What is the largest n dimensional box that can fit inside a sphere?

Thread starter rylz
Start date Dec 6, 2012
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Box Fit Sphere

In summary, an n-dimensional box is a geometric shape with length, width, and height in each dimension. The size of a box is determined by the length of its edges in each dimension and is directly related to the size of a sphere in n-dimensional space. There is no limit to the number of dimensions in which a box can exist, but it is typically only considered up to 3 or 4 dimensions. The largest n-dimensional box that can fit inside a sphere is calculated by finding the diameter of the sphere and dividing it by the square root of n.

Dec 6, 2012

rylz

in ℝⁿ what is the largest n-dimensional box that can fit into the (n-1) sphere

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Dec 7, 2012

mfb

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Insights Author

Can you find a solution for n=2, n=3?
This should give you a hint how it works for n dimensions.

1. What is the definition of an n-dimensional box?

An n-dimensional box is a geometric shape that exists in n-dimensional space and is defined by its length, width, and height in each dimension. For example, a 3-dimensional box has length, width, and height, while a 4-dimensional box has length, width, height, and depth.

2. How is the size of a box determined in n-dimensional space?

The size of a box in n-dimensional space is determined by the length of its edges in each dimension. For example, a 3-dimensional box with edges of 1 unit in length would have a volume of 1 cubic unit, while a 4-dimensional box with edges of 1 unit in length would have a hypervolume of 1 tesseract unit.

3. What is the relationship between the size of a box and the size of a sphere in n-dimensional space?

In n-dimensional space, the size of a box is directly related to the size of a sphere. The largest n-dimensional box that can fit inside a sphere will have edges that are equal to the diameter of the sphere. This means that the box will have a volume that is exactly half of the volume of the sphere.

4. Is there a limit to the number of dimensions in which a box can exist?

There is technically no limit to the number of dimensions in which a box can exist. However, as the number of dimensions increases, the concept of a box becomes more abstract and difficult to visualize. In practical terms, a box is typically only considered up to 3 or 4 dimensions.

5. How is the largest n-dimensional box that can fit inside a sphere calculated?

The largest n-dimensional box that can fit inside a sphere is calculated by finding the diameter of the sphere and dividing it by the square root of n. This will give the length of each edge of the box. The volume of the box can then be calculated by multiplying the length of each edge together.