Math Art
Septic with 16 Triple Points

Septic with 16 Triple Points

Septic with 16 Triple Points is an algebraic surface, defined by an implicit equation, immersed with singularities, with Td symmetry (a point group).

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algebraic aperiodic def-implicit implemented singular tradition-classical

Formula

−2⁢((x+y+z+1)2⁢(x⁢y+x⁢z+x+y⁢z+y+z)⁢(x⁢y⁢z+x⁢y+x⁢z+y⁢z)−(x+y+z+1)⁢(x⁢y⁢z+x⁢y+x⁢z+y⁢z)2−(x+y+z+1)3⁢x⁢y⁢z)−24⁢(x+y+z+1)⁢(x⁢y+x⁢z+x+y⁢z+y+z)3+21⁢(x⁢y+x⁢z+x+y⁢z+y+z)2⁢(x⁢y⁢z+x⁢y+x⁢z+y⁢z)+15⁢(x+y+z+1)⁢(x⁢y+x⁢z+x+y⁢z+y+z)⁢x⁢y⁢z−3⁢(x⁢y⁢z+x⁢y+x⁢z+y⁢z)⁢x⁢y⁢z=0-2 \left(\left(x + y + z + 1\right)^{2} \left(x y + x z + x + y z + y + z\right) \left(x y z + x y + x z + y z\right) - \left(x + y + z + 1\right) \left(x y z + x y + x z + y z\right)^{2} - \left(x + y + z + 1\right)^{3} x y z\right) - 24 \left(x + y + z + 1\right) \left(x y + x z + x + y z + y + z\right)^{3} + 21 \left(x y + x z + x + y z + y + z\right)^{2} \left(x y z + x y + x z + y z\right) + 15 \left(x + y + z + 1\right) \left(x y + x z + x + y z + y + z\right) x y z - 3 \left(x y z + x y + x z + y z\right) x y z = 0

Properties

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

Definition

A one-parameter family of S_4-symmetric septics; the general element carries sixteen ordinary triple points. Imposing the symmetry is what reduces the problem enough to solve (paper, Theorem 5.1).

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 1.14e-13).