This is one of the eleven possibilities of Archimedean tilings of neck structures, which I've finished coding lately. Each vertex of the tiling is composed of three equilateral triangles and two squares.
The ms that I am currently working on is titled "Doubly and Triply Periodic Porous Graphitic Structures". I wonder if you have better suggestion to the title. I was first considering doubly and triply periodic quasi-minimal surfaces, since I am not sure that mathematicians would agree that these are minimal surfaces realized as graphitic structures. The paper on high-genus fullerenes that we've just published uses the word "high-genus". Although the periodic structures discussed in this new article are still made of neck structures, they may not be as high-genus as their 0D analogs since the genus value per unit cell is not-so-high.
(By Chuang on byjingroup blog, Tuesday, July 21, 2009)
A beaded model for 2-d periodic tile (Platonic tile) of graphitic structures made by Chuang a few months ago: