Surface Homeomorphism Between Cubes and Spheres?

Thread starter e12514
Start date Mar 25, 2009
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Spheres Surfaces

In summary, the difference between the surface area of a cube and a sphere is that a cube has a flat surface area while a sphere has a curved surface area. To calculate the surface area of a cube, you need to find the area of one side and multiply it by six. The formula for surface area of a sphere takes into account the curved nature of its surface. It is not possible for a cube to have a greater surface area than a sphere with the same volume. The surface area of a sphere is used in calculations for physical properties such as surface tension because it represents the total area of the sphere's surface in contact with another substance.

Mar 25, 2009

e12514

Is it true that the surface of a (hyper)cube in Rⁿ is homeomorphic to S^n-1? Or only for particular n?

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Mar 25, 2009

HallsofIvy

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Yes, that's true for all n. One way to see that is to take p to be the centroid of Rⁿ. for every point x on the surface of Rⁿ, let f(x) be the point where the ray from p through x crosses the S^n-1. Show that f is a homeomorphism.

Related to Surface Homeomorphism Between Cubes and Spheres?

1. What is the difference between the surface area of a cube and a sphere?

The surface area of a cube is the total area of all six sides, whereas the surface area of a sphere is the total area of the spherical surface. In other words, a cube has a flat surface area, while a sphere has a curved surface area.

2. How do you calculate the surface area of a cube?

To calculate the surface area of a cube, you need to find the area of one side and multiply it by six (since a cube has six sides). The formula for finding the area of a square (which is the shape of a cube's side) is length x width. So, the formula for surface area of a cube is 6 x (length x width).

3. How do you calculate the surface area of a sphere?

To calculate the surface area of a sphere, you need to know the radius (r) of the sphere. The formula for surface area of a sphere is 4πr². This formula takes into account the curved nature of a sphere's surface.

4. Can the surface area of a cube be greater than the surface area of a sphere with the same volume?

No, it is not possible for a cube to have a greater surface area than a sphere with the same volume. This is because the sphere has a curved surface, which means it can "hold" more volume within a smaller surface area compared to a cube's flat surface.

5. Why is the surface area of a sphere used to calculate certain physical properties, such as surface tension?

The surface area of a sphere is used in these calculations because it represents the total area of the sphere's surface that is in contact with another substance (such as a liquid). This is important in determining the strength of surface tension, as it is affected by the amount of surface area in contact with the liquid.