Final answer:
The Euclidean geometry postulate assuming a flat space does not always accurately describe points, lines, and planes, as non-Euclidean geometries and General Relativity suggest space can be curved by gravity. Polar coordinates offer a different method of describing points without relying on Euclidean assumptions.
Step-by-step explanation:
The postulate that does not accurately describe points, lines, and planes, particularly in the context of modern physics and geometry, is the assumption within Euclidean geometry that a flat space is the default condition of space. According to Euclidian geometry, a straight line represents the shortest distance between two points, the sum of angles in a triangle must be 180 degrees, and parallel lines do not intersect. However, the development of non-Euclidean geometries and General Relativity has shown us that in presence of mass and energy, the geometry of space is not Euclidean. This discovery has led to the understanding that space can be curved, and therefore, the Euclidean postulates may not hold true in all situations, as spacetime can be bent by gravity.
Comparatively, other geometrical systems, such as polar coordinates, offer alternative methods of describing points in a plane that do not rely on the same assumptions as Euclidean geometry. Polar coordinates label a point on a plane with a pair of numbers representing the distance from a reference point and an angle from a reference direction, which can be more useful than Cartesian coordinates in certain situations.