Barth's Sextic Surface and two associated regular dodecahedra

A Barth sextic surface is a surface given by the equation

4(Φ2x2-y2)(Φ2y2-z2)(Φ2z2-x2) - (1+2Φ)(x2+y2+z2- w2)2w2=0

Φ=(sqrt(5)+1)/2 is the golden number, appearing in the golden section and w is a parameter further taken as 1.

A lot of beautiful images of this surface are available; in many cases those images are rendered using a raytracing programme such as Povray.

The following animation shows the part of the surface inside the sphere centered at the origin and radius 2.5.

Barth's sextic surface has 65 double points; 20 double points are the vertices of a regular dodecahedron (the red one) and 30 other double points are the midpoints of the edges of another regular dodecahedron (the yellow one). Both dodecahedra have the same center and parallel faces. These properties are illustrated in the following animations.

Barth's Decic Surface

A Barth decic surface is a surface given by the equation

8(x24y2)(y24z2)(z24x2) (x4+y4+z4-2x2y2-2y2z2-2z2x2)

+(3+5Φ)(x2+y2+z2-w2)2 [x2+y2+z2-(2-Φ)w2]2w2=0

Φ=(sqrt(5)+1)/2 is the golden number, appearing in the golden section and w is a parameter further taken as 1.

A lot of beautiful images of this surface are available; in many cases those images are rendered using a raytracing programme such as Povray.

The following animation shows the part of the surface inside the sphere centered at the origin and radius 45.

Barth's decic surface has 345 double points.
Herman Serras, October 2005