jimthecelt: Is Plane Geometry a subdivision of Euclidian Geometry, or is the reverse true?
I know that Euclidian Geometry is the geometry of lines and shapes. What, then, is Plane Geometry?
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Answer by rhsaunders
The terms are equivalent. Non-Euclidean geometries apply on curved surfaces, such as spheres, where the Euclidean proposition, that exactly one line can be drawn through a point so as to be parallel to another line, does not hold.
Plane geometry is two-dimensional geometry – it is geometry on a surface, and does not include solids like spheres and cubes.
Euclidean geometry is geometry where triangles always have angles that add up to 180 degrees, and two lines in a plane perpendicular to a 3rd line always remain equidistant from each other.
In the 1800’s, mathematicians discovered that there are reasonable geometric theories which violate the principles of Euclidean geometry – they are called non-Euclidean geometries. Most people never learn about non-Euclidean geometry unless they take an advanced college geometry course.
Scientists now believe that the geometry of our universe is non-Euclidean. But in very small regions of space, the difference between Euclidean and non-Euclidean geometry is so small it cannot be measured – and our solar system is a very small region of the universe. So for practical purposes in this solar system, we use Euclidean geometry because it is simpler. But if interstellar space travel ever becomes possible, we may need to use non-Euclidean geometry.
The answer to your question is that neither is a subdivision of the other. Plane geometry can be either Euclidean or non-Euclidean. And Euclidean geometry can be either 2-dimensional (plane) or 3-dimensional (solid), or even higher dimensional.
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