blackwork embroidery and algorithms for maze traversals

Blackwork embroidery and algorithms for mazetraversals

Joshua HoldenJoint work with Lana Holden

Rose-Hulman Institute of Technologyhttp://www.rose-hulman.edu/~holden

Joshua Holden (RHIT) Blackwork embroidery and algorithms 1 / 28

http://www.rose-hulman.edu/~holden

“Spanish Stitch”

Figure: Portrait of Catherine of Aragon, by Lucas Horenbout


“Holbein Stitch”

Figure: Jane Seymour, Queen of England, by Hans Holbein the Younger


The Rules

Figure: A typical piece of fabric used in blackwork


“Double Running Stitch”

9

1 42 6 8

1415

3 5 7

12 1013 11

Figure: Double running stitch


Reversible Patterns: Historical

Figure: A boy’s shirt from the 1540’s


Reversible Patterns: Modern

Figure: "Betsy", by Catherine Strickler, published by Indigo Rose


Graphs and Digraphs

Definition

A (loop-free) graph is a set of vertices, V , and a set of edges, E , whereeach edge is an unordered pair of distinct vertices.

Definition

A (loop-free) digraph is a set of vertices, V , and a set of edges, E ,where each edge is an ordered pair of distinct vertices.(The order is thought of as indicating a “direction”.)


Associating Graphs and Digraphs

Fact

An graph may be associated to a digraph by forgetting about theordering of the pairs.

FactA digraph may be associated to a graph by including both possibleorders (or directions) of each edge.


Walks

Definition

A walk on a graph is a finite alternating sequence of vertices andedges x0, e1, x1, . . . , en, xn where each ei = {xi−1, xi}.

Definition

A graph is connected if there is a walk between any two vertices.

Definition

A directed walk on a digraph is a finite alternating sequence of verticesand edges x0, e1, x1, . . . , en, xn where eachei = (xi−1, xi).


Connected and Strongly Connected

Definition

A digraph is strongly connected if there is a directed walk between anytwo vertices.

Lemma

If a digraph is strongly connected then the associated graph isconnected.

Lemma

If a graph is connected then the associated digraph is stronglyconnected.


Trails and Circuits

Definition

A directed trail is a directed walk where no edge is repeated. (Althoughvertices may be.)

DefinitionA directed circuit is a directed trail where x0 = xn.

Definition

An Eulerian circuit of a digraph is a directed circuit which includes all ofthe edges of the digraph.

Definition

A digraph is Eulerian if it has an Eulerian circuit.


Degrees

Definition

For each vertex v of a digraph, the out degree of v is the number ofedges with v as the starting point.

Definition

For each vertex v of a digraph, the in degree of v is the number ofedges with v as the ending point.


Eulerian Graphs

Theorem

A directed graph is Eulerian if and only if it is strongly connected andthe in degree of each vertex is equal to the out degree.

Lemma

The digraph associated to a graph has the in degree of each vertexequal to the out degree.

Corollary

The digraph associated to a graph is Eulerian if and only if the originalgraph is connected.


Holbeinian Graphs

Definition

A digraph is symmetric if it is the digraph associated to some graph.

Definition

If x0, e1, x1, . . . , en, xn is a directed trail on a digraph, we say that theparity of each edge ei is the parity of i .

Definition

An Holbeinian circuit of a symmetric digraph is an Eulerian circuitwhere all edges (x , y) and (y , x) have opposite parities.

Definition

A symmetric digraph is Holbeinian if it has an Holbeinian circuit.


Characterizations of Holbeinian Graphs

Theorem (Theorem 1)

A symmetric digraph is Holbeinian if and only if it is strongly connected.

Corollary

A symmetric digraph is Holbeinian if and only if the associated graph isconnected.


Trémaux’s Algorithm

Suppose we have a strongly connected symmetric digraph.

1. Start at an arbitrary vertex x0.

2. Proceed along any edge.


Trémaux’s Algorithm, continued

3. At each later step, suppose we have just traversed an edge (x , y)and arrived at a vertex y .

a. If we have not visited y before:i. If there is an edge leaving y other than (y , x), proceed along any

such edge.ii. If there are no such edges, turn back along (y , x).

b. If we have visited y before:i. If we have not traversed (y , x), turn back along (y , x).ii. If we have traversed (y , x):A. If there is an edge (y , z) for which neither (y , z) nor (z, y) has been

traversed, proceed along any such edge.B. If every edge (z, y) has been traversed but there is an edge (y , z)

which has not been traversed, proceed along any such edge.C. If every edge (y , z) has been traversed, terminate the algorithm.


Example of Trémaux’s Algorithm

Example

4

18

8

15

7

1617

3

1 2

19 56

9

1011

12

13 14


Facts About Trémaux’s Algorithm

Lemma

Trémaux’s algorithm terminates at the initial vertex.

Theorem

Trémaux’s algorithm traverses every directed edge exactly once.

Theorem (Theorem 2)

Trémaux’s algorithm traverses (x , y) and (y , x) with opposite parity.

Corollary

Trémaux’s algorithm produces a Holbeinian circuit of the digraph.


Tarry’s Algorithm (with parity)

Suppose we have a strongly connected symmetric digraph.

1. Start at an arbitrary vertex x0.

2. Proceed along any edge.


Tarry’s Algorithm, continued

3. At each later step, suppose we have arrived at a vertex y . If y isnot x0, let (x1, y) be the edge that first reached y (“entry edge”).

a. If there is an edge (y , z) other than (y , x1) which has not beentraversed (and such that (z, y) either has not been traversed or wastraversed with parity opposite the current parity) proceed along anysuch edge.

b. If every edge (y , z) other than (y , x1) has been traversed (or (z, y)was traversed with the current parity), leave along (y , x1) (“reverseof entry edge”) (if (x1, y) was traversed with parity opposite thecurrent parity).

c. If there are no allowed moves as above, terminate the algorithm.


Facts About Tarry’s Algorithm

Lemma

Tarry’s algorithm (with parity) terminates at the initial vertex.

Lemma

Tarry’s algorithm (with parity) visits every vertex, and leaves it alongthe reverse of its entry edge.

Theorem (Theorem 3)

Tarry’s algorithm (with parity) traverses every directed edge exactlyonce, and traverses (x , y) and (y , x) with opposite parity.

Corollary

Tarry’s algorithm (with parity) produces a Holbeinian circuit of thedigraph.


A False Conjecture

Conjecture

Every Holbeinian circuit can be produced from Trémaux’s algorithm.

Counterexample

13 14

15 16

1718

5

9

1

4

11

12

8

10

62 3

19

7


Tarry Is More General

Theorem (Theorem 4)

Every circuit which can be produced from Trémaux’s algorithm can beproduced from Tarry’s algorithm (with parity).


Another False Conjecture

Conjecture

Every Holbeinian circuit can be produced from Tarry’s algorithm.

Counterexample

5

4

15 16

1718

14

13

12

11

10

9

7

3

1

8

62

19


An Open Conjecture

Fact

It is not possible for a “local” algorithm to distinguish between a graphwith an Eulerian circuit and one without.

Conjecture

There is a “local” algorithm with an “Eulerian oracle” which producesevery Holbeinian circuit.


Thanks, and happy stitching!

[1] Joshua Holden, The Graph Theory of Blackwork Embroidery, Making Mathematicswith Needlework (sara-marie belcastro and Carolyn Yackel, eds.), A K Peters,2007, pp. 136–153.

Figure: A modern blackwork pattern, by the author


blackwork embroidery and algorithms for maze traversals

Technology

hans holbein

holbein stitch figure

queen of england

jane seymour

portrait of catherine

spanish stitchfigure