Barth Sextic

Barth sextic surface

Barth Sextic is a algebraic surface of degree 6. Its formula is:

b = 0;
4*(ϕ^2*x^2 - y^2)*(ϕ^2*y^2 - z^2)*
    (ϕ^2*z^2 - x^2) - (1 + 2*ϕ)*(x^2 + y^2 + z^2 - b^2)^2*b^2,
  {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, PlotPoints -> {6, 6}]

where ϕ is the golden ratio (ϕ = (1 + Sqrt[5])/2 ≈ 1.61803) and b is the parameter. In the above plot, b=1.

It's interesting because it has the maximum of double points for a degree 6 surface. It has 65 double points.

a double point is like the tip of a cone, a point that satisfies all of the following:

D[f[x,y,z],x] means derivative with respect to x.

Barth Sextic is found by Wolf Barth in 1996.

• Wolf Barth, Two projective surfaces with many nodes, admitting the symmetries of the icosahedron, Journal of Algebraic Geometry 5 (1994), 173–186.

The related surface is Barth decic of degree 10 with 345 double points.

Barth Sextic has icosahedral symmetry.

Barth sextic

When b is 0, the surface is 6 intersecting planes, arranged in a way like 3 sets of x-crossed planes intersecting from mutually orthogonal directions. In the above image, the cleavages is an artifact of plotting software.

Barth Sextic parameter change, b goes from 0 to 1.

Mathematica file

Barth sextic rotated cutaway view 46634
Barth sextic: A view obtained by applying a rotation involving the w (homogeneous) coordinate in order to make the 15 ordinary double points (the cone-like features) that are located on the plane at infinity visible. The cut is along the (transformed) plane at infinity. 〔by Abdelaziz Nait Merzouk Apr 21, 2016 at
barth sextic golden
Barth sextic. 〔by Abdelaziz Nait Merzouk Apr 21, 2016 at

Barth Sextic Mapped to a Unit Sphere

barth sextic mapped to sphere barth sextic mapped to sphere cutaway view
Barth sextic. The inverse of the mapping: R^3 → R^3 : p → 2 p / ( 1 p²) maps the plane at infinity onto the unit sphere. We also end up with two copies of the euclidean space: One inside the unit sphere and the other outside. In these pictures, the 〔by Abdelaziz Nait Merzouk Apr 24, 2016 at

Barth Decic

barth decic infinity cut03
Barth Decic Now cut by the plane at inifinity 〔by Abdelaziz Nait Merzouk Apr 21, 2016 at


Barth Sextic By John Baez. @