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        riddles >> putnam exam (pure math) >> Volume of a Cube and Its Inscribed Sphere
        
(Message started by: Whiskey Tango Foxtrot on Dec 5^th, 2006, 7:42pm)

Title: Volume of a Cube and Its Inscribed Sphere
Post by Whiskey Tango Foxtrot on Dec 5^th, 2006, 7:42pm

A sphere is inscribed inside a five-dimensional cube. What fraction of the volume of the cube lies inside the sphere?

What fraction lies inside the sphere for a ten-dimensional cube?

What fraction for a 5n-dimensional cube?

Title: Re: Area of an Inscribed Sphere
Post by THUDandBLUNDER on Dec 7^th, 2006, 9:02am

What has this to do with the title of the thread? ;)

Title: Re: Area of an Inscribed Sphere
Post by Whiskey Tango Foxtrot on Dec 7^th, 2006, 9:09am

Nothing at all. :-/ Should be something like "Volume of a Cube and Its Inscribed Sphere." Sorry, it's the end of the term and I've been sleeping too little. Maybe a moderator will stop by and fix this.

Title: Re: Area of an Inscribed Sphere
Post by towr on Dec 7^th, 2006, 9:18am

You can edit your own posts, you know ::)
Maybe you should get some sleep ;)

Title: Re: Area of an Inscribed Sphere
Post by SMQ on Dec 7^th, 2006, 11:04am

Some research (http://mathworld.wolfram.com/Hypersphere.html) on Mathworld gives the volume of a unit n-sphere as 2^(n+1)/2[pi]^(n-1)/2/n!! for odd n, and 2[pi]^n/2/(n(n/2 - 1)!) for even n, where n!! is the "double factorial" 1*3*5*...*(n-2)*n. The volume of the enclosing cube is simply 2ⁿ.

Thus for n odd, the fraction inside the sphere is [pi]^(n-1)/2/(2^(n-1)/2n!!), and for n even is [pi]^n/2/(2^n-1n(n/2-1)!).

Plugging in n = 5 gives 0.164493407, n = 10 gives 0.002490395. In general the fraction of the volumn of the n-cube within the n-sphere decreases somewhat faster than 10^-n/2. :o

--SMQ

Title: Re: Volume of a Cube and Its Inscribed Sphere
Post by Whiskey Tango Foxtrot on Dec 7^th, 2006, 11:43am

Oof, yes sleep is what I need. That's right SMQ. I thought this was a pretty cool problem, if I do say so myself.

Title: Re: Volume of a Cube and Its Inscribed Sphere
Post by THUDandBLUNDER on Dec 7^th, 2006, 12:46pm

on 12/07/06 at 11:43:35, Whiskey Tango Foxtrot wrote:

I thought this was a pretty cool problem, if I do say so myself.

Yeah, nice one and here is another. .:P

http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1102657337

Title: Re: Volume of a Cube and Its Inscribed Sphere
Post by Barukh on Dec 8^th, 2006, 1:31am

on 12/07/06 at 11:43:35, Whiskey Tango Foxtrot wrote:

I thought this was a pretty cool problem

The cool thing is to derive the formula for the volume of n-dimensional sphere. The coolest thing is to notice that this has a global maximum at some unexpected dimension.