Endraß octic (deg 8)

 
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nextstep
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Joined: 06 Jan 2007
Posts: 539

PostPosted: Fri Sep 05, 2014 4:28 am    Post subject: Endraß octic (deg 8) Reply with quote

Hi all,
Endraß surfaces are a pair of octic surfaces which have 168 ordinary double points. This is the maximum number known to exist for an octic surface, although the rigorous upper bound is 174.
Endraß constructed it in 1997 during the writing of his Ph.D. thesis (advisor W. Barth).
The surfaces were discovered in a five-dimensional family of octics with 112 nodes.
The illustrated surface take w=1 and has 144 real ordinary double points.

Code:
{
    "Iso3D": {
        "Cnd": [
            "(x^2+y^2+z^2)>25"
        ],
        "Component": [
            "Endraß-Octic"
        ],
        "Const": [
            " w = 1.0",
            "a1 = 1.0",
         "a2 = 1.0",
         "a3 = 1.0",
         "a4 = -1.0"
        ],
        "Fxyz": [
            "64 * (x^2 - w^2)*(y^2 - w^2)*((x + y)^2 -2*w^2)*((x - y)^2 -2*w^2)  - (-4*(1 + a1*2^(1/2))*(x^2 + y^2)^2 +
         (8*(2 + a2*2^(1/2))*z^2 + 2*(2 + a3*7*2^(1/2))*w^2 )* (x^2 + y^2) -
         16*z^4+ 8*(1 + a4*2*2^(1/2))*z^2*w^2 - (1 + 12*2^(1/2))*w^4)^2"
        ],
        "Name": [
            "Endraß-Octic"
        ],
        "Xmax": [
            "5"
        ],
        "Xmin": [
            "-5"
        ],
        "Ymax": [
            "5"
        ],
        "Ymin": [
            "-5"
        ],
        "Zmax": [
            "5"
        ],
        "Zmin": [
            "-5"
        ]
    }
}


EndraßOctic by taha_ab, on Flickr
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Abderrahman
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nextstep
Site Admin


Joined: 06 Jan 2007
Posts: 539

PostPosted: Sat Sep 20, 2014 11:28 pm    Post subject: Sarti's Octic Reply with quote

Hi all,
This is the Alessandra Sarti's Octic version with 144 nodes (see also her Dodecic (deg 12) 600 nodes)

Quote:
{
"Iso3D": {
"Component": [
" SartiOctic"
],
"Fxyz": [
"-1728*x^4*y^2*z^2+928.0*z^4*x^4+9024.0*z^2*x^4+928.0*z^4*y^4+9024.0*z^2*y^4+9024.0*x^2*y^2+2720.0*x^4*y^4-1728*x^2*z^2-1728*y^2*z^2-1728*x^4*y^2-1728*x^2*y^4-1728*x^2*z^4-1728*y^2*z^4-576*x^6*y^2-576*x^6*z^2-576*x^2*y^6-576*x^2*z^6-576*y^6*z^2+9024.0*z^4*x^2*y^2-24960.0*z^2*x^2*y^2-1728*x^2*y^4*z^2+2720.0*z^4+112.0*z^8-576*z^2+928.0*x^4+112.0*x^8+928.0*y^4+112.0*y^8-576*x^2-576*y^2-576*x^6-576*y^6-576*z^6+112.0-576*y^2*z^6"
],
"Name": [
"SartiOctic"
],
"Xmax": [
"5.6"
],
"Xmin": [
"-5.6"
],
"Ymax": [
"5.6"
],
"Ymin": [
"-5.6"
],
"Zmax": [
"5.6"
],
"Zmin": [
"-5.6"
]
}
}

SartiOctic by taha_ab, on Flickr
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Abderrahman
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