Understanding Manifolds: Exploring Dimensions and Shapes in Mathematics

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In summary, a manifold is a mathematical space that resembles Euclidean space on a smaller scale, with a specific dimension. Examples include lines, circles, planes, and spheres. A sphere is a perfectly round three-dimensional object. Boundaries of shapes and objects have one less dimension, such as the sphere being the boundary of a solid ball. Manifolds and boundaries can be understood in terms of spatial dimensions, such as the circle being the boundary of a sphere. In topology, a circle is the boundary of a solid disc. It is a weird and abstract concept, but can be explored through books like "Topology: A Geometric Approach" by Terry Lawson.
  • #1
Kidphysics
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Homework Statement


Taken from Wiki:
a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus, a line and a circle are one-dimensional manifolds, a plane and sphere (the surface of a ball) are two-dimensional manifolds, and so on into high-dimensional space.

Homework Equations



A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

The Attempt at a Solution



The dimensions of a manifold are=n-1 dimensions of shapes and objects of reality?
 
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  • #2
What you're referring to in actual fact are the boundaries of trhese shapes. For example the sphere is the boundary of a solid ball. The reason for this is because the solid ball is contractable to a point. When you take the boundary you reduce the dimension by one.
 
  • #3
a solid ball is contractible to a point? I do not understand that since a sphere is supposed to exist in 3 dimensions and and a point has only one.

Is there a way to think of manifolds and boundaries in terms of spatial dimensions?

The circle is the boundary of a sphere making it a 2 dimensional manifold?
 
  • #4
I think it means that the sphere is simply connected therefore a loop on the surface can be compressed to a point unlike a torus. This was shown in Perelman's Ricci Flow with Surgery proof of the Poincare Conjecture to be true for a 3-dimensional closed surface.
"Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere." - Wikipedia
Solution:
http://en.wikipedia.org/wiki/Solution_of_the_Poincaré_conjecture
 
  • #5
The circle id the boundary of a solid disc. Topology is weird, get used to it.
 
  • #6
Kevin_Axion said:
I think it means that the sphere is simply connected therefore a loop on the surface can be compressed to a point unlike a torus.

Thank you Kevin, I have seen pictures of this and I know exactly what you're referring to.



hunt_mat said:
The circle id the boundary of a solid disc. Topology is weird, get used to it.

Lol yes it is very abstract, can anyone point me in a good direction for elementary reading on the subject?
 
  • #7
Topology: A Geometric Approach (Oxford Graduate Texts in Mathematics) by Terry Lawson
 

Related to Understanding Manifolds: Exploring Dimensions and Shapes in Mathematics

1. What is a manifold?

A manifold is a mathematical concept that describes a space that is locally similar to Euclidean space. In simpler terms, it is a type of geometric structure that is smooth and can be described using coordinates.

2. What are the different types of manifolds?

There are several types of manifolds, including smooth manifolds, topological manifolds, Riemannian manifolds, and complex manifolds. These differ based on the specific properties and structures they possess.

3. How are manifolds used in science?

Manifolds have various applications in science, particularly in physics and engineering. They are used to study and model complex systems and phenomena, such as fluid dynamics, electromagnetism, and general relativity.

4. What is the difference between a manifold and a surface?

A surface is a two-dimensional object, while a manifold can have any number of dimensions. Additionally, a manifold is a more general concept that encompasses surfaces and other higher-dimensional objects.

5. Can manifolds be visualized?

Yes, manifolds can be visualized through mathematical representations and diagrams. However, this may be limited to lower-dimensional manifolds, as it becomes more difficult to visualize higher-dimensional ones beyond three dimensions.

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