Math Art
Van Straten Octic (124 nodes)

Van Straten Octic (124 nodes)

Van Straten Octic (124 nodes) is an algebraic surface, defined by an implicit equation, immersed with singularities, with D8h symmetry (a point group).

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algebraic aperiodic def-implicit implemented singular tradition-classical

Formula

(x2)3⁢y2−2⁢(x2)2⁢(y2)2+x2⁢(y2)3−(x2)3⁢z2−3⁢(x2)2⁢y2⁢z2−3⁢x2⁢(y2)2⁢z2−(y2)3⁢z2+5⁢(x2)2⁢(z2)2+10⁢x2⁢y2⁢(z2)2+5⁢(y2)2⁢(z2)2−8⁢x2⁢(z2)3−8⁢y2⁢(z2)3−508⁢(z2)4+1024⁢(z2)3−640⁢(z2)2+128⁢z2−8=0\left(x^{2}\right)^{3} y^{2} - 2 \left(x^{2}\right)^{2} \left(y^{2}\right)^{2} + x^{2} \left(y^{2}\right)^{3} - \left(x^{2}\right)^{3} z^{2} - 3 \left(x^{2}\right)^{2} y^{2} z^{2} - 3 x^{2} \left(y^{2}\right)^{2} z^{2} - \left(y^{2}\right)^{3} z^{2} + 5 \left(x^{2}\right)^{2} \left(z^{2}\right)^{2} + 10 x^{2} y^{2} \left(z^{2}\right)^{2} + 5 \left(y^{2}\right)^{2} \left(z^{2}\right)^{2} - 8 x^{2} \left(z^{2}\right)^{3} - 8 y^{2} \left(z^{2}\right)^{3} - 508 \left(z^{2}\right)^{4} + 1024 \left(z^{2}\right)^{3} - 640 \left(z^{2}\right)^{2} + 128 z^{2} - 8 = 0

Properties

Family
algebraic
Given by
an implicit equation
Curvature
no condition imposed
Periodicity
not periodic
Ends
0
Embedding
immersed with singularities
Symmetry
D8h
Symmetry kind
point group
Fidelity
exact
Exactness
elementary

Sources

Closed form reproduced numerically against the shipped implementation (math_art/surfaces/algebraic.py) over 240 sample points: matches oracle x 1 over 240 points (worst 1.42e-14).