Truchet curves and surfaces

Spanning tree contours, a special class of Truchet contour based upon a random spanning tree of a Truchet tiling's underlying graph, are presented. This spanning tree method is extended to three dimensions to define a Truchet surface with properties similar to its two-dimensional counterpart. Both contour and surface are smooth, have known minimum curvature and known maximum distance to interior points, and high ratios of perimeter to area and area to volume, respectively. Expressions for calculating contour length, contour area, surface area and surface volume directly from the spanning tree are given.