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Topic: Volume of a Cube and Its Inscribed Sphere (Read 1402 times) |
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Whiskey Tango Foxtrot
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Volume of a Cube and Its Inscribed Sphere
« on: Dec 5th, 2006, 7:42pm » |
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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?
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| « Last Edit: Dec 7th, 2006, 11:42am by Whiskey Tango Foxtrot » |
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"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
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ThudnBlunder
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Re: Area of an Inscribed Sphere
« Reply #1 on: Dec 7th, 2006, 9:02am » |
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What has this to do with the title of the thread? 
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| « Last Edit: Feb 25th, 2010, 4:49pm by ThudnBlunder » |
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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Whiskey Tango Foxtrot
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Re: Area of an Inscribed Sphere
« Reply #2 on: Dec 7th, 2006, 9:09am » |
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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.
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| « Last Edit: Dec 7th, 2006, 9:10am by Whiskey Tango Foxtrot » |
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"I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use." - Galileo Galilei
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towr
wu::riddles Moderator
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Re: Area of an Inscribed Sphere
« Reply #3 on: Dec 7th, 2006, 9:18am » |
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You can edit your own posts, you know 
Maybe you should get some sleep
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Wikipedia, Google, Mathworld, Integer sequence DB
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SMQ
wu::riddles Moderator
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Re: Area of an Inscribed Sphere
« Reply #4 on: Dec 7th, 2006, 11:04am » |
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Some research 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/(2n-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. 
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Whiskey Tango Foxtrot
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Re: Volume of a Cube and Its Inscribed Sphere
« Reply #5 on: Dec 7th, 2006, 11:43am » |
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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.
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| « Last Edit: Dec 7th, 2006, 11:44am by Whiskey Tango Foxtrot » |
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ThudnBlunder
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Re: Volume of a Cube and Its Inscribed Sphere
« Reply #6 on: Dec 7th, 2006, 12:46pm » |
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on Dec 7th, 2006, 11:43am, Whiskey Tango Foxtrot wrote:| I thought this was a pretty cool problem, if I do say so myself. |
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Yeah, nice one and here is another. .
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_med ium;action=display;num=1102657337
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| « Last Edit: Feb 25th, 2010, 4:53pm by ThudnBlunder » |
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Barukh
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Re: Volume of a Cube and Its Inscribed Sphere
« Reply #7 on: Dec 8th, 2006, 1:31am » |
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on Dec 7th, 2006, 11:43am, Whiskey Tango Foxtrot wrote:| I thought this was a pretty cool problem |
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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.
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