Archimedes was an ancient Greek mathematician, physicist, engineer, inventor, and astronomer who lived in Syracuse, a Greek city-state in Sicily, during the 3rd century BC. He is considered one of the greatest and most influential mathematicians of all time. Some key details about his life and work:


- Archimedes was born around 287 BC in Syracuse, and died in 212 BC during the Second Punic War when Syracuse was conquered by Roman general Marcus Claudius Marcellus. Archimedes was killed by a Roman soldier despite orders to capture him unharmed due to his valuable scientific knowledge.


- His father, Phidias, was an astronomer which likely influenced Archimedes' interest in mathematics and mechanics from a young age. Not much else is known about his family background or early life.


- Archimedes studied in Alexandria, Egypt which was a great center of scholarship and the foremost Greek city for mathematics and science at the time. He likely studied with successors of Euclid and learned a great deal of geometry and other mathematics while there.


- Once back in Syracuse, Archimedes devoted his life to research and made many groundbreaking discoveries in math and physics. However, he viewed practical engineering applications with as much interest as pure mathematical inquiries.


- His key contributions include calculating an approximate value for pi, developing a method for determining volumes and surface areas of solids from immersing them in fluids, formulating explanations for the principles of levers and pulleys, creating innovative machines of war to defend Syracuse, and making observations and theories about astronomy.


- The story goes that Archimedes discovered the concept of water displacement and buoyant forces when he stepped into a bath and noticed the water level rising. This inspired him to formulate the famous Archimedes Principle regarding buoyancy which proved very useful in his later work.


- Another famous legend is when the king asked Archimedes to determine whether a newly crafted crown was pure gold without damaging it. Weighing the crown was inconclusive, but while stepping into a bath, Archimedes supposedly realized he could submerge the crown and measure the precise amount of water displacement to calculate its density and composition. He was so excited that he immediately ran home naked yelling "Eureka!" (I have found it!).


- Archimedes made many important discoveries in geometry as well, establishing formulas for calculating the volumes of spheres, cylinders, pyramids and cone-shaped figures, as well as the surface areas of spheres and cylinders. He showed great ingenuity in enclosing solids between circular and triangular boundaries to better approximate their areas.


- He was also a pioneer in infinitesimals and limits, applying the method of exhaustion to effectively integrate areas and volumes by approximation through summing an infinite sequence of areas or volumes of enclosing shapes. This was a key step toward integral calculus.


- In physics, Archimedes elucidated the principles behind levers, pulleys and compound pulley systems. He analyzed balances and centers of gravity. One treatise entitled On Floating Bodies laid out what later came to be known as Archimedes' Principle regarding the mechanics of buoyancy.


- Fascinated by astronomy as well, Archimedes formulated theories to explain the movement of the planets which may have influenced Copernicus's later heliocentric model. He also studied the nature of spheres and circles related to astronomical phenomena.


- Apart from his brilliant mind as a theoretician, Archimedes also had a profoundly inventive and practical side. During the Second Punic War when Rome attacked Syracuse by sea, Archimedes developed a wide array of mechanical war machines such as claws that could grapple and overturn ships, crane-like devices to drop heavy weights onto ships, mirrors that could concentrate sunlight to set Roman ships aflame, catapults, and automated arrow launchers. These inventions held the Roman fleet at bay for 2 years.


- Only a few written works by Archimedes have survived antiquity - mainly mathematical treatises like On the Sphere and the Cylinder, Measurement of the Circle, On Conoids and Spheroids, On Spirals, On the Equilibrium of Planes, The Sand-Reckoner, and The Method of Mechanical Theorems. These show the depth and sophistication of his mathematical skill.


- Archimedes viewed applied mechanics with equal interest as pure geometry, stating: “Give me a place to stand on, and I will move the Earth." His inventions were created as much for intellectual stimulation as practical use.


- Plutarch wrote that Archimedes would become so absorbed in drawing geometrical figures in the ashes on the ground that his servants would have to snap him out of it just to get him to go bathe. This shows his utterly obsessive immersion in mathematics.


- Archimedes died during the conquest of Syracuse when he was killed by a Roman soldier - in spite of direct orders from General Marcellus to take Archimedes alive. Archimedes was working on mathematical diagrams in the sand when a soldier confronted him, and he angrily told the soldier not to disturb his circles. Infuriated, the soldier killed one of history's greatest geniuses.


- General Marcellus, an admirer of Greek culture, was outraged at Archimedes’ death and held an honorable funeral for the legendary mathematician. His tomb was adorned with a sculpture of a sphere inscribed in a cylinder - one of the key breakthroughs in proving the formulas for their volumes which Archimedes had achieved.


- Archimedes' voluminous works and knowledge were mostly lost during antiquity. Over time, his writings were scattered, copied in bits and pieces, or referenced in limited passages by ancient scholars. It took centuries for the legacy of Archimedes to gradually come back into focus during the Renaissance and beyond.


- Today, Archimedes' surviving works and the legends of his ingenious devices continue to inspire admiration for his supreme mathematical talents and almost superhuman favored insights. He pushed geometry, infinitesimals, mechanics and physics far beyond anything achieved before him.