The Byzantine Empire, also referred to as the Eastern Roman Empyre or Byzantium, was the continuation of the Roman Empire in its eastern provinces during Late Antiquity and the Middle Ages, that is from 330 to 1453 AD.

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Surviving the Fall of the West
The Eastern Roman Empire survived the fragmentation and fall of the Western Roman Empire in 476 AD and continued to exist for an additional thousand years until it fell to the Ottoman Turks in 1453. During most of its existence, the empire was the most powerful economic, cultural and military force in Europe. Both “Byzantine Empire” and “Eastern Roman Empire” are terms created after the end of the realm; its citizens continued to refer to their empire simply as the Roman Empire or Romania, and to themselves as “Romans”.

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Byzantine Architecture and its Characteristics
Byzantine architecture is a style of building that flourished under the rule of Roman Emperor Justinian. In addition to an extensive use of interior mosaics, its defining characteristic is a heightened dome, the result of the latest sixth-century engineering techniques. Byzantine architecture dominated the eastern half of the Roman Empire during the reign of Justinian the Great, but the influences spanned centuries, from 330 until the fall of Constantinople in 1453 and on into today’s church architecture.
Byzantine architecture blended Western and Middle Eastern architectural details and ways of doing things. Builders renounced the Classical Order in favour of columns with decorative impost blocks inspired by Middle Eastern designs. Mosaic decorations and narratives were common. For example, the mosaic image of Justinian in the Basilica of San Vitale in Ravenna, Italy, honours the Roman Christian Emperor.
The early Middle Ages was also a time of experimentation with building methods and materials. Clerestory windows became a popular way for natural light and ventilation to enter an otherwise dark and smoky building.

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Construction and Engineering Techniques
Byzantine builders experimented with different methods of construction. Art historian Hans Buchwald writes that “sophisticated methods for assuring structural solidity were developed, such as well-built deep foundations, wooden tie-rod systems in vaults, walls and foundations, and metal chains placed horizontally inside masonry”.
Byzantine engineers turned to the structural use of pendentives to elevate domes to new heights. With this technique, a dome can rise from the top of a vertical cylinder, like a silo, giving height to the dome. Like the Church of Hagia Irene, the exterior of the Church of San Vitale in Ravenna, Italy is characterized by the silo-like pendentive construction. A good example of pendentives seen from the inside is the interior of the Church of Hagia Sophia in Istanbul, one of the most famous Byzantine structures in the world.

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The Extent of the Empire
With its capital founded at Constantinople by Constantine I (r. 306-337 CE), the Empire varied in size over the centuries, at one time or another, possessing territories located in Italy, Greece, the Balkans, Levant, Asia Minor and North Africa. A Christian state with Greek as the official language, the Byzantines developed their own political systems, religious practices, art and architecture, which, although significantly influenced by the Greco-Roman cultural tradition, were distinct and not merely a continuation of ancient Rome. The Byzantine Empire was the longest-lasting medieval power, and its influence continues today, especially in the religion, art, architecture, and law of many Western states, Russia, Eastern and Central Europe.

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Economic and Social Policies
The empire’s economy had prospered in a spotty fashion. Certain provinces, or parts of provinces such as northern Italy, flourished commercially as well as agriculturally. Constantinople, in particular, influenced urban growth and the exploitation of agricultural frontiers. Balkan towns along the roads leading to the great city prospered, while others not so favoured languished and even disappeared. For instance, untilled land in the hilly regions of northern Syria fell under the plow to supply foodstuffs for the masses of Constantinople and western Africa territories were spoiled to provide gold to the Empire.

Image source: https://en.wikipedia.org/wiki/Roman_Syria
Relations with the Barbarians
Differences between Eastern and Western social structures, together with certain geographical features, account for the different reception found by the Germanic invaders of the 4th and 5th centuries in the East and the West. The initial interaction between Roman and barbarians was far from amicable; the Romans seemed to have exploited their unwelcome guests, and the Goths rose in anger, defeating an East Roman army at Adrianople in 378 and killing the Eastern emperor in command. Emperor Theodosius I (ruled 384–395) adopted a different policy, granting the Goths lands and according them the legal status of allies, who fought within the ranks of the Roman armies as autonomous units under their own leaders.

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Info sources: https://www.ancient.eu/Byzantine_Empire/
https://en.wikipedia.org/wiki/Byzantine_Empire
https://www.britannica.com/place/Byzantine-Empire
https://www.thoughtco.com/what-is-byzantine-architecture-4122211