Thursday, August 25, 2011

Truchet tiles for the map operations important in basket weaving




DUAL OPERATIONS

A pair of map operations A() and B() are considered dual to each other if (refer to George W. Hart's page on the Conway Notation for Polyhedra):

A(M)=Du(B(Du(M))

In this chart the dual operations are:

Id() and Du(),

Ra() and Me(),

Ki() and Tr(),

Ko() and Le().


ROTATION PAIRS

Also of interest are pairs of map operations that we will call rotation pairs, i.e., pairs for which:

A(M) = B(Du(M)) and B(M) = A(Du(M))

Rotation pairs in the chart above are:

Ki() and Ko(),

Tr() and Le().


ROTATION SELF-PAIRS

Map operations with four-fold rotational symmetry are in a sense their own rotation pairs. The ones in the chart above are:

radial, Ra();

medial, Me();

expand, Ex().

The map operation dual, Du(M), rotates the edges of the base map by 90 degrees (compare against the identity operation, Id(M).) Because they have four-fold rotational symmetry, the self-paired operations are unaffected by such a rotation in the base map. They produce the same resultant map whether they operate on a given base map or on its dual.

MAP OPERATIONS THAT YIELD BASKETS

Medial, Me(), and expand, Ex() (expand being the same as medial taken twice,) always yield four-regular, chess-colorable maps. Such a map specifies a chiral pair of plain-woven baskets. Rarely do we need to specify the chirality of weave we have in mind, so I use the term loosely and refer to every four-regular, chess-colorable map as a basket.

Every map can be operated on by medial, Me(), and therefore every map specifies a basket. We will call that basket its medial image.

Every four-regular, chess-colorable map is the medial image of some other map and of that map's dual. We will call those other maps its medial bases.


MAP OPERATIONS THAT YIELD TRIVALENT MAPS

In genetic (a.k.a., undip) fabric construction we are interested in trivalent (a.k.a., cubic) maps, which are maps with three edges meeting at every vertex. The map operations in the chart above that always yield trivalent maps are:

truncate, Tr();

leapfrog, Le();

bevel, Be().


MAP OPERATIONS THAT YIELD HAMILTONIAN MAPS

In genetic fabric construction we are also interested in hamiltonian maps, which are maps that possess a closed path that visits every vertex exactly once. In the chart above the map operations that always yield hamiltonian maps are:

expand, Ex();

bevel, Be().

The argument presented in the previous post shows that this is true for Mrs. Stott's expansion operation, Ex(). When seen as a further development of Ex(M), the action of bevel, Be(M), is to split each vertex of Ex(M) into two connected vertices, thus further enlarging the connected ring of vertices that surround the former location of a single vertex in M. In the image below, the former location of a vertex in M is shown by a dotted circle, and nearby portions of a hamiltonian circuit are shown in red.



Bevel's splitting and reconnection of vertices neither interrupts hamiltonian circuits that were in Ex(M), nor creates any new vertices that are not incidentally visited by the corresponding circuit in Be(M). We conclude that the map operation bevel, Be(), also always results in a hamiltonian map.

Maps produced by medial and bevel are actually quite abundantly hamiltonian since they possess a a hamiltonian circuit for every spanning tree in the base map. That allows freedom in choosing a working order for an undip basket.

PLACING TILES

To place truchet tiles, we must first perform Ra(M) or Ki(M) algorithmically to locate the boundaries of the tiles. The difference between Ra() and Ki() is that Ki() includes the original edges of the base map.






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