Barth Decic (deg 10)

 
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PostPosted: Sun Sep 07, 2014 2:09 am    Post subject: Barth Decic (deg 10) Reply with quote

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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