Skeletal Graphs of Triply Periodic Surfaces

Although the notion of a skeletal graph of a surface does not have a precise mathematical definition, several experiments suggest that certain surfaces do have unique skeletal graphs. The skeletal graph can be thought of as the end result of expanding or shrinking a surface along the direction of its normal vectors, while avoiding any pinching off that would change the topology of the surface, until all that remains is a connected graph of arcs and nodes. Examples of surfaces for which there are obvious skeletal graphs are the triply periodic P, D, G, and iWp surfaces. These graphs are identical whatever the 'flavor' (minimal, CMC, or level) of the surface.

Here are skeletal graphs for all of the surfaces listed on the Triply Periodic Level Surfaces page, where the thickened representations of the skeletal graphus were generated using level set methods. These surfaces, being oriented, have two skeletal graphs, which are designated primary and secondary. The primary and secondary graphs of the P and D surfaces are congruent; those of the G surface have mirror symmetry with eachother; and the primary and secondary graphs of the W and N surfaces are different. The domain of 2 pi (meaning the cube enclosing the volume from -pi to pi in X, Y, and Z) gives the unit cell for all the surfaces. The unit cell is the smallest portion of space which can be repeated in the X, Y, and Z directions with translations to form the complete surface. For the statistics, the unit cell's dimensions are normalized to occupy a cube whose sides are length 2. (NOTE: The secondary skeletal graph for the 'W' (iWp) surface is incorrect.)

Controls Key
Display Miller Indices Domain
primary skeletal graph
surface
secondary skeletal graph
: length of arcs
: number of arcs per unit cell
: total arclength per unit cell
: number of arcs meeting at nodes
: number of nodes per unit cell
: surface area in unit cell
surface view statistics
G
0.707
12
8.48
3
8
12.40
D
0.866
16
13.86
4
8
15.50
P
2.0
3
6.0
6
1
9.42
W
1.732; 0.707
8; 24
13.86; 16.97
8; 4
2; 12
14.20
N
1.414; 1.414
12; 12
16.97; 16.97
12,4; 6,4
1,3; 1,3
14.13

The thichened representations of skeletal graphis depicted here were generated using as level surfaces of some rather complicated expressions. A much simpler approach to approximating the same skeletal graph also using level surfaces, illustrated here.