Barth Sextic
![Barth sextic surface](barth_sextic.png)
Barth Sextic is a algebraic surface of degree 6. Its formula is:
b = 0; phi=GoldenRatio; ContourPlot3D[ 4*(phi^2*x^2 - y^2)*(phi^2*y^2 - z^2)* (phi^2*z^2 - x^2) - (1 + 2*phi)*(x^2 + y^2 + z^2 - b^2)^2*b^2, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]
where phi is the golden ratio 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:
f[x,y,z] == 0
D[f[x,y,z],x] == 0
D[f[x,y,z],y] == 0
D[f[x,y,z],z] == 0
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](barth_sextic_plane.png)
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 rotated cutaway view 46634](Barth_sextic_rotated_cutaway_view_46634.jpg)
![barth sextic golden](barth_sextic_golden.jpg)
Barth Sextic Mapped to a Unit Sphere
![barth sextic mapped to sphere](barth_sextic_mapped_to_sphere.png)
![barth sextic mapped to sphere cutaway view](barth_sextic_mapped_to_sphere_cutaway_view.png)
Barth Decic
![barth decic infinity cut03](barth_decic_infinity_cut03.png)
Reference
[Barth Sextic By John Baez. At http://blogs.ams.org/visualinsight/2016/04/15/barth-sextic/ , accessed on 2016-05-02 ]