About golden ratio

The Golden Ratio – a sacred number that links the past to the present

There is one thing that ancient Greeks, Renaissance artists, a 17th century astronomer and 21st century architects all have in common – they all used the Golden Mean, otherwise known as the Golden Ratio, Divine Proportion, or Golden Section.  Precisely, this is the number 1.61803399, represented by the Greek letter Phi, and considered truly unique in its mathematical properties, its prevalence throughout nature, and its ability to achieve a perfect aesthetic composition.

According to astrophysicist Mario Livio:

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

In mathematics and the arts, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. When the Golden Mean is conceptualised in two dimensions it is typically presented as a regular spiral that is defined by a series of squares and arcs, each forming “Golden Rectangles”.

This symbolic potential arises because of the way the mean’s spiral shape resembles growth patterns observed in nature and its proportions are reminiscent of those in human bodies. Thus, these simple spirals and rectangles, which served to suggest the presence of a universal order underlying the world, were thereby dubbed “golden” or “divine”.

The Golden Ratio in History

The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. The earliest known monuments believed to have been built according to this alluring number are the statues of the Parthenon in Greece, dating back between 490 and 430 BC.  However, there are many who have argued that it goes back much further than this and that the Egyptians were well versed in the properties of this unique number.

According to some historians, the Egyptians thought that the golden ratio was sacred.  Therefore, it was very important in their religion.  They used the golden ratio when building temples and places for the dead.  In addition, the Egyptians found the golden ratio to be pleasing to the eye.  They used it in their system of writing and in the arrangement of their temples.  The Egyptians were aware that they were using the golden ratio, but they called it the «sacred ratio.»

The first recorded definition of the golden ratio dates back to the period when Greek mathematician, Euclid (c. 325–c. 265 BC), described what he called the «extreme and mean ratio». However, the ratio’s unique properties became popularised in the 15th century when aesthetics were a vital component of Renaissance art and geometry served both practical and symbolic purposes.  As the famous mathematician, astronomer, and astrologer, Johannes Kepler (1571 – 1630) wrote:

Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel.

The Golden Ratio in Architecture

Many artists and architects have proportioned their work to approximate the golden ratio, with the belief that the outcome will be more aesthetically pleasing.  Using any of these ratios, an architect can design a door handle that has a complementary relationship to its door, which in turn has a similar relationship to its enclosing wall, and so on.  But more than this, the golden ratio has been used for the façade of great buildings from the Parthenon to the Great Mosque of Kairouan and all the way through to modern landmarks such as the Sydney Opera House and the National Gallery in London.

The Golden Ratio in Nature

Perhaps what is most surprising about the Golden Ratio is that it can be seen as a naturally occurring phenomenon in nature.   The golden ratio is expressed in the arrangement of branches along the stems of plants and the veins in leaves.   It can be seen in the skeletons of animals and humans and the branching of their veins and nerves.  It can even be seen in the proportions of chemical compounds and the geometry of crystals. Essentially, it is all around us and within us and for this reason, German psychologist Adolf Zeising (1810 – 1876) labelled it a ‘universal law’:

in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.

As a result of the unique properties of this golden proportion, many view the ratio as sacred or divine and as a door to a deeper understanding of beauty and spirituality in life, unveiling a hidden harmony or connectedness in so much of what we see.

By April Holloway

– See more at: http://www.ancient-origins.net/unexplained-phenomena/golden-ratio-sacred-number-links-past-present-001091#sthash.wE7t6PVy.dpuf

Poem: Alexander the Great By Christa Wehner Radeburg

article-2401057-0D5D720400000578-514_306x423

It was as if the dawn broke forth in song

as Alexander lay concealed in Olympias’ womb.

The forests of Macedonia rustled in accord,

surrounding Pella’s grove at Phillip’s castle.

 

The young prince saddled his horse and rode

toward the East,

transforming the world.

Bucephalus strained,

Bucephalus stamped,

Bucephalus struggled

against the youth’s strength.

 

The Iliad ran through his dreams,

engrossed … soon he stood upon the hill of Troy …

Achilles was near.

We remember Issus Battle, Darius’ death,

the battlefield at Gaugamela,

Persepolis reduced to ruins…

Still Susa’s fountains add their melodious melody

to the joyous sound of strings and wedding dance,

as Orient’s splendor with the legion was entwined.

 

The young man’s path led on to the heights of Hindu Kush

where battle he did win, but lost Bucephalus.

 

Babylon, Babylon, Babylon cries silently

for a young son of Greece,

the friend of his youth, Hephaestion,

Iskander even…

 

A king’s golden throne is shattered,

scattering stardust through the night for light.

 

For centuries the desert sands remain aglow,

borne within a mind as old as time.

Twilight falls, spreading its glow over the Euphrates,

from whose depths Hellas’ legacy emerges,

harbored in the dawn as it breaks forth in song..

Alexander the Great – Pakistan

If you want to follow in the footsteps of Alexander the Great in Pakistan and build up a sort of travelogue. I have marked in red on the map below Alexander’s journey through what is now Pakistan. It has numbers on it to show you where we are in Pakistan. To remove any ambiguity I shall limit my journey within Pakistan only.

Of course the journey will begin on the Afghan-Pak border because Alexander with his armies marched in from what is now Afghanistan. It will end near the Arabian Sea on the Iran-Pakistan border where the route leaves Pakistan and enters Iran. This is almost the exact route Alexander took. To prevent any possibility of insult to anybodies version of history I will use terms like ‘Native’ or ‘South Asian’ when talking about the region.

I would ask please that this thread is not derailed. Please read the title of the thread. Can we please stay with the ambit of this thread which will be a journey through Pakistan following the footsteps of the mighty Alexander.

I would be grateful if people can contribute with photo’s or maps but onlyu as long as they avoid the tricky subject of Pak-India sensibilities by employing neutral nomenclature like as described – native or South Asian.

I begin the journey on the Khyber Pass through which runs the main road from Kabul, Afghanistan into Pakistan. As he his army through the Khyber Pass his men had entered what is now Pakistan past what is now the Afghan-Pak Border Checkpoint where today stands the Bab R Khyber translated means Khyber Gate in English.

Click the image to open in full size.

On the Map above the red line represent approximate route followed by Alexander and his armies. Khyber Pass is marked No. 1 red star on the map.

As we go along the road which snakes down the pass the fertile Peshawar valley would have become visible which then was dominated by the former Persian Satrapy of Gandhara. This area today is source of lot of trouble in Pakistan as effects of the Afghan war spill over and suffers from regular Taliban attacks. All those centuries ago Alexander would not have known that another mighty army from the West – NATO would arrive in this part of the world.

This road that comes from the city of Peshawar is main supply route for NATO and is full of trucks carrying supplies for NATO. Below the Babe Khyber border between Afghan-Pak that Alexander would have marched.

Alexander would been subjected to continous trouble from the local tribes even today are source of so much trouble. As we move deeper into Pakistan. The local cheiftain from Taxila marked green No.2 on the map ( which is only about 15 miles from the present Pakistani capital Islamabad met Alexander and effectively surrundered his territory. The hill tribes to the north still refused but Alexander and his Greeks marched victorious into Taxila a magnificient city at that time adjacent to modern capital of Pakistan Islamabad.

Alexander subsequently had to go up north into the wild mountains each with hidden valleys to subdue the tribes who refused to bow to his power. This area of Pakistan to this day has people who display extreme western features and the common myth ( true or not ) is they are the decendants of the Greeks although I have seen blue and blond hair here which is very rare in Greece. In Pakistan entire population groups within certain areas display extreme features. Most people of this province look Mediteranean although some as below are almost north western European.

Click the image to open in full size.

Click the image to open in full size.

Click the image to open in full size.

We will move on Taxila next. For now can I request people from across the border not to destroy this thread please. I am tired of my country hitting the headlines for the wrong reason. I want to show people this ancient land with a history going back 5,000 years plus. Is there anything wrong with this ? Thank you. Will be back for the next part of the journey through Pakistan.

Taxila which I have marked as No 2 Green on the map is adjacent to the Pakistan capital of Islamabad. This is where Alexander was recieved. This area would subsequently come under the rule of Greek Kingdom. Taxila today is in ruins like Parthenon but it is rich in history. Regularly Greek coins are found which were minted in the Greek era. There is Taxila Musuem which houses large amount of Greek era artifects including coins. Unfortunatly lot of smuggling goes on and precious heritage is sold to rich outsiders to be kept in private collections.

Click the image to open in full size.

Click the image to open in full size.

Click the image to open in full size.

Click the image to open in full size.

Click the image to open in full size.

Double headed Eagle shrine, Taxila, Pakistan showing Greek influence around about 2nd BC. About 15 miles west of Pakistan capital Islamabad.

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Pakistan Paedia – Archaeological Treasure of Pakistan (Taxila)

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Double headed Eagle shrine, Taxila, Pakistan showing Greek influence around about 2nd BC. About 15 miles west of Pakistan capital Islamabad.

Click the image to open in full size.

Pakistan Paedia – Archaeological Treasure of Pakistan (Taxila)

After Taxila we leave Khyber Pakhtunwa province ( where the people speak a Persian related language called Pashto ) and pass the federal capital Islamabad on toward the plians of Punjab province. On the map we are moving east and the land starts dropping.

This area is called the Salt Range and the colours are amazing. At times the eroded hills look like mars. As the road cuts through the hills we start droppind down toward the Indus Plain.

Click the image to open in full size.
Click the image to open in full size.

Click the image to open in full size.
As you drive through this extremly difficult terain you realise how difficult it must have been for Alexander’s army who marched through along this very route. Further down we drop into the plains we hit the a major river which is now known as Jhelum river but to the Greeks it was known as the Hydaspes. Near the river is the modern town of Jhelum in Punjab Province.

It was here on the banks of Hydaspes – marked No 3 blue on the map that a major battle was fought on the banks of Hydaspes or as it is called now River Jhelu

Anyway we are now on No 3 marked blue on map. On the banks of Jhelum river ( to Greeks the Hydaspes ) Alexander had one of the tough battles he had faced against the ruler of Punjab – Porus. Although the Greeks won but the Punjabi’s under Porus fought valiantly but lost to Alexander’s skill.

However Alexander’s horse Bucephalus was killed. Alexander subsequently set up two towns in the vicinty called Alexandia Bucephalous in honour of his horse.

Near where the Battle of Hydaspes was fought today a monument has been built called the Alexandria Monument.

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Click the image to open in full size.

Foundation stone laid by the Greek Ambassador to Pakistan 1997.

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From Greece to Pakistan – Alexander.

In addition Alexander set up another town in the area called Nicea which possiblity might be the Pakistani town of Mong not to far from this place. After this Alexander moved east through the flat plains of the Indus valley in today’s Punjab, Pakistan generally heading toward Lahore city.

http://historum.com

The fantastical beasts of ancient Greece

Alastair Sooke

Le mystérieux tombeau d’Amphipolis avait plusieurs occupants

Ce n’est pas un, mais au moins cinq défunts qui gisaient dans le gigantesque tumulus situé dans le nord de la Grèce : deux hommes, une femme et un bébé.
Voici certains des ossements découverts dans le tombeau d'Amphipolis, présentés sur une esquisse de la chambre funéraire.
Voici certains des ossements découverts dans le tombeau d’Amphipolis, présentés sur une esquisse de la chambre funéraire. © Ministère grec de la Culture
(avec AFP)

Le mystérieux tombeau d’Amphipolis n’est pas celui d’un seul individu ! C’est ce qu’a récemment révélé un premier examen des ossements qui y ont été découverts. Les restes trouvés dans le plus grand tombeau jamais fouillé en Grèce appartiennent à «au moins» cinq personnes dont une femme d’une soixantaine d’années, deux hommes plus jeunes et un nouveau-né. «Le nombre minimum de squelettes identifiés est de cinq, dont quatre ont été enterrés et l’un incinéré», a indiqué le ministère grec de la Culture.

Toutefois, cette analyse préliminaire des quelque 550 ossements découverts dans ce tombeau, situé dans le nord de la Grèce près de Thessalonique, ne résout toujours pas l’énigme de l’identité des dépouilles qui font l’objet de nombreuses spéculations. Puisque le monument funéraire, manifestement bâti pour accueillir un ou des personnages importants, date de l’époque d’Alexandre le Grand (356-323 av. J.-C.) dont la tombe, réputée se trouver en Égypte à Alexandrie, n’a cependant jamais été retrouvée. Depuis le début de ces fouilles, différentes hypothèses sur les personnes inhumées circulent : Roxane, l’épouse perse d’Alexandre le Grand, Olympias, la mère du roi, ou un de ses compagnons et généraux. Les chances que le tombeau soit celui d’Alexandre lui-même paraissent elles quasi nulles.

Test ADN

Au terme d’une progression des archéologues vers le coeur du tumulus, le ministère de la Culture avait fait état en novembre de la découverte «d’un» squelette. Après examen, ce sont finalement cinq personnes au moins qui reposent dans le monument. Un seul crâne a été découvert, «en assez bon état». Il appartient «avec certitude» à la dépouille d’une femme «âgée de plus de 60 ans», dont la taille estimée est de «1,57 mètre». Les restes de deux hommes âgés «de 35 à 45 ans», mesurant 1,68 m et 1,62 m ou 1,63 m, ont également été reconstitués. L’un d’eux portait des traces d’une blessure infligée par un objet tranchant. Les deux autres dépouilles sont celles d’un «nouveau-né» et «probablement d’un adulte», les os de ce dernier étant les seuls à présenter des traces d’incinération.

«Ces annonces posent plus de questions qu’elles n’apportent de réponse», estime Michalis Tiverios, professeur d’archéologie à l’université de Thessalonique. À présent, des comparaisons d’ADN devraient être menées entre les trois dépouilles suffisamment complètes pour permettre ce type de tests. L’objectif sera de déterminer s’il existe entre eux un possible lien de parenté. D’autres analyses essaieront également de déterminer le type d’alimentation des défunts ou l’existence d’éventuelles pathologies. Mais une chose est sûre, la disposition des ossements indique qu’ils ont été bougés. Rien de très surprenant puisque les précédentes découvertes faites dans le tombeau indiquaient clairement que l’édifice avait été pillé, probablement dans un lointain passé. Très peu de reliques précieuses ont d’ailleurs été trouvées dans les chambres funéraires.

http://www.lepoint.fr