Kummer quartic surface (deg 4)

 
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nextstep
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Joined: 06 Jan 2007
Posts: 538

PostPosted: Sat Aug 16, 2014 7:06 am    Post subject: Kummer quartic surface (deg 4) Reply with quote

Hi all,
In algebraic geometry, a Kummer quartic surface, first studied by Kummer (1864), is an irreducible algebraic surface of degree 4 in P^3 with the maximal possible number of 16 double points.
The Kummer involution has 16 fixed points: the 16 2-torsion point of the Jacobian, and they are the 16 singular points of the quartic surface. Resolving the 16 double points of the quotient of a (possibly nonalgebraic) torus by the Kummer involution gives a K3 surface with 16 disjoint rational curves; these K3 surfaces are also sometimes called Kummer surfaces.

Code:
{
    "Iso3D": {
        "Component": [
            "Kummer"
        ],
        "Const": [
            " mu=2/3"
        ],
        "Fxyz": [
            "(x^2+y^2+z^2-mu)^2-((3*mu-1)/(3-mu))*(1-z-x*sqrt(2))*(1-z+x*sqrt(2))*(1+z-y*sqrt(2))*(1+z+y*sqrt(2))"
        ],
        "Name": [
            "Kummer"
        ],
        "Xmax": [
            "1"
        ],
        "Xmin": [
            "-1"
        ],
        "Ymax": [
            "1"
        ],
        "Ymin": [
            "-1"
        ],
        "Zmax": [
            "1"
        ],
        "Zmin": [
            "-1"
        ]
    }
}


kummer by taha_ab, on Flickr
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Cheers,
Abderrahman


Last edited by nextstep on Sat Sep 20, 2014 5:44 pm; edited 1 time in total
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nextstep
Site Admin


Joined: 06 Jan 2007
Posts: 538

PostPosted: Sun Sep 14, 2014 5:29 am    Post subject: Reply with quote

Hi all,
"Nordstrand's weird surface" is an attractive quartic surface given by the implicit equation:
Quote:

{
"Iso3D": {
"Name": [
"Nordstrand"
],
"Cnd": [
"((x-.1)^2+(y-.1)^2+(z-.1)^2)>1"
],
"Component": [
" Nordstrand"
],
"Fxyz": [
" 25*(x^3*(y+z)+y^3*(x+z)+z^3*(x+y))+50*(x^2*y^2+x^2*z^2+y^2*z^2)-125*(x^2*y*z+y^2*x*z+z^2*x*y)+60*x*y*z-4*(x*y+x*z+y*z)"
],
"Xmax": [
" 1.1"
],
"Xmin": [
"-1.1"
],
"Ymax": [
" 1.1"
],
"Ymin": [
"-1.1"
],
"Zmax": [
" 1.1"
],
"Zmin": [
"-1.1"
]
}
}

Nordstrand by taha_ab, on Flickr
_________________
Cheers,
Abderrahman
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nextstep
Site Admin


Joined: 06 Jan 2007
Posts: 538

PostPosted: Sun Sep 14, 2014 4:21 pm    Post subject: Reply with quote

Hi,
In mathematics, Fresnel's wave surface, found by Augustin-Jean Fresnel in 1821, is a quartic surface describing the propagation of light in an optically biaxial crystal.
Wave surfaces are special cases of tetrahedroids which are in turn special cases of Kummer surfaces.
The images correspond to what is known as Fresnel's wave-surface, for particular elasticity parameters.
Quote:

{
"Param3D": {
"Component": [
"Fresnel_2"
],
"Description": [
"Description of the model"
],
"Fx": [
"cos(u)*cos(v)/(-2.*sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))*cos((acos(-(-0.941/6.+0.374*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4)-1.309/6.*((cos(u)^6+sin(u)^6)*cos(v)^6+sin(v)^6)-1.221*cos(u)^2*cos(v)^4*sin(u)^2*sin(v)^2)/sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))^3)-pi)/3.)+0.8 )"
],
"Fy": [
"sin(u)*cos(v)/(-2.*sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))*cos((acos(-(-0.941/6.+0.374*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4)-1.309/6.*((cos(u)^6+sin(u)^6)*cos(v)^6+sin(v)^6)-1.221*cos(u)^2*cos(v)^4*sin(u)^2*sin(v)^2)/sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))^3)-pi)/3.)+0.8 )"
],
"Fz": [
"sin(v)/(-2.*sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))*cos((acos(-(-0.941/6.+0.374*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4)-1.309/6.*((cos(u)^6+sin(u)^6)*cos(v)^6+sin(v)^6)-1.221*cos(u)^2*cos(v)^4*sin(u)^2*sin(v)^2)/sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))^3)-pi)/3.)+0.8 )"
],
"Name": [
"Fresnel_2"
],
"Umax": [
"2*pi"
],
"Umin": [
"0"
],
"Vmax": [
"pi/2"
],
"Vmin": [
"-pi/2"
]
}
}


Fresnel by taha_ab, on Flickr
_________________
Cheers,
Abderrahman
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