Monday, April 22, 2013

A generic truchet tile for fabric structures

A generic quadrilateral truchet tile for fabric structures. (This tile is "inked-through": flipping the tile over shows its mirror image.)

A closed surface of any complexity can be cut or broken into simple pieces (i.e., topological disks.) Let's look at this in more detail, considering only orientable surfaces---those surfaces where we can consistently distinguish "outside" and "inside."

For example, if we cut a piece from a surface that is simple except for having a hole in its middle, we can make an additional cut from its periphery to the boundary of the hole to eliminate the problem. Another example: if the piece is simple except for having a handle in its middle, cutting all the way across the piece, underneath its handle, converts the piece into a cylinder. A cylinder (topologically speaking) is just a disk with a hole in its middle, so we can make a second cut to eliminate the problem.

Place a vertex everywhere a cut ends, or rejoins itself, or intersects another cut, then we can see our simple pieces as polygons in a broad sense: we allow the sides of a polygon to be nonplanar curves, we allow a polygon to have as few as one sides, and we allow a polygon to have monovalent as well as the usual multivalent vertices on its periphery.

As a simple example, here are cuts that separate a sphere into simple pieces (a 12-sided piece and nine 4-sided pieces.)

Cuts dividing a sphere into a 12-sided piece and nine 4-sided pieces.

So now we have all polygonal pieces. Proceed to chop all the polygons into triangles: create a new vertex in the center of each polygon, and cut the piece like a pie, making cuts from the vertices on the periphery to the new vertex in the center. (Do this procedure even if a polygon happens to be a triangle already.)

Secondary cuts (dashed lines) cutting all ten polygonal pieces into triangles.

Now we have exclusively triangular pieces. We proceed to glue the primary cuts back together again. We do this in a way that records the former location of each primary cut with a solid line. Since each repaired cut joins two triangular pieces, the result is a polygon with 2 * 3 - 2 = 4 sides. Now we have cut the surface exclusively into quadrilaterals. Each repaired cut (marked by a solid line) is a diagonal of a quadrilateral.

Every orientable, closed surface can be cut into quadrilaterals with distinguished diagonals. The distinguished diagonals comprise the edges of a map of the surface, the remaining diagonals comprise the edges of the dual map.

The above is a well known mathematical fact, but it is nonetheless technologically exciting because it means that any technology that can be described by quadrilateral truchet tiles can populate any closed, orientable surface.

We now repair the secondary cuts as well, recording their former location with dashed lines. The surface is now fully reassembled, and looks like the diagram immediately above. We will now use this diagram as the template for placing quadrilateral truchet tiles.

As drawn, the dashed-line triangles composing the 12-sided polygon are very distorted. In that portion of the surface, we will not attempt to distort the truchet tiles correctly; we will simply show them with white gaps between them. It will be evident in the following diagrams how portions of the truchet tiles need to be distorted to mate up across the gaps.

The quadrilateral truchet tile we are going to place (first image above) represents a generic fabric structure. It can represent crochet, net tying, linking, looping, peyote stitch, balloon tying, unit weaving and others. (Among those it cannot represent are knitting and nalebinding.) The procedure described here is for making a surface from a single thread, working along an advancing front of finished fabric, the work ending exactly where it started.

The generic truchet tile is an "inked through" tile: flipping it over reveals its mirror image. In each quadrilateral opening there are eight ways this tile can be played. The choices are: whether or not to flip the tile over, and, then, which of four possible orientations to choose in placing it.

Because working order is a concern, we must begin with a simple operation that fixes a working order. To do this: find primary cut that is part of a cycle of primary cuts and gray it out; repeat the previous step until no black-edged cycles remain. The black edges remaining are a spanning tree. Here is a spanning tree for the example.
A spanning tree (black edges.)

Placement rules:
Rule 1: Threads must not cross black edges; they must cross gray edges. 

Rule 2: The first pass of the thread through the tile follows the route and direction indicated by the drop's point. (The indicated direction is into the concavity of the drop.)

Tiling procedure:
0. Arbitrarily choose some edge as the origin. 
1. Place a tile on the origin edge in an orientation satisfying Rule One. 
2. Guided by the drop's point, find the next edge. (If the next edge is already occupied by a tile, continue along the indicated route of the thread until you come to an unoccupied next edge, or until the fabric is complete.) 
3. Place a tile on the next edge in an orientation satisfying Rule One. 
4. Flip the tile (i.e., rotate it 180 degrees around either of its diagonals, or both) until the established route and direction comports with that indicated by the tile per Rule Two.
5. Go to step 2.

The first few tile placements in this example:

First tile.

Second tile.

Third tile.

Fourth tile.

Fifth tile.
Sixth tile.
Seventh tile.
Eighth tile.
Ninth tile.

Completed fabric.

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