K3 Surfaces

The following images are visualisations of the K3 surfaces that I studied for my thesis. They are all screenshots taken from the program MathMod.

The notation [A,B,C,D,E] means the K3 surface is defined by the equation

This means that the following images only shows the projection of the K3 surface onto an affine patch (w=1) of the real space. The actual K3 surfaces live in a 4 complex-dimensional world (or a 8 dimensional real world).

The surface [425,0,-1207,-1207,1025]

This surface is exciting as all 320 conics (i,e, circles, ellipses and hyperbolas) on it occurs over the rationals, and hence can be seen. On the left, you can scroll through 3 different view of the surface, while on the right there are 10 views of the surface with plane cuts to highlight various pairs of conics (all conics come in pairs lying on the same plane) in different configurations.

The configuration notation ((a,b),(c,d)) means that branch one of hyperbola 1 intersect hyperbola 2 in a points on branch one and b points on branch two; while branch two of hyperbola 1 intersect hyperbola 2 in c points on branch one and d points on branch two .

The surface [1,-15,-3,-3,-3]

Very symmetrical

Other surfaces to show different points of view of [425,...]??