Curved space is any space that isn’t completely flat, with examples being a sphere and a saddle. Gravity is caused by the curvature of space, with mass curving space and forcing objects to come together. The Pythagorean theorem is used to test whether space is flat or curved. Euclidean geometry is the study of shapes in flat space, while curved space plays an important role in modern astronomy. Non-Euclidean geometry can be used to predict Mercury’s orbit and true Euclidean forms only exist in spaces away from any gravitational body.
Any space that isn’t completely flat is called a curved space. The surface of a sphere is a curved space, as is the surface of a saddle. A sphere is an example of positive curvature, which means that if a triangle is made with straight lines in curved space, the angles add up to more than the normal 180 degrees. A saddle is an example of negative curve spacing. Gravity is caused by the curvature of space: Mass curves space, forcing objects to come together.
The Pythagorean theorem is often used to test whether space is flat or curved. This mathematical formula uses the length of each side of a triangle instead of angles. If the lengths match what the theorem states, then the triangle is in flat space. If the lengths do not match the theorem exactly, then the triangle is in curved space. Angles are difficult to measure over long distances, but measuring the sides or perimeter of a triangle can easily show the nature of space.
Euclidean geometry is the study of shapes in flat space. It is based on a list of basic pieces of information, called axioms, and demonstrates many mathematical concepts such as the Pythagorean theorem. Axioms are often disproved, in the sense that they are shown not always to be true, in curved space or in non-Euclidean geometry. All triangles have 180 degrees in Euclidean geometry, which is easy to disprove in curved space by measuring each angle with a protractor.
Curved space plays an important role in modern astronomy. Gravity is considered to be the curved space surrounding a large body causing smaller objects to orbit or collide with the large body. This was not discovered until Einstein published his Theory of General Relativity which first described gravity as curved space. Prior to this, astronomers computed orbits inaccurately because space was treated as a three-dimensional Euclidean shape. Modern astronomers can calculate and predict much more with non-Euclidean space, such as black holes and the way galaxies move.
The father of physics, Isaac Newton, also used Euclidean geometry. It has been the only way to study shapes for over 2,000 years. Then, at the end of the 19th century, the axiom that parallel lines never cross was disproven by Janos Bolyai. Einstein was able to understand non-Euclidean geometry and how it could be used to correctly predict Mercury’s bizarre orbit. The modern view is that true Euclidean forms exist only in spaces away from any gravitational body.
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