All right angles are equal 5. Any two straight lines equidistant from one another at two points are infinitely parallel Euclidean geometry is of great practical value. Non-Euclidean Geometry shown on a sphere. Share this: Twitter Facebook. Like this: Like Loading Leave a Reply Cancel reply Enter your comment here Fill in your details below or click an icon to log in:.
Email required Address never made public. Name required. About Author. Categories mathematics pmri. Blog Stats 48, hits. Free counters.
Follow Following. What precisely is the difference between Euclidean Geometry, and non-Euclidean Geometry? Ask Question. Asked 8 years ago. Active 6 years, 3 months ago. Viewed 5k times. Seraphina Seraphina 1 1 gold badge 5 5 silver badges 18 18 bronze badges. In the non - euclidean geometry it doesn't.
It means in the euclidean geometry to a point outside of a straight line passes exactly one line parallel to the line. In non - euclidean geometry this isn't true. You may want to investigate Klein's Erlangen Program , a treatise that characterizes geometries by their isometry groups. Speaking of homogeneous spaces, it is the isometry group of the space that defines it.
Euclidean space has a certain group of isometries distinct from the other spaces. There are flat homogeneous spaces that are non-Euclidean. Add a comment. Active Oldest Votes. What about in three dimensions, which corresponds to the space we actually live in? It has been shown that in three dimensions there are eight possible geometries. There is a 3-dimensional version of Euclidean geometry, a 3-dimensional version of spherical geometry and a 3-dimensional version of Hyperbolic geometry.
There is also a geometry which is a combination of spherical and Euclidean, and a geometry which is a combination of hyperbolic and Euclidean. The three other geometries are a bit more exotic and are harder to describe.
Since all these geometries look the same at small scales, we cannot tell the shape of the space that we live in without studying difficult questions about the universe itself. In particular, there are still two very fundamental questions about the universe that remain unknown:.
When we ask if the universe is finite, we are really asking if it closes up like a sphere, or extends infinitely like the plane or hyperbolic space. Another way to ask this question is to think about a rocket traveling through space in a straight line: If the universe is finite, it will eventually wrap around and return. On a 2 dimensional surface, if we travel in a straight path and never return we would be on something like an infinite plane.
If we did manage to return, even though we travel in a straight line, then we would have been on something like a sphere. Some scientists believe our universe is more like the 3-dimensional version of the sphere.
Our rocket would eventually return to Earth after an impossibly long time. What about the geometry of the universe? Euclidean, spherical and hyperbolic geometry are different on small scales. The sum of the angles in a triangle is different, for example. However, for really small triangles in spherical and hyperbolic geometry, the triangles begin to look a lot like their Euclidean cousins.
One would have to be able to do very precise measurement to measure the angle defects. We run into a similar problem when trying to measure the geometry of the universe. So far, measurements are not accurate enough or large enough to decide the issue. The universe could be what we call flat which corresponds to Euclidean or it could have some small amount of curvature which could make it have some other geometry.
How do we picture possible 3-dimensional spaces? Think about some computer games where our screen in a square or a rectangle, but if we leave the screen on the right hand side, we re-appear on the left. Similarly if we were to leave the screen at the top, we would show up again at the bottom.
This really means that the left is connected to the right and the top to the bottom. I little more thought would show that we were actually playing the game on a torus a doughnut like shape. We can do similar things in 3-space. And you definitely should since it will force you to think about questions like "How many degrees are in a triangle?
Why is that? And, as a simple balloon can show you, the implications of this other kind of geometry are rather surprising. Balloons, Triangles, and Angles A few months ago, my daughter got her first balloon at her first birthday party. Ever since that day, balloons have become just about the most amazing thing in her world. As I described last time , you can get a glimpse at one of these properties by performing a simple maths-and-crafts project.
All you have to do is get an uninflated balloon, lay it on a flat surface, and draw as close to a perfect triangle on it as you can. If you have a protractor, this would be a good time to measure its angles and make sure they add up to approximately 0.
Now, blow up the balloon and take a look at your once-perfect triangle. What happened to it? Do its angles still add up to 0?
What Is Euclidean Geometry? The type of geometry we typically learn in school is known as Euclidean geometry. Why such a proper name?
0コメント