puzzle: Sphere in a cube

Higher-dimensional geometry (previously "Polyshapes").

puzzle: Sphere in a cube

Postby A_Square » Thu Jun 17, 2010 12:53 pm

Here's a puzzle :mrgreen:

If you have a cube of side 1 and put a sphere of radius 1 centered at the corner in 2D the center of the cube (square) is inside the sphere (circle) and in 3D it is, but what about 4D and higher?
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Re: puzzle: Sphere in a cube

Postby Keiji » Fri Jun 18, 2010 1:54 pm

I can't say anything exact, but there will be some dimension for which the center of the hypercube is no longer inside the hypersphere.
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Re: puzzle: Sphere in a cube

Postby wendy » Sat Jun 19, 2010 8:12 am

A sphere in 4d of diameter 2 would exactly emclose a 4d cube (tesseract), but higher dimensions, the dimanete of a unit cube is larger than the sphere. In some higher dimension (i think, 19), the volume of the unit-edge cube exceeds that of a sphere of diameter 2.
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Re: puzzle: Sphere in a cube

Postby Tamfang » Thu Jun 24, 2010 8:50 am

Easy. The square of the distance between the center of the sphere and the center of the cube is n times (1/2) squared. So in 4space the cube center is on the surface, and in higher dimensions it's outside.
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