شاید باور نکنید اما بسیاری از طراحان و
معماران بزرگ برای طراحی محصولات خود
امروز از این نسبت طلایی استفاده می کنند. چرا که بنظر میرسد ذهن انسان با این نسبت انس دارد و راحت تر آنرا می پذیرد.
این نسبت نه تنها توسط معماران و مهندسان
برای طراحی استفاده می شود. بلکه در طبیعت نیز
کاربردهای بسیاری دارد.
برش اهرام و نسبت طلایی اهرام مصر یکی از قدیمی ترین ساخته های بشری است که در آن هندسه و ریاضیات بکار
رفته شده است. مجموعه اهرام Giza در مصر که قدمت آنها به بیش از 2500
سال پیش از میلاد می رسد یکی از شاهکارهای بشری
است که در آن نسبت طلایی بکار رفته است. به این شکل نگاه کنید که در آن بزرگترین هرم از مجموعه اهرام Giza خیلی ساده کشیده شده است.
مثلث قائم الزاویه ای که با
نسبت های این هرم شکل گرفته شده باشد به مثلث قائم مصری یا Egyptian Triangle معروف هست و جالب اینجاست که بدانید نسبت وتر به ضلع هم کف هرم معادل با نسبت طلایی یعنی دقیقا" 1.61804 می
باشد. این نسبت با عدد طلایی تنها
در رقم پنجم اعشار اختلاف دارد یعنی چیزی حدود یک صد هزارم. باز توجه شما را به این نکته جلب می کنیم که اگر
معادله فیثاغورث را برای
این مثلث قائم الزاویه بنویسم به معادله ای مانند phi2=phi+b2 خواهیم رسید که حاصل جواب آن همان عدد معروف طلایی خواهد
بود. (معمولا" عدد طلایی را با phi نمایش
می دهند. ) طول وتر برای هرم واقعی حدود
356 متر و طول ضلع مربع قاعده حدودا" معادل 440 متر می باشد بنابر این نسبت 356 بر 220 (معادل نیم
ضلع مربع) برابر با عدد 1.618 خواهد شد.
عدد طلائی از دیدگاه کپلر
کپلر (Johannes Kepler 1571-1630) منجم معروف نیز علاقه بسیاری به نسبت طلایی داشت بگونه ای که در یکی از کتابهای خود اینگونه نوشت :
"هندسه دارای دو گنج بسیار با
اهمیت می باشد که یکی از آنها قضیه فیثاغورث و دومی
رابطه تقسیم یک پاره خط با نسبت طلایی می باشد. اولین گنج را می توان به طلا و دومی را به جواهر تشبیه کرد".
تحقیقاتی که کپلر راجع به مثلثی که اضلاع آن به
نسبت اضلاع مثلث مصری باشد به حدی بود که
امروزه این مثلث به مثلث کپلر نیز معروف می باشد.همچنین
کپلر پی به روابط بسیار زیبایی میان اجرام آسمانی و این نسبت طلایی پیدا کرد.
همچنین ببینید:
پیوندهای خارجی
ستاره پنج‌پر یا پنتاگرام شکلی هندسی است که
از پنج خط تشکیل شده و به شکل ستاره است.
کلمهٔ «پنتاگرام» از کلمهٔ یونانی
πεντάγραμμον منشا گرفته‌است که به معنی پنج
خط است.
پنتاگرم به عنوان سمبلی ارزشمند در یونان و
بابل قدیم مورد استفاده قرار میگرفته،
پنتاگرم دارای گرایشات جادویی نیز می‌باشد. بسیاری از کسانی که پیرو نوپگانیسم یا مهرپرستی هستند از این سمبل استفاده
می‌کنند.
پنتاگرم ید طولایی در رابطه با سیاره ونوس
(زهره)♀ دارد و آن را سمبل این
سیاره می‌شمارند، سیاره زهره نماد مادینه مقدس است که در اسلام آن را فاطمه در مسیحیت آن را مریم مجدلیه (که بعد از افشا گری‌ها
کتاب راز داوینچی اثر دن براون بیشتر
مرسوم شد) و در ایران باستان آناهیتا (که بعضی به
صورت کلی آن را ننه یا ننه خاتون نیز مینامند)، و همچنین رابطه
مستقیمی با فراماسونی‌ها یا شوالیه‌های
معبد (Knights Templar) دارد که
آنها نگه دارنده راز جام مقدس هستند)
و همچنین سیاره ونوس هر هشت سال شکل ستاره پنج پر
را می‌پیماید یعنی چرخشش چنین شکلی را به وجو می‌آورد.
اولین استفاده از ستاره پنج پر به پنج هزار
سال پیش در بین‌النهرین در نزد
سومریان بر میگردد. سومریان آن را به عنوان شکل تصویر
(Pictogram) کلمه UB به معنی گوشه به کار میبردند. در شکل تصویر سومریان ستاره
پنج پر در شماره ۳۰۶
بود، پنج گوشه ستاره پنج پر به معنی جوپیتر، عطارد، مریخ♂، ساترن، و زهره♀ است به معنی ملکه بهشت
(Ishtar).
ادیان
در قرون وسطا مسیحی‌ها ستاره پنج پر را به
عنوان طلسم زد شیطان و عجوزه‌ها به
کار میبردند ستاره پنج پر در زمانی نشانه شهر اورشلیم بوده‌است همچنین شیطان پرست‌ها نیز آن را با یک بز که ریشش
در ضلع پایین گوش‌هایش در دو ضلع وسط و
شاخش در دو ضلع بالا بود به کار میبرده‌اند.در نو
پگانیسم نیز به عنوان سمبل اصلی شناخته شده‌است و به عنوان نشانه‌ای
از پنج عنصر خاک،آب،باد،آتش و
روح است.
هندسه
φ = (۱+√۵)/۲ ≈ ۱٫۶۱۸ نسبت طلایی: نسبت طلایی گاهی نیز نسبت الهی شناخته میشود که همان عدد فی است که جزو دنباله‌های
فیبوناچی است۱٫۶۱۸. اگر تعداد زنبور‌های ماده را بر زنبور‌های نر در هر کندویی
در هر کجای دنیا بر هم تقسیم کنیم نسبت
الهی به دست می‌آید، ناتیلوس یک نرم تن است که نسبت
قطر هر مارپیچ نسبت به بعدی در صدفش برابر با عدد فی است، مارپیچ گل برگ‌های مخروط کاج، آرایش برگ‌ها روی ساقه گیاهان، بند‌های
حشرات همگی فرمانبرداری از نسبت الهی
هستند، همچنین فاصله سر انسان تا زمین تقسیم بر فاصله
شکم تا زمین نسبت الهی را به دست می‌آورد، فاصله شانه‌ها تا نوک انگشت بر فاصله آرنج تا نوک انگشت نیز نسبت الهی را حاصل
میشود و همچنین مثال‌ها دیگری
مانند: باسن تا زمین بر زانو تا زمین و همچنین تقسیمات ستون فقرات و ... همگی عدد فی را حاصل می‌شوند.
نوترینو0:
در آرایش دانه های آفتابگردان ، منحنی جدایش
معمولا از یکسو یک عدد فیبوناچی و از
سوی دیگر عدد دیگر فیبوناچی و نسبت آنها طلایی است. البته این آرایش بخاطر اینست که تعداد بیشتری دانه در واحد سطح
کنار یکدیگر قرار میگیرند و
مسلما بعد از تکثیر چند نسل ، نسلی بیشتر میشود که دانه های بیشتری تولید کرده باشد. یعنی این نسبت بخاطر آنست که
بخاطر تعداد دانه های بیشتر این
نوع آفتابگردان بیشتر تکثیر شده و ما اکنون آنرا میبینیم و ممکن است از دیدن این نسبت عجیب شگفتزده شویم.
*****
|
مطلبی که در زیر می آید بخشی از مقاله ای است که در آغاز این نوشته
یاداور شدم " فرداي آن روز كه در بخش داكومنت كامپيوترم به
دنبال چيزي مي‌گشتم چشمم به مقاله‌اي به زبان انگليسي افتاد كه مدتي پيش
و به هنگام جست‌وجوي شكل حلزوني در اينترنت آن را ديده و از آن يك كپي
گرفته‌بودم تا بعد آن را بخوانم ولي فراموشش كرده بودم."
Read Chapter 8 from:
Esoteric
Anatomy: The Body as Consciousness
by Bruce Burger, MA
We invite you to contemplate an ancient vision of the universe as a
living, breathing, conscious Being . . . and the ubiquitous Golden Spiral as the
mathematical expression of the life breath of a living universe.
Chapter 8
The Golden
Spiral:
a Key To Understanding Energy in Nature
All creation is the interweaving of cycles. From Galactic
manifestation to subatomic waves, the universe is a vast spectrum of
cycles. The cycles of birth and death, summer and winter, day and
night, in-breath and out-breath weave the fabric of life. The ancient
rishis (Yogi's who purified their body/minds and directly
experienced the fundamental forces of creation) experienced the
underlying unity of all cycles as the breath of Brahma and the
ubiquitous periodicity of the universe as the rhythm of the life
breath of a single harmonious Living Being.
All bodies in nature or organized around the
proportional harmonics of the rhythms of the Earth's breath.
There is a profound body of scientific evidence that points to the
fact that the universe is a single harmonious system. A key to
understanding this unity is found in an aspect of natural law known
variously as the “Divine Proportion,” “Golden Section,” “Golden
Ratio,” or Golden Spiral.”
Bruce Rawles explains:
The Divine Proportion was
closely studied by the Greek sculptor, Phidias, hence, it was given
the name Phi. Also known as the Golden Mean, the Magic Ratio, the
Fibonacci Series, etc., Phi can be found throughout the universe;
from the spirals of galaxies to the spiral of a Nautilus seashell;
from the harmony of music to the beauty in art. A botanist will find
it in the growth patterns of flowers and plants, while the zoologist
sees it in the breeding of rabbits. The entomologist views it in the
genealogy of a bee, and the physicist observes it in the behavior of
light and atoms. A Wall
Street analyst finds it in the rising
and falling patterns of a market, the mathematician in the
examination of the pentagram. . . . The ancient Egyptians used it in
the construction of the great pyramids and in the design of hieroglyphs
found on tomb walls . . . . Plato in his Timaeus considered it the
most binding of all mathematical relations and makes it the key to
the physics of the cosmos.1
The Golden Spiral can be described mathematically through a principle
known in the West as the Fibonacci progression which is named after
an Italian mathematician Leonardo Fibonacci da Pisa. Fibonacci, the
father of western mathematics, learned this principle from traders of
the Aryan civilizations of Asia. The
Fibonacci progression is a mathematical sequence that is produced by
starting with 1 and adding the last two numbers in the progression to
arrive at the next. The Fibonacci sequence begins: 1, 1, 2, 3, 5, 8,
13, 21, 34, 55, 89, 144, . . . Each number is the sum of the previous
two numbers.
The Fibonacci numbers, and especially ratios of two successive
Fibonacci numbers, show up extensively in nature. For example, the
petals of a Monterey
pine cone are arranged in spirals crossing in both directions: eight
spirals in one direction and thirteen in the other. Similar patterns
arise in the seeds of sunflowers and in other plants whose leaves
grow in a spiral around a central stem; each successive leaf may be
on the opposite side (1/2 way around) or may be 2/3 of the way
around, or 3/5, etc.2
The ratio of any two successive Fibonacci numbers from three on is
about 1:1.618. This ratio occurs ubiquitously throughout nature, in
logarithmic spirals that underlie the process of growth.
The Golden Spiral describes the radiation of energy from a center.
In all natural processes energy radiates from a center in the
logarithmic proportions of the Golden Spiral. One can witness the
Golden Spiral in the curve of an elephant’s tusk, the horns of wild
sheep, the curve of a canary’s claw, the spiral in a pineapple or
daisy. The spiral of your fingerprint or curl in your eye lash follow
the Divine Proportion. The planets of our solar system radiate from
our Sun, and galaxies manifest following the rhythms of the Golden
Spiral. The Fibonacci progression describes the law that underlies
the radiation of energy in nature. “It governs, for example, the laws
involved with the multiple reflections of light through mirrors, as
well as the rhythmic laws of gains and losses in the radiation of
energy.”3
We find
the same proportional harmonics in the average distance of the
planets from the sun as the chambers of a nautilus.
The breath of Brahma sustains the harmony of the Golden Spiral. The
source of all vibration is the word of God. The song of God animates
the cosmic dance of Shiva and Shakti, centrifugal and centripetal
currents that undulate through every spiraling atom of creation The
vibration of these spiraling yang and yin currents weaves the fabric
of creation. David Tame in his valuable book, The Secret Power of
Music, in a section titled “The vocal range of the one Singer,”
points out the unity of the electromagnetic spectrum:
Not only “solid” matter but all forms of energy, are composed of
waves: which is to say, vibrations. All of the different kinds of
electromagnetic energy—including radio waves, heat, X rays, cosmic
rays, visible light, infra-red, and ultraviolet—are composed of wave
like or vibratory activity, these vibrations traveling through the
universe at 186,000 miles per second. The only difference between
each of these phenomena is their frequency of vibration or
wavelength. Each merges into the other at a certain wavelength: which
obviously means, when one gets down to it, they are each one and the
same thing.4
All vibration is aligned with the harmony of the electromagnetic
spectrum. The mathematics that describe the radiation of energy
throughout the vast spectrum of vibration of the universe is the
Golden Spiral. Every level of macrocosm and microcosm of vibration is
attuned to the harmony of the one singer.
(Picture this spiral three-dimensionally as a spiraling vortex,
undulating in both directions, centrifugally, and centripetally
simultaneously at the speed of life.)7
The ancient rishis (seers)
purified their bodies and concentrated their minds. They communed
with the intelligence of the cosmic being through prayer and
devotion. They actualized micro clairvoyant powers (siddhi) in which
they directly experienced the fundamental forces of creation. They
witnessed “spirals, within, spirals, within spirals . . . ” as they
described the vibratory nature of energy. They described the gunas,
the fundamental modes of nature, which they likened to a rope spun of
three threads. Centrifugal and centripetal spiraling around an
unchanging core. Spiraling through all vibration are a balance of
centrifugal and centripetal currents fractally emanating from an
infinite core. These forces radiate from the core, universally, in
the proportional harmonics defined by the Golden Spiral.5
Like wheels within wheels, fields undulate within fields in the
universal electromagnetic process of centrifugal and
centripetal forces.
All energy in
creation vibrates as a Logarithmic Spiral.6
Undulating through every spiral of periodicity in creation is the
profound balance of the universal breath of ascending and descending
currents of yang and yin, expansion and contraction, Fire and Water.
Energy fields in nature are organized following the proportional
harmonics of the Golden Spiral
Understanding this undulating “dance of Shiva” is a key to the
vibratory basis of creation. The infinity archetype (∞ ) embodies the
key to this universal process in which balanced opposites sustain a
unity. In this case the opposites are the balance of centrifugal and centripetal
forces whose logarithmic undulations fractally weave the fabric of
creation. Following the harmonic proportions defined by the Golden
Spiral, the opposite undulating forces weave a pattern of seven
intervals within eight phases. Esoterically this is called The Law
of Seven.
The Cosmic Octave: The Law of Seven
On every level of macrocosm and microcosm, vibration spirals as the
cosmic octave. An octave describes a process with eight steps and
seven intervals. Seven is a fundamental level of organization in the
universe. The seven whole notes of the tones of the musical scale are
a natural measure of the harmony of universal law. The musical octave
is so “natural” that it defines the proportional distances between
the planets in the solar system! The seven colors of the rainbow
define the spectrum of light.
The physicist Hermann von Helmholtz devised a correspondence between
the visible spectrum and the musical scale.8
Musical Notes
Colors
G
Ultraviolet
F, F#
Violet
E
Indiago Blue
D#
Cyanogen Blue
D
Greenish Blue
C#
Green
C
Yellow
A#
Orange-red
G#, A Red
G
Infrared
Tame reveals:
Nature herself also
indicates the close link between sound and light, the solar spectrum
of colors displaying a number of the properties of tones. The
resemblance is just as though the one phenomenon—light—were a higher
state of the other. Just as audible tone organizes itself naturally
into the seven notes of the diatonic scale, so too does the visible
solar spectrum form the seven colors of the rainbow.9
In
Sanskrit the letters of the alphabet that reflect the spectrum of vibration
underlying creation are called varnas, or colors. Sanskrit letters
varnas coordinate with universal basic principles that build and
unfold creation.
Out of the one white light
radiate the solar spectrum of the seven colors of the rainbow.
Tradition holds that all of the universe is formed by and thus
mirrors the seven major (and five minor) cosmic tones.
All growth in nature is ruled by the lunar cycle. “From one cell to
two there is a cycle of change in eight phases with seven intervals,
analogous to the musical octave, or the spectrum of light . . . The
lunar month, a perfect example of graduated phases within a
continuous process, is dominated by seven and its multiples.”10 The
twenty-eight days of the lunar cycle, of the two fortnights of the waxing
and waning Moon, map a cycle of four seven-day cycles and
twenty-eight graduated steps that underlie all biological processes.
The cycle spiraling through the lunar month of four-times-seven is
fundamental to all growth. Cells divide in an octavian process.
The planets spiral out from the nucleus of the solar system following
what in astronomy is called Bodes law, in which the distance between
the planets follows the Fibonacci progression. The planets of the
solar system follow the proportional harmonics of the Golden Spiral
and parallel our earthly musical scale. The seven days of the week is
a “natural process” of the rhythms of the cosmos. Sun-day, Moon-day,
Mars-day, Mercur-day, Thors-day (Jupiter), Fri-day (Venus), and
Saturn-day reflect the same cosmic octave.
The candelabra of seven lights (menorah) held as a sacred symbol by
the Jews, offers a model of the universe. The Sun is represented by
the center light, Mars and Venus, Mercury and Earth, Saturn and
Jupiter are represented in the polarity of candles on each side.
Seven is a fundamental number in nature. There are seven rows of
stable elements in the periodic table of the elements, which is
understood to be the scientific basis of matter. Indeed in
Leadbetter’s Occult Chemistry, the periodic table is portrayed as a
Golden Spiral. There are seven types of crystal systems—cubic,
rhombohedral, hexagonal, triclinic, monoclinic, trigonal, and
orthorhombic—the crystallization within the mineral kingdom of the
seven elemental harmonics of creation. There are seven major hormonal
glands in the human body and seven ventricles or cavities of the
skull.11
" Looking down from
above, from the perspective of the nucleus of our energy system the
Sun, we see that the distribution of leaves marking the growth of
plants follows the spiraling Fibonacci progression: 3 leaves in 5
turns, 5 leaves in 8 turns."
In Sacred Geometry, Robert Lawler asserts:
“Numerous studies document
the ubiquity of the Golden Ratio and the Fibonacci Series, which
describe the rhythmic laws of the gains and losses in the radiation
of energy.”13 For example, Professor Amstutz of the Mineralogical
Institute at the University
of Heidelberg
maintains: “Matter’s latticed waves are spaced at intervals
corresponding to the frets on a harp or guitar with analogous
sequences of overtones arising from each fundamental. The science of
musical harmony is in these terms practically identical with the
science of crystals.”14 Wilfried Krüger, in his book Das Universium
Singt (The Universe Sings) combines a knowledge of musical theory and
atomic physics. He demonstrates that the structure of atoms contain
ratios and numbers that parallel the harmonious principles of music.
Through a wealth of highly detailed and painstakingly prepared notes
and diagrams he demonstrates that it is impossible for chance alone
to account for the consistently musical patterns of the song of the
universe.15
The ancient cultures
understood the profound harmony of the universe. Their science was
based on this fundamental aspect of universal law. Universally,
energy emanates from a center in a “step-down” process that follows
the Golden Spiral of the Divine Proportion. Seven fields of force
radiate from a center, in a quantum step-down of graduated phases,
within a continuous process. The first two steps are the inner
harmonics of the causal and mental plane; the next five steps are the
elemental harmonics of resonance that underlie matter: Ether, Air,
Fire, Water, and Earth. The Law of Seven is a fundamental principle
in the theomorphic perspective.
"Theomorphic" step down of
energy. All forms in Nature embody the process of creation!
Universally, energy currents
spiral in an octave of eight nodes that sustain seven steps. The
seven chakras where idea and pingala undulate around sushumna
personify this cosmic process.
The Golden Ratio and the logarithmic succession of the Fibonacci
progression are fundamental expressions of universal law. The Golden
Spiral is a key to looking beyond the physical to the unified field
that is the substratum underlying manifestation. All physical
phenomena exist in alignment with this underlying field. The
fundamental features, structure, and function of all organisms are
reflections of the fact that they are subsystems within the larger
energy systems of nature and the cosmos from which they derive their
energy and being.
