The Gyroid triply-periodic labyrinth

A homogeneous three-coordinated labyrinth where maxima of the Euclidean distance map are the midpoint of edges and not the nodes of the network

A mathematical model labyrinth with a well-defined three-connected network and the peculiar property that the nodes of the network are inflection points (rather than maxima) of the Euclidean distance map whereas the maxima are located at the mid-points of the edges. The Gyroid labyrinth is periodic in three dimensions, symmetry and it is very homogeneous in terms of channel diameter variations. The interface between solid and pore space is a minimal surface.
Gyroid labyrinth



Description

... under construction ....

What is pathological about it?

The Gyroid is remarkable in that there is no doubt that the network shown above is the correct homotopy-equivalent network [2], yet the nodes of the network (red spheres) are not the widest points of the network. The widest points are at the mid-points of the edges (yellow spheres). This is all the more unusual in that the Gyroid is one of the most homogeneous labyrinthine space partitions of its class [2,3].

References

[1] A. Schoen, Infinite Periodic minimal surfaces without self-intersections, Technical Report, NASA (1970)
[2] G.E. Schroeder, S.J. Ramsden, A.G. Christy, and S.T. Hyde, Medial Surfaces of hyperbolic structures, European Physical Journal B, 35, 551-564 (2003)
[3] G.E. Schroeder, Skeletons in the labyrinths, PhD thesis, Australian National University, Canberra, Australia (2005)


Datasets

The following data sets represent the cubic translational unitcell, in I4(1)32 space group, of the Gyroid labyrinth. Periodic boundary conditions can be applied in all three directions.

The dataset is generated by creating a triangulation of the Gyroid interface (from Weierstrass integration) in its unit cube 0 < x,y,z < 1. Then all points on a grid of n3 points that are on the positive side of the surface (in normal direction) are set to 1, all others to 0. The points on that grid are {(i+0.5)*dx, (j+0.5)*dy, (k+0.5)*dz} with i,j,k = 0, ..., n-1. The grid values are then written to the data file.

Gyroid triply-periodic labyrinth (n=50)