nextstep Site Admin
Joined: 06 Jan 2007 Posts: 539
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Posted: Sun Sep 07, 2014 2:09 am Post subject: Barth Decic (deg 10) |
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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"
]
}
}
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BarthDecic by taha_ab, on Flickr[/quote] _________________ Cheers,
Abderrahman |
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