Flight through the cubic Gyroid triply-periodic minimal surface

The cubic Gyroid surface is a “bicontinuous” minimal surface that divides space into two intertwined network- or labyrinth-like subdomains. The two subdomains are enantiomorphic, that is identical apart from different handedness. This animation shows a rotation of a fairly large subsection of this periodic surface, and a flight through one of the two domains. The flight path first follows a three-fold screw axis, and then turns into a four-fold screw axis. In a second run of the same animation, the same surface is shown together with the skeletal graphs that are often used to represent the Gyroid.

Gyroid FlyThru, AVI version of the animation (90MB)

This animation was produced at the ANU during my PhD. Special thanks to Stuart Ramsden for a lot of help!

Bonnet transformation between the D, G and P minimal surfaces

The three cubic triply-periodic minimal Diamond, Gyroid and Primitive surfaces are related to each other by the so-called Bonnet transformation. That means they are specific members of a single one-parameter family of surfaces, called the Bonnet family with free parameter t for the specific values t=0 (Diamond), t=38.????^o (Gyroid) and t=90^o (Primitive). However, in contrast to the D, G and P surfaces, all other members of that family have self-intersections. The Animation shows the transformation of a single asymetric patch in E³ (top left), of an extended patch where the coloring of the asymmetric patch has been retained and also showing the three-fold rotation axis common to all members (bottom left) and of a large enough patch of the surface that illustrates the self-intersections (bottom right). In that last image one side of the surface is orange, and the other green. Also shown are the tiles of the complex plane that (via the Weierstrass equation) give rise to the asymmetric unit patch. (Details...)

Bonnet transformation plus rotations of cubic cases, as .mpeg for Apple Quicktime players (40 MBytes)