Math Art
Sarti Dodecic (600 nodes)

Sarti Dodecic (600 nodes)

Sarti Dodecic (600 nodes) is an algebraic surface, defined by an implicit equation, immersed with singularities.

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algebraic aperiodic def-implicit implemented record-holder singular tradition-classical

Formula

243⁢(73.79024325749306⁢((x2⁢y2+z2)2⁢(x2⁢z2+y2)−(x2⁢y2+z2)⁢(x2⁢z2+y2)2+(x2⁢z2+y2)2⁢(x2+y2⁢z2)−(x2⁢z2+y2)⁢(x2+y2⁢z2)2+(x2+y2⁢z2)2⁢(x2⁢y2+z2)−(x2+y2⁢z2)⁢(x2⁢y2+z2)2)+19⁢((x2⁢y2+z2)2⁢(x2⁢z2+y2)+(x2⁢y2+z2)⁢(x2⁢z2+y2)2+(x2⁢z2+y2)2⁢(x2+y2⁢z2)+(x2⁢z2+y2)⁢(x2+y2⁢z2)2+(x2+y2⁢z2)2⁢(x2⁢y2+z2)+(x2+y2⁢z2)⁢(x2⁢y2+z2)2)+10⁢((x2⁢y2+z2)3+(x2⁢z2+y2)3+(x2+y2⁢z2)3)−14⁢(x4+y4+z4+1)⁢((x2⁢y2+z2)⁢(x2⁢z2+y2)+(x2⁢y2+z2)⁢(x2+y2⁢z2)+(x2⁢z2+y2)⁢(x2+y2⁢z2))+2⁢(x4+y4+z4+1)2⁢(x2⁢y2+z2+x2⁢z2+y2+x2+y2⁢z2)−6⁢(x4+y4+z4+1)⁢((x2⁢y2+z2)2+(x2⁢z2+y2)2+(x2+y2⁢z2)2)−352⁢(x⁢y⁢z)2⁢(x2⁢y2+z2+x2⁢z2+y2+x2+y2⁢z2)+336⁢(x⁢y⁢z)2⁢(x4+y4+z4+1)+48⁢(x2⁢y2+z2)⁢(x2⁢z2+y2)⁢(x2+y2⁢z2))−22⁢(x2+y2+z2+1)6=0243 \left(73.79024325749306 \left(\left(x^{2} y^{2} + z^{2}\right)^{2} \left(x^{2} z^{2} + y^{2}\right) - \left(x^{2} y^{2} + z^{2}\right) \left(x^{2} z^{2} + y^{2}\right)^{2} + \left(x^{2} z^{2} + y^{2}\right)^{2} \left(x^{2} + y^{2} z^{2}\right) - \left(x^{2} z^{2} + y^{2}\right) \left(x^{2} + y^{2} z^{2}\right)^{2} + \left(x^{2} + y^{2} z^{2}\right)^{2} \left(x^{2} y^{2} + z^{2}\right) - \left(x^{2} + y^{2} z^{2}\right) \left(x^{2} y^{2} + z^{2}\right)^{2}\right) + 19 \left(\left(x^{2} y^{2} + z^{2}\right)^{2} \left(x^{2} z^{2} + y^{2}\right) + \left(x^{2} y^{2} + z^{2}\right) \left(x^{2} z^{2} + y^{2}\right)^{2} + \left(x^{2} z^{2} + y^{2}\right)^{2} \left(x^{2} + y^{2} z^{2}\right) + \left(x^{2} z^{2} + y^{2}\right) \left(x^{2} + y^{2} z^{2}\right)^{2} + \left(x^{2} + y^{2} z^{2}\right)^{2} \left(x^{2} y^{2} + z^{2}\right) + \left(x^{2} + y^{2} z^{2}\right) \left(x^{2} y^{2} + z^{2}\right)^{2}\right) + 10 \left(\left(x^{2} y^{2} + z^{2}\right)^{3} + \left(x^{2} z^{2} + y^{2}\right)^{3} + \left(x^{2} + y^{2} z^{2}\right)^{3}\right) - 14 \left(x^{4} + y^{4} + z^{4} + 1\right) \left(\left(x^{2} y^{2} + z^{2}\right) \left(x^{2} z^{2} + y^{2}\right) + \left(x^{2} y^{2} + z^{2}\right) \left(x^{2} + y^{2} z^{2}\right) + \left(x^{2} z^{2} + y^{2}\right) \left(x^{2} + y^{2} z^{2}\right)\right) + 2 \left(x^{4} + y^{4} + z^{4} + 1\right)^{2} \left(x^{2} y^{2} + z^{2} + x^{2} z^{2} + y^{2} + x^{2} + y^{2} z^{2}\right) - 6 \left(x^{4} + y^{4} + z^{4} + 1\right) \left(\left(x^{2} y^{2} + z^{2}\right)^{2} + \left(x^{2} z^{2} + y^{2}\right)^{2} + \left(x^{2} + y^{2} z^{2}\right)^{2}\right) - 352 \left(x y z\right)^{2} \left(x^{2} y^{2} + z^{2} + x^{2} z^{2} + y^{2} + x^{2} + y^{2} z^{2}\right) + 336 \left(x y z\right)^{2} \left(x^{4} + y^{4} + z^{4} + 1\right) + 48 \left(x^{2} y^{2} + z^{2}\right) \left(x^{2} z^{2} + y^{2}\right) \left(x^{2} + y^{2} z^{2}\right)\right) - 22 \left(x^{2} + y^{2} + z^{2} + 1\right)^{6} = 0

Properties

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

Definition

Invariant under the reflection group of the 600-cell.

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 4.66e-10).