetric. For example, the windpipe divides into two main bronchi, one long (the left) and the other short (the right). This asymmetrical division continues into the subsequent subdivisions of the bronchi.[6] It was determined that in all these divisions the proportion of the short bronchus to the long was always 1/1.618.

The Golden Rectangle and the Design in the Spiral

A rectangle, the proportion of whose sides is equal to the golden ratio is known as a “golden rectangle”
A rectangle, the proportion of whose sides is equal to the golden ratio is known as a “golden rectangle.” A rectangle whose sides are 1.618 and 1 unit long is a golden rectangle. Let us assume a square drawn along the length of the short side of this rectangle and draw a quarter circle between two corners of the square. Then, let us draw a square and a quarter circle on the remaining side and do this for all the remaining rectangles in the main rectangle. When you do this you will end up with a spiral.

The British aesthetician William Charlton explains the way that people find the spiral pleasing and have been using it for thousands of years stating that we find spirals pleasing because we are easily able to visually follow them.[7]

The spirals based on the golden ratio contain the most incomparable designs you can find in nature. Examples we can give of this are the spiral sequences on the sunflower and the pine cone.

The Design in Sea Shells

When investigating the shells of mollusks, which live at the bottom of the sea, the form and the structure of the internal and external surfaces of the shells attracted scientists’ attention.

The internal surface is smooth, while the outside surface is fluted. The mollusk’s body is inside the shell. The outside edges of the shell augment its rigidity and, thus, increase its strength. The shell’s form is astonishing in its perfection highlighting the beauty of its creation. The spiral idea in shells is expressed in the perfect geometrical form, in a surprisingly beautiful, “sharpened” design.[8]

The shells of most mollusks grow in a logarithmic spiral manner. There can be no doubt, of course, that these animals are unaware of even the simplest mathematical calculation, let alone logarithmic spirals. So how is it that the creatures in question can know that this is the best way for them to grow? How do these animals, that some scientists describe as “primitive,” know that this is the ideal form for them? It is impossible for growth of this kind to take place in the absence of a consciousness or intellect. That consciousness exists neither in mollusks nor, despite what some scientists would claim, in nature itself. It is totally irrational to seek to account for such a thing in terms of chance. This design can only be the product of the Almighty Allah.

An example of perfect geometry
Growth of this kind was described as “gnomic growth” by the biologist Sir D’Arcy Thompson, an expert on the subject, who stated that it was impossible to imagine a simpler system, during the growth of a seashell, than which was based on widening and extension in line with identical and unchanging proportions. As he pointed out, the shell constantly grows, but its shape remains the same.[9]

One can see one of the best examples of this type of growth in a nautilus, just a few centimetres in diameter. C. Morrison describes this growth process, which is exceptionally difficult to plan even with human intelligence, stating that along the nautilus shell, an internal spiral extends consisting of a number of chambers with mother-of-pearl (calcium carbonate and conchiolin secreted by the mantle of mollusks) lined walls. As the animal grows, it builds another chamber at the mouth of the spiral shell larger than the one before it, and moves forward into this larger area by closing the door behind it with a layer of mother-of-pearl.[10]

Growth in a spiral form in the animal world is not restricted to the shells of mollusks. Animals such as antelopes, goats and rams complete their horn development in spiral forms based on the golden ratio.[11]

The Golden Ratio in the Hearing and Balance Organ

The cochlea in the human inner ear serves to transmit sound vibrations. This bony structure, filled with fluid, has a logarithmic spiral shape with a fixed angle of α=73°43´ containing the golden ratio.

The Golden Ratio in DNA

The molecule in which all the physical features of living things are stored, too, has been created in a form based on the golden ratio. The DNA molecule, the very program of life, is based on the golden ratio. DNA consists of two intertwined perpendicular helixes. The length of the curve in each of these helixes is 34 angstroms and the width 21 angstroms. (1 angstrom is one hundred millionth of a centimetre.) 21 and 34 are two consecutive Fibonacci numbers.

The Golden Ratio in Snow Crystals

The golden ratio also manifests itself in crystal structures. Most of these are in structures too minute to be seen with the naked eye. Yet you can see the golden ratio in snow flakes. The various long and short variations and protrusions that comprise the snow flake all yield the golden ratio.[12]

The Golden Ratio in Space

In the universe there are many spiral galaxies containing the golden ratio in their structures.

The Golden Ratio in Physics

You encounter Fibonacci series and the golden ratio in fields that fall under the sphere of physics. When a light is held over two contiguous layers of glass, one part of that light passes through, one part is absorbed, and the rest is reflected. What happens is a “multiple reflection.” The number of paths taken by the ray inside the glass before it emerges again depends on the number of reflections it is subjected to. In conclusion, when we determine the number of rays that re-emerge, we find that they are compatible with the Fibonacci numbers.[13]

The fact that a great many unconnected animate or inanimate structures in nature are shaped according to a specific mathematical formula is one of the clearest proofs that these have been specially designed. The golden ratio is an aesthetic rule well known and applied by artists. Works of art based on that ratio represent aesthetic perfection. Plants, galaxies, micro-organisms, crystals and living things designed according to this rule imitated by artists are all examples of Allah’s superior artistry. Allah reveals in the Qur’an that He has created all things with a measure. Some of these verses read:

[Allah has appointed a measure for all things.] (Qur’an, 65: 3)

[Everything has its measure with Him.] (Qur’an, 13: 8)


**The author, who writes under the pen-name Harun Yahya, has published many books on political, faith-related and scientific issues. Some of the books of the author have been translated into English, German, French, Spanish, Italian, Portuguese, Albanian, Arabic, Polish, Russian, Bosnian, Indonesian, Turkish, Tatar, Urdu and Malay and published in the countries concerned

[1] Mehmet Suat Bergil, Doğada/Bilimde/Sanatta, Altın Oran (The Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari, 2nd Edition, 1993, p. 155.

[2] Guy Murchie, The Seven Mysteries of Life, First Mariner Boks, New York, pp. 58-59.

[3] J. Cumming, Nucleus: Architecture and Building Construction, Longman, 1985.

[4] Mehmet Suat Bergil, Doğada/Bilimde/Sanatta, Altın Oran (The Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari, 2nd Edition, 1993, p. 87.

[5] A. L. Goldberger, et al., “Bronchial Asymmetry and Fibonacci Scaling.” Experientia, 41 : 1537, 1985.

[6] E. R. Weibel, Morphometry of the Human Lung, Academic Press, 1963.

[7] William Charlton, Aesthetics: An Introduction, Hutchinson University Library, London, 1970.

[8] "The ‘Golden’ spirals and ‘pentagonal’ symmetry in the alive Nature,

[9] D’Arcy Wentworth Thompson, On Growth and Form, C.U.P., Cambridge, 1961.

[10] C. Morrison, Along The Track, Withcombe and Tombs, Melbourne.

[11] “The ‘Golden’ spirals and ‘pentagonal’ symmetry in the alive Nature,” o

[12] Emre Becer, “Biçimsel Uyumun Matematiksel Kuralı Olarak, Altın Oran” (The Golden Ratio as a Mathematical Rule of Formal Harmony), Bilim ve Teknik Dergisi (Journal of Science and Technology), January 1991, p.16.

[13] V.E. Hoggatt, Jr. and Bicknell-Johnson, Fibonacci Quartley, 17:118, 1979.

The works posted on this page reflect solely the opinions of the authors.

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                         Prayer (Salat),
            According to Five Islamic Schools of Law

                            (Part I)

             By: 'Allamah Muhammad Jawad Maghniyyah
          Translated from the Arabic by Mujahid Husayn

Salat (prayer) is either obligatory (wajib) or supererogatory (mandub).
The most important of prayers are the obligatory prayers performed daily
five times, and there is consensus among Muslims that a person who denies
or doubts their wujub is not a Muslim, even if he recites the shahadah,
for these prayers are among the ‘pillars’ (arkan) of Islam. They are the
established necessity of the faith (al-Din) that doesn’t need any
ijtihad or study, taqlid.

The Daily Supererogatory Prayers (Rawatib):

Supererogatory prayers are of various kinds, and among them are those

which are performed along with the obligatory daily prayers (fara’id). The
schools differ regarding the number of their rak’ahs. The Shafi’is consider
them to be eleven rak’ahs: two before the morning (subh) prayer, two before
the noon (zuhr) prayer and two after it, two after the sunset (maghrib)
prayer,
two after the night (‘isha’) prayer and a single rak’ah called ‘al-watirah’

 The Hanbalis consider them to be ten rak'ahs; two rak'ahs before and

after the noon prayer, two after the sunset and The night prayer, and two
rak’ahs before The morning prayer.

 According to the Malikis there is no fixed number for the

supererogatory (nawafil) prayers performed with the obligatory salat,
Though it is best to offer four rak’ahs before the zuhr and six after the
maghrib prayer.

 The Hanafis classify the nawafil performed along with the fara'id

into ‘masnunah’ and ‘mandubah’.(1) The ‘masnunah’ are five: two rak’ahs
before the subh; four before the zuhr, and two after it, except on Friday;
two after the maghrib and two after the ‘isha’ prayer.
The ‘mandubah’ are four: four -or two- rak’ahs before the 'asr, six
after the maghrib, and four before and after the ‘isha’ prayer.

 The Imamis observe:  The rawatib are 34 rak'ahs: eight before the

zuhr, eight before the ‘asr, four after the maghrib, two after the ‘isha’
(recited while sitting and counted as a single rak’ah; it is called ‘al
watirah’), eight rak’ahs of the midnight prayer (salat al-layl), two
rak’ahs of al-shaf’, a single of al-watr,(2) and two rak’ahs before the
morning prayer, called ‘salat al-fajr’.

The time of Zuhr and 'Asr Prayers:

The fuqaha' begin with salat al-zuhr, because it was the first salat

to be declared obligatory, followed by the 'asr, the maghrib, the ‘isha’
and the subh prayer, in that order. All the five prayers were made
obligatory on the night of Prophet’s cosmic journey (al ‘Isra’),
nine years after the beginning of his mission (bi’thah). Those who hold
this opinion cite as proof verse 78 of the Surat al-‘Isra’ which
stipulates all the five prayers:

" Perform salat from the declining of the sun to the darkening of the
night and the recital of the dawn; surely the recital of the dawn is
witnessed." (17:78)

The schools concur that salat is not valid if performed before its

appointed time and that the time of the zuhr prayer sets in when the sun
passes the meridian. They differ concerning its duration.

 The Imamis say: The specific period of the zuhr prayer extends from

the moment the sun crosses the meridian up to a period required to perform
it, and the specific period of the ‘asr prayer is the duration required to
perform it just before sunset. The time between these two specific periods
is the common period for the two salats. This is the reason they consider
it valid to perform both the prayers successively during their common
period.(3) But if the time remaining for the end of the day is sufficient
only for performing the zuhr prayer, the ‘asr prayer will be offered first
with the niyyah of ada’ and later the zuhr prayer will be performed as
qada’.

 The four Sunni schools observe: The time of the zuhr prayer begins

when the sun crosses the meridian and continues till the shadow of an
object becomes as long as its height; and when the length of the shadow
exceeds the height of the object, the time for the zuhr prayer comes to an
end. Here the Shafi’is and the Malikis add: These limits are for an
unconstrained person (mukhtar), and for one who is constrained (mudarr),
the time for zuhr prayer extends even after an object’s shadow equals its
height. The Imamis consider the time when an object’s shadow equals its
height as the end of the time of fadilah (honor) for the zuhr, and when it
equals twice the height of the object as the time of fadilah for the 'asr
prayer.

 The Hanafis and the Shafi'is state: The time of 'asr prayer begins

when the length of an object’s shadow exceeds its height and continues up
to sunset.

 The Malikis say: For the 'asr prayer there are two times, the first

for ordinary circumstances and the second for exigencies. The former
begins with an object’s shadow exceeding its height and lasts until the
sun turns pale. The latter begins from when the sun turns pale and
continues until sunset.

 The Hanbalis observe: One who delays offering the 'asr prayer till

after an object’s shadow exceeds twice its height, his salat will be
considered ada’ if performed before sunset, though he will have sinned
because it is haram to delay it until this time. They are alone in all the
schools in holding this opinion.

The Time of Maghrib and ‘Isha’ Prayers:

 The Shafi'i and the Hanbali schools (in accordance with the view of

their respective Imams) state: The time for the maghrib prayer begins when
the sun sets and ends when there radish after glow on the western horizon
vanishes.

 The Malikis say: The duration for the maghrib prayer is narrow

and confined to the time required after sunset to perform the maghrib
prayer along with its preliminaries of taharah and adhan, and it is not
permissible to delay it voluntarily. But in an emergency, the time for the
maghrib prayer extends until dawn. The Malikis are alone in considering it
impermissible to delay the maghrib prayer beyond its initial time.

The Imamis observe: The period specific to the maghrib prayer extends

from sunset(4) for a duration required to perform it, and the specific
period of the ‘isha’ prayer is the duration required to finish it before
midnight. The time between these two specific periods is the common time
for both maghrib and ‘isha’ prayers. Hence they allow the joint
performance of these two salats during this common time.

That was with respect to someone who is in a position to act out of

free choice (mukhtar), but as to a person constrained by sleep or
forgetfulness, the time for these two salats extend until dawn, with the
period specific for the ‘isha’ prayer becoming the time required to
perform it just before dawn and the specific period for the maghrib prayer
becoming the time required to perform it just after midnight.

The Time of subh Prayer:

 There is consensus among The schools, with the exception of the

Maliki, that the time for the morning prayer begins at day-break (al fajr
al-sadiq) and lasts until sunrise. The Malikis say: The subh prayer has
two times: for one in a position to act out of free choice it begins with
daybreak and lasts until there is enough twilight for faces to be
recognized; for one in constrained circumstances it begins from the time
when faces are recognizable and continues up to sunrise.


NOTE: THE NUMBERS FOLLOWING SOME TERMS ARE MARKS FOR FOOTNOTES THAT WILL
BE APPENDED IN SHAA ALLAH IN THE NEXT PART.


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                         Prayer (Salat),
            According to Five Islamic Schools of Law

                           (Part II)

             By: 'Allamah Muhammad Jawad Maghniyyah

Ignorance of the Qiblah (direction):

It is wajib for a person ignorant of the qiblah to inquire and
strive to determine its exact or approximate direction, and in case
neither of the two is possible, the four Sunni schools and a group from
among the Imamis say: He may perform salat in any direction; his salat
will be valid and it will not be wajib for him to repeat it except in the
opinion of the Shafi’is. Most Imamis observe: He will perform Salat in
four directions to comply with the command for salat and to ascertain its
proper performance. But if there isn’t sufficient time for performing
salat four times or if one is incapable of performing it in four
directions, he may perform, salat in the directions that he can.

A Subsidiary Issue:

If a person prays not facing the qiblah and comes to know about
his mistake, the Imamis state: If the error is known during the salat and
the correct qiblah lies between his two hands, the part of the salat
already performed will be valid and he will have to correct his direction
for the remaining part of the salat. But if it is known that he has been
praying facing the right or the left (90 degree off the direction) or his
back towards the qiblah (180 degree off the direction), the salat will be
invalid and he will perform it anew. If the error is known after
performing the salat, it should be performed again if its time
is still there, not otherwise. Some Imamis say: The salat will not be
repeated if there is only a little deviation from the qiblah, irrespective
of whether its time is still there or not. But if it has been performed
facing the right or the left (90 degree off), it should be repeated if its
time is there, not otherwise. If the salat is performed with one’s back to
the qiblah (180 degree off), it should be repeated regardless of whether
its time is still there or has passed

The Hanafis and the Hanbalis observe: If after inquiring and
striving to find the qiblah one is unable to ascertain its approximate
direction and performs salat in a direction which turns out to be wrong,
he must change his direction accordingly if the mistake is known during
the salat, and if it is known afterwards his salat is valid and he has no
further obligation.

The Shafi’is say: If it becomes certain that there has been a
mistake in determining the qiblah, it is wajib to repeat the salat, but if
there is only a likelihood of mistake, the salat is valid Irrespective of
whether the probability arises during the salat or after it.

As to one who neither makes an inquiry nor an effort to determine
the qiblah, but by chance performs the salat in the right direction, the
Malikis and Hanbalis consider his salat to be invalid. The opinion of the
Imamis and the Hanafis is that his salat is valid provided he has no
doubts while praying and was sure about the direction of the qiblah at the
time of starting the salat, because, as pointed out by the Imamis, in such
a situation it is correct for him to make the niyyah of acquiring nearness
(qurbah) to God.


The Numbers that follow some of the words
refer to notes that will appended in the coming issues in shaa Allah.
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                         Prayer (Salat),
            According to Five Islamic Schools of Law

                           (Part III)

             By: 'Allamah Muhammad Jawad Maghniyyah

Wajib Covering During Salat:

 The schools concur that it is wajib (necessary) upon both men and

women
to cover those parts of their bodies during salat which should ordinarily
be
kept covered before ‘strangers’. Beyond that their positions differ. Is it
wajib for a woman to cover, fully or partly, her face and hands during
salat, although she is not required to do so outside salat? Is it wajib for
a man to cover other parts of his body during salat apart from the area
between the navel and the knees, though it is not wajib to do so outside
salat?

 The Hanafis observe: It is wajib upon a woman to cover the back

of her hands and the soles of her feet as well, and upon a man to cover
his knees in addition to the area between the navel and the knees.
The Shafi’is and Malikis say: It is permissible for a woman to keep
her face and both the palms and the back of her hands uncovered
during salat.

The Hanbalis state: It is not permissible for her to expose any

part except the face.

The Imamis observe: It is wajib for both men and women to cover only

those parts of their body during salat which they are supposed to cover
ordinarily in the presence of a ‘stranger’. Hence it is permissible for a
woman to expose during salat that part of her face which is washed
during wudu’; her hands up to the wrists, and her feet up to the ankles
both the back as well as the palms of hands and the soles of feet. For
a man, it is wajib to cover the rear and the private parts, though better
to cover the entire area between the navel and the knees.

The Material Used for Covering During Salat:

The covering should meet the following requirements where the

ability and freedom to meet them exist:

  1. Taharah: The purity of The covering and the body are necessary for
    the validity of salat in the opinion of all the schools, although each of
    them concedes certain exceptions in accordance with the following
    details:

    The Imamis state: Blood from wounds and sores, irrespective of
    its quantity, is considered excusable on The dress as well as the body if
    its removal entails difficulty and harm (mashaqqah and haraj). A blood spot
    smaller than the size of a dirham coin, regardless of its being due to
    one’s blood or that of someone else, is also excusable provided that: it is
    in a single place and not in different places; it is not the blood of hayd,
    nifas and istihadah; it is not the blood of anything intrinsically najis,
    such as dog and pig, or the blood of a dead body. Also excusable is the
    impurity of anything that does not constitute part of essential dress
    during salat, e.g. a sash, cap, socks, shoes, ring, anklet and that which
    one carries with oneself, e.g. knife or currency. The impurity of the dress
    of a woman rearing a child, irrespective of whether she is the mother or
    someone else, is exempted on condition that it be difficult for her to
    change it and that she washes it once every day. In other words, in their
    opinion every najasah on dress or body is exempted in conditions of
    emergency (idtirar).

    The Malikis observe: Cases of uncontrolled discharge of urine or
    excrement, as well as piles, are excusable; so is any impurity on the
    body or clothes of a woman suckling an infant that may be soiled by
    the infant’s urine or feces. So also are exempted the body and clothes
    of a butcher, surgeon and scavenger. Also exempted is: blood -even
    that of a pig- if it is less than the size of a dirham coin: the discharge
    from boils, the excrement of fleas, and other things which need not be
    mentioned because they occur rarely.

    The Hanafis say: Najasah, blood or anything else, if less than the
    size of a dirham coin is exempted. Also exempted in emergencies is the
    urine and excrement of a cat and mouse. Tiny splashes- as small as the
    point of a needle- of urine, the blood that unavoidably stains a butcher,
    and the mud on roads- even if it is usually mixed with najasah and
    provided the najasah itself is not visible- are exempted. Consequently,
    they consider najasah in a small quantity as exempted, such as the urine
    of an animal eating which is halal, if it covers a fourth of the clothes
    and less than one-fourth of the body.

    According to the Shafi’is, every najasah which is in such a small
    quantity that the eye cannot see it is exempted. So is the mud on roads
    which is mixed with a small quantity of najasah, worms present in
    fruits and cheese, najis liquids added in medicines and perfumes,
    excrements of birds, najis hair in small quantity if they do not belong
    to a dog or a pig, and other things as well which are mentioned in
    detailed works.

     The Hanbalis say: Minute quantities of blood and pus are ex-
    

empted, and so is the mud on roads whose najasah is certain, as well as
the najasah that enters the eyes and washing which is harmful.

  1. Wearing Silk: There is consensus among the schools that wearing silk
    and gold is haram for men both during and outside salat, while it is
    permissible for women. This is in accordance with this statement of the
    Prophet (S):

Wearing silk and gold is unlawful for the men of my ummah, while it is
lawful for its women.

Accordingly, the Imamis observe: A man’s salat is not valid if he wears
pure silk and any clothing embroidered with gold during it, regardless
of whether it is a waistband, cap, socks, or even a gold ring. They allow
wearing silk during salat in times of illness and during war.

The Shafi’is state: If a man performs salat while wearing silk or over
something made of it, it will be considered a haram act, though his salat
will be valid (al-Nawawi, Sharh al-Muhadhdhab, iii, 179). I have not found
an express statement in the books of the remain- ing schools concerning the
validity or invalidity of salat performed in silk, though the Hanafis as
well as the Hanbalis (in accordance with one of two narrations) concur with
the Shafi’is regarding the general rule that if there is any command
prohibiting something which is not directly connected with salat such as
the command prohibiting usurpa- tion- the salat will be valid if it is not
observed and The person will be considered as having performed a wajib and
a haram act together. Accordingly the salat performed in a dress of silk is
valid. The author of al-fiqh 'ala al-madhahib al-'arba’ah reports a
consensus for the Sunni schools that that it is valid for a man constrained
to perform salat while wearing silk, and it is not wajib for him to repeat
it.

  1. Lawfulness of the Clothing: The Imamis consider it necessary that
    the clothing worn to be lawfully owned. Hence if a person performs salat
    in usurped clothes with the knowledge of their being so, his salat is
    invalid. This is also the opinion of Ibn Hanbal in one of the two state-
    ments narrated from him.

    The other schools regard salat in usurped clothes as valid on the
    grounds that the prohibition does not directly relate to salat so as to
    invalidate it.

    The Imamiyyah are very strict concerning usurpation, and some
    of them even observe: If a person performs salat in clothes in
    which a single thread is usurped, or carries with him an usurped knife,
    dirham, or any other thing, his salat will not be valid. But they also
    say: If one performs salat in usurped clothes out of ignorance or
    forgetfulness, his salat is valid.

  2. The Skin of Uneatable Animals: The Imamis are alone in holding that
    it is invalid to perform salat while wearing the skin (even if tanned)
    of an animal whose flesh is not allowed to be eaten, as well as anything
    consisting its hair, wool, fur or feathers. The same is true of clothes
    bearing any secretion from its body- eg. sweat and saliva- as long as
    it is wet. Hence, even if a single hair of a cat or any such animal happens
    to be present on the dress of a person performing salat and if he performs
    it with the knowledge of its presence, his salat is invalid.
    They exclude wax, honey, the blood of bugs; lice, fleas and other
    insects which have no flesh, as well as the hair, sweat and saliva of
    human beings.

    They also consider salat invalid if any part of a dead animal
    (maytah) happens to be on the clothes irrespective of whether the
    animal is one used for food or not, whether its blood flows when cut
    or not, and its skin is tanned or not.

A Subsidiary Issue: If there is only a single clothing to cover the body
and that too is najis to an extent that is not excusable, what should
one do if he has no alternative other than either performing salat in the
najis clothing or in the state of nature?

The Hanbalis say: He should perform salat in the najis clothing,

but it is wajib upon him to repeat it later.

The Malikis and a large number of Imamis observe: He should

perform salat in the najis clothing and its repetition is not wajib upon
him.

The Hanafis and the Shafi'is state: He should perform salat naked

and it is not valid for him to cover himself with the najis clothing.

The Place of Salat:

An Usurped Place: The Imamis consider salat performed in an usurped
place and usurped clothing as invalid provided it is done voluntarily and
with the knowledge of the usurpation. The other schools observe: The
salat performed in an usurped place is valid, though the person per-
forming it will have sinned, since the prohibition does not relate directly
to salat; rather, it relates to dispensations (of property). Their position
in this regard is the same as in the case of usurped clothing. Furthermore
in the opinion of the four schools the Salat of usurper himself is valid
in usurped property.

  The Imamis also consider as valid the salat of the true owner and

anyone whom he permits, and regard as invalid the salat of the usurper
and anyone whom the owner has not granted permission. The Imamis however
permit salat in vast stretches of (owned) land which are either impossible
or difficult for people to avoid, even if the permission of the owner has
not been acquired.

Taharah (purity) of the Place: The four Sunni schools observe: The place
should be free from both wet and dry najasah (impurity). The Shafi’is
overdo by saing: The taharah of all that which touches and comes into
contact with the body or clothes of the performer is wajib. Therefore, if
he rubs himself against a najis wall or cloth or holds a najis object or a
rope laying over najasah, his salat will be invalid. The Hanafis require
only the location of the feet and the forehead to be tahir. The Imamis
restrict it to the loca- tion of the forehead, i.e. the place of sajdah. As
to the najasah of other locations, the salat will not be invalid unless the
najasah is transmitted to the body or clothing of the performer (the person
performing salat).

Salat Performed on a Mount: The Hanafis and the Imamis require the
place to be stationary; hence it is not valid in their opinion to perform
salat while riding an animal or something that swings back and forth,
except out of necessity, because one who has no choice will perform
salat in accordance with his capacity.

 The Shafi'is, Malikis and Hanbalis observe: Salat performed on a

mount is valid even during times of peace and despite the ability to
perform it on the ground, provided it is performed completely and
meets all the requirements.

Salat Inside the Ka’bah: The Imamis, Shafi’is and Hanafis state: It is
valid to perform salat, faridah or nafilah, inside the Ka’bah.
The Malikis and the Hanbalis say: Only nafilah, not faridah, is
valid therein.

A Woman’s Prayer Beside a Man: A group of Imami legists observe: If
a man and a woman perform salat in a single place so that she is either
in front of him or beside him, and there is neither any screen between
them nor does the distance between the two exceed 10 cubits, the
salat of the one who starts earlier will not be invalid, and if both star
simultaneously, the salat of both will be invalid.

The Hanafis say: If the woman is in front or beside a man, the
salat will be invalid if performed in a single place with no screen at
least
a cubit high between them, the woman has sex appeal, her shanks and
ankles are adjacent to his, the salat is not a funeral prayer, and the
salat is being jointly performed, i.e. either she is following him or both
are following a single imam.

The Shafi’is, the Hanbalis and most Imamis are of the view that
the salat is valid, though the manner of performance is makruh.

The Locale of Sajdah: The schools concur that the place where the
forehead is placed during prostration should be stationary and should
not be inordinately higher than the location of the knees (during
sajdah). They differ regarding that on which sajdah is valid.

The Imamis state: It is valid to perform sajdah only on earth and those
things which grow on it which are not used for food or clothing.
Therefore, a person cannot perform sajdah on wool, cotton, minerals
and that which grows on the surface of water, for water is not earth.
They permit sajdah on paper because it is made of a material
which grows on earth. They argue their position by pointing out that
sajdah is an 'ibadah (obedience) prescribed by the Shari’ah that depends
for
its particulars on textual evidence (nass). The legists of all the schools
concur regarding the validity of sajdah on earth and that which grows
on it, thus Imamis restrict it to that because there is certainty. They
offer as further evidence these traditions of the Prophet (S):

The salat of any of you will not be valid unless he performs wudu’ as in-
structed by God and then performs sajdah by placing his forehead on the
earth.

The earth has been created a masjid (a place for performing sajdah) and
a purifier .

Khabbab says: “We complained to The Prophet (S) regarding the
excessive heat of sun-baked ground on our foreheads, but he did not
accept our complaint.”

Had it been valid to perform sajdah on carpets, why would they have
complained?! However Imamis permit sajdah on cotton and linen in the case
of emergency.

The four schools observe: It is valid to perform sajdah on anything,
including even a part of one’s turban, provided it is tahir. Rather,
the Hanafis permit sajdah on one’s palm even without an emergency,
though it is considered as makruh.

To be continued-insha’Allah


FOOTNOTES:

  1. The Hanafis use two terms (‘fard’ and ‘wajib’) for something whose
    performance is obligatory and whose omission is impermissible. Hence
    they divide obligation into two kinds: fard and wajib. 'Fard ‘is a duty
    for which there is definite proof, such as Qur’anic text, mutawatir
    sunnah, and ijma’ (consensus). ‘Wajib’ is a duty for which there is a
    Dhanni (non-definite) proof, such as qiyas (analogy) and khabar
    al-wahid (isolated tradition). That whose performance is preferable to its
    omission is also of two kinds: ‘masnun’ and ‘mandub’. ‘Masnun’ is an act
    which the Prophet (S) and the ‘Rashidun’ caliphs performed regularly, and
    ‘mandub’ is an act ordered by the Prophet (S) though not performed
    regularly by him (S). That which it is wajib to avoid and whose
    performance is not permissible is ‘muharram’ if it is established by a
    definite proof. If based on a Dhanni proof, it is ‘makruh’, whose
    performance is forbidden.

  2. According to the Hanafis, the salat al-watr consists of three rak’ahs
    with a single salam. Its time extends from the disappearance of twilight
    after sunset to dawn. The Hanbalis and Shafi’is say: At minimum it is one
    rak’ah and at maximum eleven rak’ahs, and its time is after the ‘isha’
    prayer. The Malikis observe: It has only one rak’ah.

  3. There are among ‘ulama’ of the Sunni schools those who agree with
    the Imamis on performing the two salats together even when one is not
    travelling. al-Shaykh Ahmad al-Siddiq al-Ghumari has written a book on
    this topic, Izalat al khatar 'amman jama’a bayn al-salatayn fi al-hadar.

  4. There is no difference regarding the definition of sunset between the
    Imamis and the other four schools. But the Imamis say that the setting of
    the sun is not ascertained simply by the vanishing of the sun from sight,
    but on the vanishing of the reddish afterglow from the eastern horizon,
    for the east overlooks the west and the eastern afterglow, which is a
    reflection of sun’s light,

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

In mathematics, the Fibonacci numbers are the numbers in the following sequence:

0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots.

By definition, the first two Fibonacci numbers are 0 and 1, and each remaining number is the sum of the previous two. Some sources omit the initial 0, instead beginning the sequence with two 1s.

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

F_n = F_{n-1} + F_{n-2},\!\,

with seed values

F_0 = 0 \quad   ext{and}\quad F_1 = 1.

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonaccio, "son of Bonaccio"). Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics.[2][3]

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

The Fibonacci sequence was well known in ancient India, where it was applied to the metrical sciences (prosody), before it was known in Europe. Developments have been attributed to Pingala (200 BC), Virahanka (6th century AD), Gopāla (c.1135 AD), and Hemachandra (c.1150 AD).[4]

The motivation came from Sanskrit prosody, where long syllables have duration 2 and short syllables have duration 1. Any pattern of duration n can be formed by adding a short syllable to a pattern of duration n − 1, or a long syllable to a pattern of duration n − 2; thus the prosodists showed that the number of patterns of duration n is the sum of the two previous numbers in the sequence. Later authors gave algorithms for ranking and unranking these patterns (e.g. finding the kth pattern of duration n), and discovered the higher-order Fibonacci numbers. Donald Knuth reviews this work in The Art of Computer Programming.[5][6]

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

In the West, the sequence was studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202).[7] He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that: a newly-born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?

* At the end of the first month, they mate, but there is still one only 1 pair.
* At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
* At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
* At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

At the end of the nth month, the number of new pairs of rabbits is equal to the number of pairs in month n-2 plus the number of pairs alive last month. This is the nth Fibonacci number

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

List of Fibonacci numbers

The first 21 Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for n = 0, 1, 2, ... ,20 are:[9][10]

F0     F1     F2     F3     F4     F5     F6     F7     F8     F9     F10     F11     F12     F13     F14     F15     F16     F17     F18     F19     F20
0     1     1     2     3     5     8     13     21     34     55     89     144     233     377     610     987     1597     2584     4181     6765

Using the recurrence relation, the sequence can also be extended to negative index n. The result satisfies the equation

F_{-n} = (-1)^{n+1} F_n. \!\,

Thus the complete sequence is

\ldots,\;-8,\;5,\;-3,\;2,\;-1,\;1,\;0,\;1,\;1,\;2,\;3,\;5,\;8,\;\ldots

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

Divisibility properties

Every 3rd number of the sequence is even and more generally, every kth number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property

\gcd(F_m,F_n) = F_{\gcd(m,n)}.\

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

Relation to the golden ratio
Approximate and true golden spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of one square divided by that of the next smaller square is the golden ratio.
[edit] Closed-form expression

Like every sequence defined by linear recurrence, the Fibonacci numbers have a closed-form solution. It has become known as Binet's formula, even though it was already known by Abraham de Moivre:[11]

F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}={{\varphi^n-(-1/\varphi)^{n}} \over {\sqrt 5}}\, ,

where

\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\dots\,

is the golden ratio (sequence A001622 in OEIS).

That

1-\varphi=-1/\varphi,

follows from the defining equation above.

The Fibonacci recursion

F(n+2)-F(n+1)-F(n)=0\,

is similar to the defining equation of the golden ratio in the form

x^2-x-1=0,\,

which is also known as the generating polynomial of the recursion.

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

Proof by induction

Any root of the equation above satisfies \begin{matrix}x^2=x+1,\end{matrix}\, and multiplying by x^{n-1}\, shows:

x^{n+1} = x^n + x^{n-1}\,

By definition \varphi\, is a root of the equation, and the other root is 1-\varphi\,=-1 / \varphi\, . Therefore:

\varphi^{n+1} = \varphi^n + \varphi^{n-1}\, 

and

(1-\varphi)^{n+1} = (1-\varphi)^n + (1-\varphi)^{n-1}\, .

Both \varphi^{n} and (1-\varphi)^{n}=(-1/\varphi)^{n} are geometric series (for n = 1, 2, 3, ...) that satisfy the Fibonacci recursion. The first series grows exponentially; the second exponentially tends to zero, with alternating signs. Because the Fibonacci recursion is linear, any linear combination of these two series will also satisfy the recursion. These linear combinations form a two-dimensional linear vector space; the original Fibonacci sequence can be found in this space.

Linear combinations of series \varphi^{n} and (1-\varphi)^{n}, with coefficients a and b, can be defined by

F_{a,b}(n) = a\varphi^n+b(1-\varphi)^n for any real a,b\, .

All thus-defined series satisfy the Fibonacci recursion

\begin{align} F_{a,b}(n+1) &= a\varphi^{n+1}+b(1-\varphi)^{n+1} \\ &=a(\varphi^{n}+\varphi^{n-1})+b((1-\varphi)^{n}+(1-\varphi)^{n-1}) \\ &=a{\varphi^{n}+b(1-\varphi)^{n}}+a{\varphi^{n-1}+b(1-\varphi)^{n-1}} \\ &=F_{a,b}(n)+F_{a,b}(n-1)\,. \end{align}

Requiring that Fa,b(0) = 0 and Fa,b(1) = 1 yields a=1/\sqrt 5 and b=-1/\sqrt 5, resulting in the formula of Binet we started with. It has been shown that this formula satisfies the Fibonacci recursion. Furthermore, an explicit check can be made:

F_{a,b}(0)=\frac{1}{\sqrt 5}-\frac{1}{\sqrt 5}=0\,\!

and

F_{a,b}(1)=\frac{\varphi}{\sqrt 5}-\frac{(1-\varphi)}{\sqrt 5}=\frac{-1+2\varphi}{\sqrt 5}=\frac{-1+(1+\sqrt 5)}{\sqrt 5}=1,

establishing the base cases of the induction, proving that

F(n)={{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}} for all n\, .

Therefore, for any two starting values, a combination a,b can be found such that the function F_{a,b}(n)\, is the exact closed formula for the series.

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

Computation by rounding

Since \begin{matrix}|1-\varphi|^n/\sqrt 5 < 1/2\end{matrix} for all n\geq 0, the number F(n) is the closest integer to \varphi^n/\sqrt 5\, . Therefore it can be found by rounding, or in terms of the floor function:

F(n)=\bigg\lfloor\frac{\varphi^n}{\sqrt 5} + \frac{1}{2}\bigg\rfloor,\ n \geq 0.

Similarly, if you already know that the number F is a Fibonacci number, you can determine its index within the sequence by

n = \bigg\lfloor \log_\varphi \left(F\cdot\sqrt{5}\right) + \frac{1}{2} \bigg\rfloor

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

Limit of consecutive quotients

Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost”, and concluded that the limit approaches the golden ratio \varphi\, .[12]

\lim_{n o\infty}\frac{F(n+1)}{F(n)}=\varphi,

This convergence does not depend on the starting values chosen, excluding 0, 0. For example, the initial values 19 and 31 generate the sequence 19, 31, 50, 81, 131, 212, 343, 555 ... etc. The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.
[edit] Proof

In brief, Fibonacci numbers are approximately exponential: F_{a,b}(n) \approx k\phi^n, where the constant depends on starting values – as the remaining term in the exact formula for the Fibonacci numbers becomes exponentially close to zero as n grows. Taking the ratio yields F(n+1)/F(n) \approx (k\phi^{n+1})/(k\phi^n) = \phi.

More formally, it must always follow from the explicit formula that for any real a
e 0, \, b
e 0 \,

\begin{align} \lim_{n   o\infty}\frac{F_{a,b}(n+1)}{F_{a,b}(n)} &= \lim_{n  o\infty}\frac{a\varphi^{n+1}-b(1-\varphi)^{n+1}}{a\varphi^n-b(1-\varphi)^n} \\ &= \lim_{n   o\infty}\frac{a\varphi-b(1-\varphi)(\frac{1-\varphi}{\varphi})^n}{a-b(\frac{1-\varphi}{\varphi})^n} \\ &= \varphi \end{align}

because \bigl|{ frac{1-\varphi}{\varphi}}\bigr| < 1 and thus \lim_{n o\infty}\left( frac{1-\varphi}{\varphi}\right)^n=0 .

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa,Italy (see Pisa on Google Earth), the city with the famous Leaning Tower, about 1175 AD.

Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the present-day Algerian town of Béjaïa, (see Bejaia on Google Earth ) formerly known as Bugia or Bougie, where wax candles were exported to France. They are still called "bougies" in French.

So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others.

D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250).

By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci! Vinci was just a few miles from Pisa on the way to Florence, but Leonardo da Vinci was born in Vinci in 1452, about 200 years after the death of Leonardo of Pisa (Fibonacci).

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

Leonardo of Pisa is now known as Fibonacci [pronounced fib-on-arch-ee] short for filius Bonacci.
There are a couple of explanations for the meaning of Fibonacci:

* Fibonacci is a shortening of the Latin "filius Bonacci", used in the title of his book Libar Abaci (of which mmore later), which means "the son of Bonaccio". His father's name was Guglielmo Bonaccio. Fi'-Bonacci is like the English names of Robin-son and John-son. But (in Italian) Bonacci is also the plural of Bonaccio; therefore, two early writers on Fibonacci (Boncompagni and Milanesi) regard Bonacci as his family name (as in "the Smiths" for the family of John Smith).
  Fibonacci himself wrote both "Bonacci" and "Bonaccii" as well as "Bonacij"; the uncertainty in the spelling is partly to be ascribed to this mixture of spoken Italian and written Latin, common at that time. However he did not use the word "Fibonacci". This seems to have been a nickname probably originating in the works of Guillaume Libri in 1838, accordigng to L E Sigler's in his Introduction to Leonardo Pisano's Book of Squares (see Fibonacci's Mathematical Books below).
* Others think Bonacci may be a kind of nick-name meaning "lucky son" (literally, "son of good fortune").

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

Other names

He is perhaps more correctly called **Leonardo of Pisa** or,   using a latinisation of his name, **Leonardo Pisano**.     Occasionally he also wrote **Leonardo Bigollo** since, in Tuscany,  *bigollo* means *a traveller*.    We shall just call him Fibonacci as do most modern authors, but if you  are looking him up in older books, be prepared to see any of the above  variations of his name.

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

** Introducing the Decimal Number system into Europe **

He was one of the first people to introduce the Hindu-Arabic number system into Europe - the positional system we use today - based on ten digits with its decimal point and a symbol for zero: 1 2 3 4 5 6 7 8 9 0
His book on how to do arithmetic in the decimal system, called Liber abbaci (meaning Book of the Abacus or Book of Calculating) completed in 1202 persuaded many European mathematicians of his day to use this "new" system. The book describes (in Latin) the rules we all now learn at elementary school for adding numbers, subtracting, multiplying and dividing, together with many problems to illustrate the methods:
1 7 4 + 1 7 4 - 1 7 4 x 1 7 4 ÷ 28
2 8 2 8 2 8 is
----- ----- -------
2 0 2 1 4 6 3 4 8 0 + 6 remainder 6
----- ----- 1 3 9 2
-------
4 8 7 2
-------

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

Let's first of all look at the Roman number system still in use in Europe at that time (1200) and see how awkward it was for arithmetic. ** Roman Numerals **

** The Numerals are letters**

The method in use in Europe until then used the Roman numerals: I = 1,
V = 5,
X = 10,
L = 50,
C = 100,
D = 500 and
M = 1000 You can still see them used on foundation stones of old buildings and on some clocks.

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

** The Additive rule**

The simplest system would be merely to use the letters for the values as in the table above, and add the values for each letter used.
For instance, 13 could be written as XIII or perhaps IIIX or even IIXI. This occurs in the Roman language of Latin where 23 is spoken as tres et viginti which translates as three and twenty. You may remember the nursery rhyme Sing a Song of Sixpence which begins Sing a song of sixpence
A pocket full of rye
Four and twenty blackbirds
Baked in a pie...
Above 100, the Latin words use the same order as we do in English, so that whereas 35 is quinque et triginta (5 and 30), 235 is ducenti triginta quinque (two hundred thirty five). In this simple system, using addition only, 99 would be 90+9 or, using only the numbers above, 50+10+10+10 + 5+1+1+1+1 which translates to LXXXXVIIII and by the same method 1998 would be written by the Romans as MDCCCCLXXXXVIII.
But some numbers are long and it is this is where, if we agree to let the order of letters matter * we can also use *subtraction.

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

The subtractive rule

The Roman language (Latin) also uses a subtraction principle so that whereas 20 is viginti 19 is "1 from 20" or undeviginti. We have it in English when we say the time is "10 to 7" which is not the same as "7 10". The first means 10 minutes before ( or subtracted from) 7 0'clock, whereas the second means 10 minutes added to (or after) 7 o'clock. This is also reflected in Roman numerals. This abbreviation makes the order of letters important. So if a smaller value came before the next larger one, it was subtracted and if it came after, it was added.

For example, XI means 10+1=11 (since the smaller one comes after the larger ten) but IX means 1 less than 10 or 9. But 8 is still written as VIII (not IIX). The subtraction in numbers was only of a unit (1, 10 or 100) taken away from 5 of those units (5, 50 or 500 or from the next larger multiple of 10 (10, 100 or 1000).

Using this method, 1998 would be written much more compactly as MCMXCVIII but this takes a little more time to interpret: 1000 + (100 less than 1000) + (10 less than 100) + 5 + 1 + 1 + 1.

Note that in the UK we use a similar system for time when 6:50 is often said as "ten to 7" as well as "6 fifty", similarly for "a quarter to 4" meaning 3:45. In the USA, 6:50 is sometimes spoken as "10 of 7".

  • Look out for Roman numerals used as the date a film was made, often recorded on the screen which gives its censor certification or perhaps the very last image of the movie giving credits or copyright information. *

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

** Arithmetic with Roman Numerals **

 Arithmetic was not easy in the Roman system: 

CLXXIIII added to XXVIII  is CCII
CLXXIIII less     XXVIII  is CXXXXVI

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

** The Decimal Positional System **

The system that Fibonacci introduced into Europe came from India and Arabia and used the Arabic symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 with, most importantly, a symbol for zero 0.
With Roman numbers, 2003 could be written as MMIII or, just as clearly, it could be written as IIIMM - the order does not matter since the values of the letters are added to make the number in the original (unabbreviated) system. With the abbreviated system of IX meaning 9, then the order did matter but it seems this sytem was not often used in Roman times.
In the "new system", the order does matter always since 23 is quite a different number to 32. Also, since the position of each digit is important, then we may need a zero to get the digits into their correct places (columns) eg 2003 which has no tens and no hundreds. (The Roman system would have just omitted the values not used so had no need of "zero".) This decimal positional system, as we call it, uses the ten symbols of Arabic origin and the "methods" used by Indian Hindu mathematicians many years before they were imported into Europe. It has been commented that in India, the concept of nothing is important in its early religion and philosophy and so it was much more natural to have a symbol for it than for the Latin (Roman) and Greek systems.

Re: SWEETSWORDS 6 Fibonacci Fitna ] Sacred geometry in nature

"Algorithm"

Earlier the Persian author Abu ‘Abd Allah, Mohammed ibn Musa al-Khwarizmi (usually abbreviated to Al-Khwarizmi had written a book which included the rules of arithmetic for the decimal number system we now use, called Kitab al jabr wa‘l-muqabala (Rules of restoring and equating) dating from about 825 AD. D E Knuth (in the errata for the second edition and third edition of his "Fundamental Algorithms") gives the full name above and says it can be translated as Father of Abdullah, Mohammed, son of Moses, native of Khwarizm. He was an astromomer to the caliph at Baghdad (now in Iraq).

(*) Al-Khowârizmî is the region south and to the east of the Aral Sea around the town now called Khiva (or Urgench) on the Amu Darya river. It was part of the Silk Route, a major trading pathway between the East and Europe. In 1200 it was in Persia but today is in Uzbekistan, part of the former USSR, north of Iran, which gained its independence in 1991.
(*) Prof Don Knuth has a picture of a postage stamp issued by the USSR in 1983 to commemorate al-Khowârizmî 1200 year anniversary of his probable birth date.
(*) From the title of this book Kitab al jabr w'al-muqabala we derive our modern word algebra.
(*) The Persian author's name is commemorated in the word algorithm. It has changed over the years from an original European pronunciation and latinisation of algorism. Algorithms were known of before Al-Khowârizmî's writings, (for example, Euclid's Elements is full of algorithms for geometry, including one to find the greatest common divisor of two numbers called Euclid's algorithm today).
(*) The USA Library of Congress has a list of citations of Al-Khowârizmî and his works.