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Topic: Identify These Surfaces (Read 791 times) |
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ThudnBlunder
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Identify These Surfaces
« on: Sep 4th, 2007, 12:38pm » |
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1) x2y2 + y2z2 + z2x2 + xyz = 0
2) (x2 + y2 + z2 + 2y - 1)[(x2 + y2 + z2 + 2y - 1)2 - 8z2] + 16xz(x2 + y2 + z2 + 2y - 1) = 0
3) 64z3(1 - z)3 - 48z2(1 - z)2(3x2 + 3y2 + 2z2) + 12z(1 - z)[27(x2 + y2)2 - 24z2(x2 + y2) + 36 2yz(y2 - 3x2) + 4z4]
+ (9x2 + 9y2 - 2z2)[-81(x2 + y2)2 - 72z2(x2 + y2) + 1082xz(x2 - 3y2) + 4z4] = 0
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| « Last Edit: Sep 5th, 2007, 1:29pm by ThudnBlunder » |
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SMQ
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Re: Identify These Surfaces
« Reply #1 on: Sep 5th, 2007, 1:06pm » |
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I recognized the first one when I plotted it: it's the Roman surface (although the traditional formulation would switch the sign of the xyz term).
My plots of the other two don't look familiar.
--SMQ
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--SMQ
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ThudnBlunder
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Re: Identify These Surfaces
« Reply #2 on: Sep 5th, 2007, 1:28pm » |
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on Sep 5th, 2007, 1:06pm, SMQ wrote:
My plots of the other two don't look familiar.
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No wonder, the 2nd one needs amending. Sorry, it should be:
(x2 + y2 + z2 + 2y - 1)[(x2 + y2 + z2 - 2y - 1)2 - 8z2] + 16xz(x2 + y2 + z2 - 2y - 1) = 0
The 3rd one is OK.
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| « Last Edit: Sep 5th, 2007, 1:31pm by ThudnBlunder » |
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Barukh
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Re: Identify These Surfaces
« Reply #3 on: Sep 6th, 2007, 1:45am » |
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on Sep 5th, 2007, 1:06pm, SMQ wrote:I recognized the first one when I plotted it: it's the Roman surface (although the traditional formulation would switch the sign of the xyz term).
--SMQ |
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Wow! Two questions:
1. How do you plot?
2. How do you recognize? 
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Barukh
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Re: Identify These Surfaces
« Reply #4 on: Sep 6th, 2007, 5:47am » |
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2) Klein Bottle
3) Boy Surface
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| « Last Edit: Sep 6th, 2007, 5:52am by Barukh » |
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ThudnBlunder
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Re: Identify These Surfaces
« Reply #5 on: Sep 19th, 2007, 10:29am » |
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on Sep 5th, 2007, 1:06pm, SMQ wrote:I recognized the first one when I plotted it: it's the Roman surface
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Correct SMQ.
Discovered by Jacob Steiner in 1844 while visiting Rome, it is one of the few mathematical objects named after a place. He constructed it using pure geometry but couldn't work out the equation for it. So he asked Weierstrass to have a go and he had no trouble coming up with the equation.
on Sep 6th, 2007, 5:47am, Barukh wrote:
Correct Barukh.
Unlike the Roman Surface, neither the Klein Bottle nor Boy's Surface have singular points. Hilbert conjectured that the projective plane could not be arranged in 3D space so that it had no singular points (only self-intersections) and asked his student Werner Boy to prove it. Being a good research student, Boy promptly disproved it!
Reference: Francois Apery, Models of the Real Projective Plane, 1987.
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| « Last Edit: Sep 19th, 2007, 10:56am by ThudnBlunder » |
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