Representation of hypermaps as flow fields

The pentakis dodecahedron hypermap as a flow field.

The distinction we make between hypervertices and hyperedges of a hypergraph amounts to a nominal orientation of the graph: hypervertices (black vertices) are nominally the sources, hyperedges (white vertices) are nominally the sinks. The hyperfaces are the saddles in this nominal flow. Taking the transpose dual (doing a black/white color swap) merely reverses the nominal direction of the flow. The representation above (lacking arrows to indicate a direction of flow) represents both of the transpose duals of the pentakis dodecahedron hypermap. The geometrically important varieties of the hextuplet of hyperduals are the ones that change the color of the saddles. The two diagrams below each represent a different pair of transpose duals from among the hextuplet of hyperduals of the pentakis dodecahedron.


The two white-saddle hyperduals of the pentakis dodecahedron hypermap.


The two black-saddle hyperduals of the pentakis dodecahedron hypermap.