Math Art
Endrass Octic Family

Endrass Octic Family

Endrass Octic Family is an algebraic surface, defined by an implicit equation, immersed with singularities.

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algebraic aperiodic def-implicit implemented singular tradition-classical

Formula

0.25⁢(x2−1)⁢(y2−1)⁢((x+y)2−2)⁢((x−y)2−2)−(−0.6035533905932737⁢(x2+y2)2+(1.7071067811865475⁢z2+1.4874368670764582)⁢(x2+y2)−z4−0.9142135623730951⁢z2−1.1231601717798214)2=00.25 \left(x^{2} - 1\right) \left(y^{2} - 1\right) \left(\left(x + y\right)^{2} - 2\right) \left(\left(x - y\right)^{2} - 2\right) - \left(-0.6035533905932737 \left(x^{2} + y^{2}\right)^{2} + \left(1.7071067811865475 z^{2} + 1.4874368670764582\right) \left(x^{2} + y^{2}\right) - z^{4} - 0.9142135623730951 z^{2} - 1.1231601717798214\right)^{2} = 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

Endrass's 5-parameter family of octics with D8 x Z2 symmetry, combining two constructions of B. Segre: a product of eight planes (four symmetric plane-pair factors) minus the square of a degree-4 form in x^2+y^2, z and w, with complex parameters a, b, c, d, e. The named members are promoted to their own records (see specimens).

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 4.44e-16).