Barth Sextic

Barth sextic surface

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

b = 0;
ϕ=GoldenRatio;
ContourPlot3D[
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 https://plus.google.com/114982179961753756261/posts/B6zWUjNTaVr
barth sextic golden
Barth sextic. 〔by Abdelaziz Nait Merzouk Apr 21, 2016 at https://plus.google.com/114982179961753756261/posts/B6zWUjNTaVr

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 https://plus.google.com/114982179961753756261/posts/VbbbqdaCbVf

Barth Decic

barth decic infinity cut03
Barth Decic Now cut by the plane at inifinity 〔by Abdelaziz Nait Merzouk Apr 21, 2016 at https://plus.google.com/114982179961753756261/posts/U4pB2yRY3um

Reference

Barth Sextic By John Baez. @ http://blogs.ams.org/visualinsight/2016/04/15/barth-sextic/