Thursday, June 27, 2013

Puck weaving the dodecahedron

The skeleton graph of the dodecahedron is not bipartite, so, despite its high symmetry, making a dodecahedral basket is a project that introduces the real housekeeping chores of puck weaving.

We have a goal to finish at an all-outboard vertex in order to make the closing moves easy and familiar. More burdensome, we must keep track of all the Petrie paths in the basket to insure that we phase them consistently (i.e., should we place the next "unprecedented" puck centrally or outboard?). As far as the absolute phase goes, we only care about the three targeted Petrie polygons that intersect at our terminal vertex. For all the other Petrie polygons, we only care that we phase them consistently. Should we fail to do so we'll come to a non-sequitur in the weaving.

A vertex on the dodecahedron happens to have an antipodal vertex where the same three Petrie polygons also intersect. Since the half circumference of a dodecahedron (taking its Petrie polygons to be geodesics) is 5 edges, an odd number, an all-outboard vertex on the dodecahedron will always have an all-central vertex at its antipodes. Unfortunately, since we must keep track of the phases of all of the Petrie polygons anyway, this property does not really make the puck weaving of a dodecahedron any easier. 

The best working method seems to be to mark up an undip word for the dodecahedron with the correct phasing of the "unprecedented" weaver to be added at each open letter. Since central phasing seems a natural default, I mark open letters with a prime if the phasing of the unprecedented weaver is outboard. The whole weaving must be calculated in advance to get this right. Below is an example worked out by hand for the dodecahedron.

The three Petrie polygons that intersect at a vertex on the dodecahedron also intersect at its antipodes.

The phasing of all of the Petrie polygons must be calculated before weaving can begin. The pink line marks the Hamilton path traced by the undip word. The Petrie paths follow the crossovers (half-twists) at the midpoint of each edge. Each Petrie polygon has been labelled with a consistent alternation of 'c' and 'o' phasing (central and outboard.)

The undip word with its housekeeping markup is unnndn'unu'uu'pppdpdpdd .

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