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        riddles >> putnam exam (pure math) >> Identify These Surfaces
        
(Message started by: ThudanBlunder on Sep 4^th, 2007, 12:38pm)

Title: Identify These Surfaces
Post by ThudanBlunder on Sep 4^th, 2007, 12:38pm

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) + 36http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2yz(y² - 3x²) + 4z⁴]
+ (9x² + 9y² - 2z²)[-81(x² + y²)² - 72z²(x² + y²) + 108http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2xz(x² - 3y²) + 4z⁴] = 0

Title: Re: Identify These Surfaces
Post by SMQ on Sep 5^th, 2007, 1:06pm

I recognized the first one when I plotted it: it's the Roman surface (http://en.wikipedia.org/wiki/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

Title: Re: Identify These Surfaces
Post by ThudanBlunder on Sep 5^th, 2007, 1:28pm

on 09/05/07 at 13:06:20, SMQ wrote:

My plots of the other two don't look familiar.

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.

Title: Re: Identify These Surfaces
Post by Barukh on Sep 6^th, 2007, 1:45am

on 09/05/07 at 13:06:20, SMQ wrote:

I recognized the first one when I plotted it: it's the Roman surface (http://en.wikipedia.org/wiki/Roman_surface) (although the traditional formulation would switch the sign of the xyz term).
--SMQ

Wow! Two questions:

1. How do you plot?
2. How do you recognize? ;D

Title: Re: Identify These Surfaces
Post by Barukh on Sep 6^th, 2007, 5:47am

2) Klein Bottle (http://mathworld.wolfram.com/KleinBottle.html)
3) Boy Surface (http://mathworld.wolfram.com/BoySurface.html)

Title: Re: Identify These Surfaces
Post by ThudanBlunder on Sep 19^th, 2007, 10:29am

on 09/05/07 at 13:06:20, SMQ wrote:

I recognized the first one when I plotted it: it's the Roman surface (http://en.wikipedia.org/wiki/Roman_surface)

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 09/06/07 at 05:47:47, Barukh wrote:

2) Klein Bottle (http://mathworld.wolfram.com/KleinBottle.html)
3) Boy Surface (http://mathworld.wolfram.com/BoySurface.html)

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.