what Manifolds are and what they’re used to understand / interpret / model?
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I thought I had an answer. However, it was in the context of cars. That’s not going to be of much use to you.
I thought I had an answer. However, it was in the context of cars. That’s not going to be of much use to you.
I thought I had an answer. However, it was in the context of cars. That’s not going to be of much use to you.
They’re spaces that can be locally approximated by normal Euclidean spaces (e.g. lines or planes) but can be non-Euclidean on a large scale. A very simple example is the surface of a sphere, which is non-Euclidean because, for example, if you try and draw a triangle, the angles do not sum to 180 degrees. However, we’re all familiar with how a 2D rectangle can very closely approximate the geometry of a small portion of the earth’s surface.
They’re used in a few areas of maths and physics, but perhaps most prominently in general relativity, which models space and time as something called a Lorentzian manifold (obviously the simpler examples, like the surface of a sphere, are used everywhere, but they usually aren’t discussed using the language of manifolds).
They’re spaces that can be locally approximated by normal Euclidean spaces (e.g. lines or planes) but can be non-Euclidean on a large scale. A very simple example is the surface of a sphere, which is non-Euclidean because, for example, if you try and draw a triangle, the angles do not sum to 180 degrees. However, we’re all familiar with how a 2D rectangle can very closely approximate the geometry of a small portion of the earth’s surface.
They’re used in a few areas of maths and physics, but perhaps most prominently in general relativity, which models space and time as something called a Lorentzian manifold (obviously the simpler examples, like the surface of a sphere, are used everywhere, but they usually aren’t discussed using the language of manifolds).
They’re spaces that can be locally approximated by normal Euclidean spaces (e.g. lines or planes) but can be non-Euclidean on a large scale. A very simple example is the surface of a sphere, which is non-Euclidean because, for example, if you try and draw a triangle, the angles do not sum to 180 degrees. However, we’re all familiar with how a 2D rectangle can very closely approximate the geometry of a small portion of the earth’s surface.
They’re used in a few areas of maths and physics, but perhaps most prominently in general relativity, which models space and time as something called a Lorentzian manifold (obviously the simpler examples, like the surface of a sphere, are used everywhere, but they usually aren’t discussed using the language of manifolds).
When you hear manifold, think surface.
Now the word surface generally refers to 2D manifolds. Meaning, if you pick a point on a surface, you only need two numbers to describe where with is. The surface of a sphere, a 2D plane, a saddle, a bow. These are surfaces. These are 2D manifolds.
Manifolds needn’t be 2D. The circumference of a circle is a 1D manifold. The boundary of a hypersphere is a 3D manifold.
When you hear manifold, think surface.
Now the word surface generally refers to 2D manifolds. Meaning, if you pick a point on a surface, you only need two numbers to describe where with is. The surface of a sphere, a 2D plane, a saddle, a bow. These are surfaces. These are 2D manifolds.
Manifolds needn’t be 2D. The circumference of a circle is a 1D manifold. The boundary of a hypersphere is a 3D manifold.
When you hear manifold, think surface.
Now the word surface generally refers to 2D manifolds. Meaning, if you pick a point on a surface, you only need two numbers to describe where with is. The surface of a sphere, a 2D plane, a saddle, a bow. These are surfaces. These are 2D manifolds.
Manifolds needn’t be 2D. The circumference of a circle is a 1D manifold. The boundary of a hypersphere is a 3D manifold.
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