Math Art
Humbert Sextic (O = (1:1:1:1) member)

Humbert Sextic (O = (1:1:1:1) member)

Humbert Sextic (O = (1:1:1:1) member) 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

(((x+0)2+(y+0)2+(z+0)2−1.6900000000000002)2−3.106870229007634⁢(1−(z+0)−1.4142135623730951⁢(x+0))⁢(1−(z+0)+1.4142135623730951⁢(x+0))⁢(1+z+0+1.4142135623730951⁢(y+0))⁢(1+z+0−1.4142135623730951⁢(y+0)))⁢(((x+1)2+(y+1)2+(z+1)2−6.760000000000001)2−3.106870229007634⁢(2−(z+1)−1.4142135623730951⁢(x+1))⁢(2−(z+1)+1.4142135623730951⁢(x+1))⁢(2+z+1+1.4142135623730951⁢(y+1))⁢(2+z+1−1.4142135623730951⁢(y+1))−2⁢(((x+0.5)2+(y+0.5)2+(z+0.5)2−3.8025)2−3.106870229007634⁢(1.5−(z+0.5)−1.4142135623730951⁢(x+0.5))⁢(1.5−(z+0.5)+1.4142135623730951⁢(x+0.5))⁢(1.5+z+0.5+1.4142135623730951⁢(y+0.5))⁢(1.5+z+0.5−1.4142135623730951⁢(y+0.5)))+2⁢(((x−0.5)2+(y−0.5)2+(z−0.5)2−0.42250000000000004)2−3.106870229007634⁢(0.5−(z−0.5)−1.4142135623730951⁢(x−0.5))⁢(0.5−(z−0.5)+1.4142135623730951⁢(x−0.5))⁢(0.5+z−0.5+1.4142135623730951⁢(y−0.5))⁢(0.5+z−0.5−1.4142135623730951⁢(y−0.5)))−(((x−1)2+(y−1)2+(z−1)2−0)2−3.106870229007634⁢(0−(z−1)−1.4142135623730951⁢(x−1))⁢(0−(z−1)+1.4142135623730951⁢(x−1))⁢(0+z−1+1.4142135623730951⁢(y−1))⁢(0+z−1−1.4142135623730951⁢(y−1))))0.256⁢(((x+1)2+(y+1)2+(z+1)2−6.760000000000001)2−3.106870229007634⁢(2−(z+1)−1.4142135623730951⁢(x+1))⁢(2−(z+1)+1.4142135623730951⁢(x+1))⁢(2+z+1+1.4142135623730951⁢(y+1))⁢(2+z+1−1.4142135623730951⁢(y+1))−2⁢(((x+0.5)2+(y+0.5)2+(z+0.5)2−3.8025)2−3.106870229007634⁢(1.5−(z+0.5)−1.4142135623730951⁢(x+0.5))⁢(1.5−(z+0.5)+1.4142135623730951⁢(x+0.5))⁢(1.5+z+0.5+1.4142135623730951⁢(y+0.5))⁢(1.5+z+0.5−1.4142135623730951⁢(y+0.5)))+2⁢(((x−0.5)2+(y−0.5)2+(z−0.5)2−0.42250000000000004)2−3.106870229007634⁢(0.5−(z−0.5)−1.4142135623730951⁢(x−0.5))⁢(0.5−(z−0.5)+1.4142135623730951⁢(x−0.5))⁢(0.5+z−0.5+1.4142135623730951⁢(y−0.5))⁢(0.5+z−0.5−1.4142135623730951⁢(y−0.5)))−(((x−1)2+(y−1)2+(z−1)2−0)2−3.106870229007634⁢(0−(z−1)−1.4142135623730951⁢(x−1))⁢(0−(z−1)+1.4142135623730951⁢(x−1))⁢(0+z−1+1.4142135623730951⁢(y−1))⁢(0+z−1−1.4142135623730951⁢(y−1))))0.256−14.143580916030535⁢(−(((x+1)2+(y+1)2+(z+1)2−6.760000000000001)2−3.106870229007634⁢(2−(z+1)−1.4142135623730951⁢(x+1))⁢(2−(z+1)+1.4142135623730951⁢(x+1))⁢(2+z+1+1.4142135623730951⁢(y+1))⁢(2+z+1−1.4142135623730951⁢(y+1)))+8⁢(((x+0.5)2+(y+0.5)2+(z+0.5)2−3.8025)2−3.106870229007634⁢(1.5−(z+0.5)−1.4142135623730951⁢(x+0.5))⁢(1.5−(z+0.5)+1.4142135623730951⁢(x+0.5))⁢(1.5+z+0.5+1.4142135623730951⁢(y+0.5))⁢(1.5+z+0.5−1.4142135623730951⁢(y+0.5)))−8⁢(((x−0.5)2+(y−0.5)2+(z−0.5)2−0.42250000000000004)2−3.106870229007634⁢(0.5−(z−0.5)−1.4142135623730951⁢(x−0.5))⁢(0.5−(z−0.5)+1.4142135623730951⁢(x−0.5))⁢(0.5+z−0.5+1.4142135623730951⁢(y−0.5))⁢(0.5+z−0.5−1.4142135623730951⁢(y−0.5)))+((x−1)2+(y−1)2+(z−1)2−0)2−3.106870229007634⁢(0−(z−1)−1.4142135623730951⁢(x−1))⁢(0−(z−1)+1.4142135623730951⁢(x−1))⁢(0+z−1+1.4142135623730951⁢(y−1))⁢(0+z−1−1.4142135623730951⁢(y−1)))6⁢(−(((x+1)2+(y+1)2+(z+1)2−6.760000000000001)2−3.106870229007634⁢(2−(z+1)−1.4142135623730951⁢(x+1))⁢(2−(z+1)+1.4142135623730951⁢(x+1))⁢(2+z+1+1.4142135623730951⁢(y+1))⁢(2+z+1−1.4142135623730951⁢(y+1)))+8⁢(((x+0.5)2+(y+0.5)2+(z+0.5)2−3.8025)2−3.106870229007634⁢(1.5−(z+0.5)−1.4142135623730951⁢(x+0.5))⁢(1.5−(z+0.5)+1.4142135623730951⁢(x+0.5))⁢(1.5+z+0.5+1.4142135623730951⁢(y+0.5))⁢(1.5+z+0.5−1.4142135623730951⁢(y+0.5)))−8⁢(((x−0.5)2+(y−0.5)2+(z−0.5)2−0.42250000000000004)2−3.106870229007634⁢(0.5−(z−0.5)−1.4142135623730951⁢(x−0.5))⁢(0.5−(z−0.5)+1.4142135623730951⁢(x−0.5))⁢(0.5+z−0.5+1.4142135623730951⁢(y−0.5))⁢(0.5+z−0.5−1.4142135623730951⁢(y−0.5)))+((x−1)2+(y−1)2+(z−1)2−0)2−3.106870229007634⁢(0−(z−1)−1.4142135623730951⁢(x−1))⁢(0−(z−1)+1.4142135623730951⁢(x−1))⁢(0+z−1+1.4142135623730951⁢(y−1))⁢(0+z−1−1.4142135623730951⁢(y−1)))6=0\frac{\frac{\frac{\frac{\left(\left(\left(x + 0\right)^{2} + \left(y + 0\right)^{2} + \left(z + 0\right)^{2} - 1.6900000000000002\right)^{2} - 3.106870229007634 \left(1 - \left(z + 0\right) - 1.4142135623730951 \left(x + 0\right)\right) \left(1 - \left(z + 0\right) + 1.4142135623730951 \left(x + 0\right)\right) \left(1 + z + 0 + 1.4142135623730951 \left(y + 0\right)\right) \left(1 + z + 0 - 1.4142135623730951 \left(y + 0\right)\right)\right) \left(\left(\left(x + 1\right)^{2} + \left(y + 1\right)^{2} + \left(z + 1\right)^{2} - 6.760000000000001\right)^{2} - 3.106870229007634 \left(2 - \left(z + 1\right) - 1.4142135623730951 \left(x + 1\right)\right) \left(2 - \left(z + 1\right) + 1.4142135623730951 \left(x + 1\right)\right) \left(2 + z + 1 + 1.4142135623730951 \left(y + 1\right)\right) \left(2 + z + 1 - 1.4142135623730951 \left(y + 1\right)\right) - 2 \left(\left(\left(x + 0.5\right)^{2} + \left(y + 0.5\right)^{2} + \left(z + 0.5\right)^{2} - 3.8025\right)^{2} - 3.106870229007634 \left(1.5 - \left(z + 0.5\right) - 1.4142135623730951 \left(x + 0.5\right)\right) \left(1.5 - \left(z + 0.5\right) + 1.4142135623730951 \left(x + 0.5\right)\right) \left(1.5 + z + 0.5 + 1.4142135623730951 \left(y + 0.5\right)\right) \left(1.5 + z + 0.5 - 1.4142135623730951 \left(y + 0.5\right)\right)\right) + 2 \left(\left(\left(x - 0.5\right)^{2} + \left(y - 0.5\right)^{2} + \left(z - 0.5\right)^{2} - 0.42250000000000004\right)^{2} - 3.106870229007634 \left(0.5 - \left(z - 0.5\right) - 1.4142135623730951 \left(x - 0.5\right)\right) \left(0.5 - \left(z - 0.5\right) + 1.4142135623730951 \left(x - 0.5\right)\right) \left(0.5 + z - 0.5 + 1.4142135623730951 \left(y - 0.5\right)\right) \left(0.5 + z - 0.5 - 1.4142135623730951 \left(y - 0.5\right)\right)\right) - \left(\left(\left(x - 1\right)^{2} + \left(y - 1\right)^{2} + \left(z - 1\right)^{2} - 0\right)^{2} - 3.106870229007634 \left(0 - \left(z - 1\right) - 1.4142135623730951 \left(x - 1\right)\right) \left(0 - \left(z - 1\right) + 1.4142135623730951 \left(x - 1\right)\right) \left(0 + z - 1 + 1.4142135623730951 \left(y - 1\right)\right) \left(0 + z - 1 - 1.4142135623730951 \left(y - 1\right)\right)\right)\right)}{0.25}}{6} \left(\left(\left(x + 1\right)^{2} + \left(y + 1\right)^{2} + \left(z + 1\right)^{2} - 6.760000000000001\right)^{2} - 3.106870229007634 \left(2 - \left(z + 1\right) - 1.4142135623730951 \left(x + 1\right)\right) \left(2 - \left(z + 1\right) + 1.4142135623730951 \left(x + 1\right)\right) \left(2 + z + 1 + 1.4142135623730951 \left(y + 1\right)\right) \left(2 + z + 1 - 1.4142135623730951 \left(y + 1\right)\right) - 2 \left(\left(\left(x + 0.5\right)^{2} + \left(y + 0.5\right)^{2} + \left(z + 0.5\right)^{2} - 3.8025\right)^{2} - 3.106870229007634 \left(1.5 - \left(z + 0.5\right) - 1.4142135623730951 \left(x + 0.5\right)\right) \left(1.5 - \left(z + 0.5\right) + 1.4142135623730951 \left(x + 0.5\right)\right) \left(1.5 + z + 0.5 + 1.4142135623730951 \left(y + 0.5\right)\right) \left(1.5 + z + 0.5 - 1.4142135623730951 \left(y + 0.5\right)\right)\right) + 2 \left(\left(\left(x - 0.5\right)^{2} + \left(y - 0.5\right)^{2} + \left(z - 0.5\right)^{2} - 0.42250000000000004\right)^{2} - 3.106870229007634 \left(0.5 - \left(z - 0.5\right) - 1.4142135623730951 \left(x - 0.5\right)\right) \left(0.5 - \left(z - 0.5\right) + 1.4142135623730951 \left(x - 0.5\right)\right) \left(0.5 + z - 0.5 + 1.4142135623730951 \left(y - 0.5\right)\right) \left(0.5 + z - 0.5 - 1.4142135623730951 \left(y - 0.5\right)\right)\right) - \left(\left(\left(x - 1\right)^{2} + \left(y - 1\right)^{2} + \left(z - 1\right)^{2} - 0\right)^{2} - 3.106870229007634 \left(0 - \left(z - 1\right) - 1.4142135623730951 \left(x - 1\right)\right) \left(0 - \left(z - 1\right) + 1.4142135623730951 \left(x - 1\right)\right) \left(0 + z - 1 + 1.4142135623730951 \left(y - 1\right)\right) \left(0 + z - 1 - 1.4142135623730951 \left(y - 1\right)\right)\right)\right)}{0.25}}{6} - \frac{\frac{14.143580916030535 \left(-\left(\left(\left(x + 1\right)^{2} + \left(y + 1\right)^{2} + \left(z + 1\right)^{2} - 6.760000000000001\right)^{2} - 3.106870229007634 \left(2 - \left(z + 1\right) - 1.4142135623730951 \left(x + 1\right)\right) \left(2 - \left(z + 1\right) + 1.4142135623730951 \left(x + 1\right)\right) \left(2 + z + 1 + 1.4142135623730951 \left(y + 1\right)\right) \left(2 + z + 1 - 1.4142135623730951 \left(y + 1\right)\right)\right) + 8 \left(\left(\left(x + 0.5\right)^{2} + \left(y + 0.5\right)^{2} + \left(z + 0.5\right)^{2} - 3.8025\right)^{2} - 3.106870229007634 \left(1.5 - \left(z + 0.5\right) - 1.4142135623730951 \left(x + 0.5\right)\right) \left(1.5 - \left(z + 0.5\right) + 1.4142135623730951 \left(x + 0.5\right)\right) \left(1.5 + z + 0.5 + 1.4142135623730951 \left(y + 0.5\right)\right) \left(1.5 + z + 0.5 - 1.4142135623730951 \left(y + 0.5\right)\right)\right) - 8 \left(\left(\left(x - 0.5\right)^{2} + \left(y - 0.5\right)^{2} + \left(z - 0.5\right)^{2} - 0.42250000000000004\right)^{2} - 3.106870229007634 \left(0.5 - \left(z - 0.5\right) - 1.4142135623730951 \left(x - 0.5\right)\right) \left(0.5 - \left(z - 0.5\right) + 1.4142135623730951 \left(x - 0.5\right)\right) \left(0.5 + z - 0.5 + 1.4142135623730951 \left(y - 0.5\right)\right) \left(0.5 + z - 0.5 - 1.4142135623730951 \left(y - 0.5\right)\right)\right) + \left(\left(x - 1\right)^{2} + \left(y - 1\right)^{2} + \left(z - 1\right)^{2} - 0\right)^{2} - 3.106870229007634 \left(0 - \left(z - 1\right) - 1.4142135623730951 \left(x - 1\right)\right) \left(0 - \left(z - 1\right) + 1.4142135623730951 \left(x - 1\right)\right) \left(0 + z - 1 + 1.4142135623730951 \left(y - 1\right)\right) \left(0 + z - 1 - 1.4142135623730951 \left(y - 1\right)\right)\right)}{6} \left(-\left(\left(\left(x + 1\right)^{2} + \left(y + 1\right)^{2} + \left(z + 1\right)^{2} - 6.760000000000001\right)^{2} - 3.106870229007634 \left(2 - \left(z + 1\right) - 1.4142135623730951 \left(x + 1\right)\right) \left(2 - \left(z + 1\right) + 1.4142135623730951 \left(x + 1\right)\right) \left(2 + z + 1 + 1.4142135623730951 \left(y + 1\right)\right) \left(2 + z + 1 - 1.4142135623730951 \left(y + 1\right)\right)\right) + 8 \left(\left(\left(x + 0.5\right)^{2} + \left(y + 0.5\right)^{2} + \left(z + 0.5\right)^{2} - 3.8025\right)^{2} - 3.106870229007634 \left(1.5 - \left(z + 0.5\right) - 1.4142135623730951 \left(x + 0.5\right)\right) \left(1.5 - \left(z + 0.5\right) + 1.4142135623730951 \left(x + 0.5\right)\right) \left(1.5 + z + 0.5 + 1.4142135623730951 \left(y + 0.5\right)\right) \left(1.5 + z + 0.5 - 1.4142135623730951 \left(y + 0.5\right)\right)\right) - 8 \left(\left(\left(x - 0.5\right)^{2} + \left(y - 0.5\right)^{2} + \left(z - 0.5\right)^{2} - 0.42250000000000004\right)^{2} - 3.106870229007634 \left(0.5 - \left(z - 0.5\right) - 1.4142135623730951 \left(x - 0.5\right)\right) \left(0.5 - \left(z - 0.5\right) + 1.4142135623730951 \left(x - 0.5\right)\right) \left(0.5 + z - 0.5 + 1.4142135623730951 \left(y - 0.5\right)\right) \left(0.5 + z - 0.5 - 1.4142135623730951 \left(y - 0.5\right)\right)\right) + \left(\left(x - 1\right)^{2} + \left(y - 1\right)^{2} + \left(z - 1\right)^{2} - 0\right)^{2} - 3.106870229007634 \left(0 - \left(z - 1\right) - 1.4142135623730951 \left(x - 1\right)\right) \left(0 - \left(z - 1\right) + 1.4142135623730951 \left(x - 1\right)\right) \left(0 + z - 1 + 1.4142135623730951 \left(y - 1\right)\right) \left(0 + z - 1 - 1.4142135623730951 \left(y - 1\right)\right)\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

Humbert's 1896 sextic, tied to abelian functions of genus 3 and built from a Kummer quartic: it inherits the Kummer surface's 16 nodes and adds a triple point at (1:1:1:1).

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 9.3e-10).