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nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Sun Sep 07, 2014 2:09 am    Post subject: Barth Decic (deg 10)

The Barth decic is a decic (degre 10) surface in complex three-dimensional projective space having the maximum possible number of
ordinary double points, namely 345. It is given by the implicit equation :
 Quote: 8(x^2-phi^4y^2)(y^2-phi^4z^2)(z^2-phi^4x^2)(x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2)+(3+5phi)(x^2+y^2+z^2-w^2)^2[x^2+y^2+z^2-(2-phi)w^2]^2w^2 =0
,
where phi is the golden ratio and w is a parameter.
The case w=1, illustrated in the plot, has 300 ordinary double points.
The Barth-Decic is invariant under the icosahedral group.
 Code: {     "Iso3D": {         "Cnd": [             "(x^2+y^2+z^2)>(1+sqrt(5))+.1"         ],         "Component": [             "Barth-Dedic"         ],         "Const": [             " w  = 1.0",             " phi= (1+sqrt(5))/2"         ],         "Fxyz": [             "8*(Ax-phi^4*Ay)*(Ay-phi^4*Az)*(Az-phi^4*Ax)*(Bx+By+Bz-2*(Ax*Ay+Ax*Az+Ay*Az)) + (3+5*phi)*(Ax+Ay+Az-w^2)^2 * (Ax+Ay+Az- (2-phi)*w^2)^2 * w^2"         ],         "Name": [             "Barth-Dedic"         ],         "Varu": [             " A  = u^2",             " B  = u^4"         ],         "Xmax": [             "(1+sqrt(5))/2 +0.2"         ],         "Xmin": [             "-(1+sqrt(5))/2-0.2"         ],         "Ymax": [             "(1+sqrt(5))/2 +0.2"         ],         "Ymin": [             "-(1+sqrt(5))/2-0.2"         ],         "Zmax": [             "(1+sqrt(5))/2 +0.2"         ],         "Zmin": [             "-(1+sqrt(5))/2-0.2"         ]     } }

BarthDecic by taha_ab, on Flickr[/quote]
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Abderrahman
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